Surveys in Differential Geometry, Vol. 2: Proceedings of the conference on geometry and topology held at Harvard University, April 23-25, 1993

o < t < 6 for some positive time 6.

~

H at least for

LEMMA 13.4. If the constant e > 0 is small enough compared to b, B, A, >. and the dimension n, it will have the following property. As long as r ~ e/VM and t ~ elM and F ~ H we will have

ID
and
Proof. In the frame {Fa} which evolves so as to stay orthonormal under the Ricci Flow we have DtDa

THE FORMATION OF SINGULARITIES

45

and for IDCPI:5A 2 at t = 0 we get

IDIt'12

~ A 2 eCMt

for t ~ o. Now ift ~ elM with e ~ (ln2)IC then We also have the formula

IDcpl2

~

2A2.

DtDoDbCP = DoDbDtcp + RocDbDcCP + RbcDaDcCP + (DcRob - DaRbc - DbRac)Dclt' with the terms in Re coming from the motion of the frame and the terms in DRe coming from the motion of the connection. This formula gives a bound

We can use the bound F ~ H. In particular

gives a bound (for t

~

IRml

~

M as before, but we get a bound on IDRml from

11M at least) IDRml

~ CM (~+ ~)

with a constant C depending on b, B, A and A. This yields the estimate

We can estimate

IDcpl2 ~ 2A2 from before.

Since cP is fixed in time and It' ~ Ar,

Now viewing this as an ordinary differential inequality at a fixed point, we see that It'ID21t'1 ~ 2A2 if cpID2cpI ~ A2 at t = 0 and t ~ elM for a suitably small constant e> O. To see this, consider the ordinary differential equation

for u

= cpID2tpi at a fixed point.

Then

RICHARD S. HAMILTON

46

Since e- CMt

:::;

1 we get

Sin~e u :::; A2 at t

= 0, we have

The latter improper integral is finite and gives

and if r :::; sired.

LEMMA

elVM and

t :::;

elM

for a suitably small e then

u :::;

2A2 as de-

0

13.5. We have F :::; H for 0

< t :::; T

on the set V where cp

> o.

Proof. Since H --t 00 for t --t 0 or cp --t 0, the set where F ~ H is a compact subset of V x (0, T]. Unless it is empty, the continuous function t assumes its minimum t* > 0 on this set at some point P*. Then F :::; H on all of V for t :::; t*, while F = H at P* at time t. This forces

at P* at time t*. But since 2 2 -aF at <- 6.F-F +M

and

we have a contradiction. Thus the set F where.

~

H is empty, and F

<

H every-

0

We conclude that

on V X (0, T] for some constant C depending only on the dimension. Since cp = r at P we are done.

47

THE FORMATION OF SINGULARITIES

14 The Harnack Inequality. There is an interesting differential Harnack inequality for the rucci Flow (see [24]). In addition to the curvature tensor Rabcd we consider the tensors

and Mab

= ~Rab -

1

2DaDbR + 2RacbdRcd - RacRbc.

For any two-form Uab and one-form Wa we form the quadratic Z

= ( Mab + ;t Rab) Wa Wb + 2Pabc Uab We + RabcdUabUcd·

THEOREM 14.1. Suppose we have a solution to the Ricci Flow for t > 0 which is either compact or complete with bounded curvature, and suppose the curvature operator is weakly positive. Then the Harnack quadratic Z is also weakly positive for all two-forms Uab and one-forms Wa for all t > o.

The proof is given in the reference quoted, and uses the maximum principle. The Harnack quadratic is found by the fact that it vanishes identically on a homothetically expanding soliton, which shows it is a delicate and precise estimate. Now there probably exists a homothetically expanding soliton which is rotationally symmetric and can be found by solving an ordinary differential equation, but no one has bothered to do this yet as far as we know. It would represent a solution emerging from a cone. There may also be non-rotationally symmetric ones, which would be more interesting. In the proof, assume for simplicity the manifold is compact. Then there will be a first time the quadratic is zero, and a point where this happens, and a choice of U and W giving the null eigenvector. We can extend U and W any way we like in space and time and still have Z ~ 0, up to the critical time and we can profit by extending them with DaUbe

1

= "2 (RabWe -

RaeWb)

1

= + 4t (gabWe -

gaeWb)

and DaWb

at the critical point where Z We also take

= O.

=0

This is an optimal choice for the following.

at the critical point. We then compute (D t - ~)Z =(PabcWe + RabedUed) (Pabe We

+ 2RaebdMcdWaWb -

+ Rabe/Ue/)

2PaedPbdcWaWb

+ 8RadcePabeUabWc + 4Raec/Rbed/UabUcd

48

RICHARD S. HAMILTON

and indeed this computation is most of the work in the proof. We then check algebraically that if Z ~ 0 then (D t - ~)Z ~ 0 and apply the maximum principle. Because of the factor lit in Z we have Z positive for small t, and then it must stay positive. There is an interesting interpretation of this formula which follows from a remark by Nolan Wallach. Suppose we have a Lie algebra 9 with Lie bracket [,] and with an inner product <, >. We can then define a system of ordinary differential equations for an element M in the symmetric tensor product 9 ®. 9 as follows. Choose any basis {¢O} for g. The Lie bracket is given by the Lie structure constants

[¢O,q,-B]

= c~{3¢-y

and the inner product is given by a matrix

while the element MEg ®. 9 is given by

for some matrix Mo{3. Then the ODE is given (independently of the choice of a basis) by d M o{3 = 9-yOMat-y M {3o + Co -y( c{3or,M-yo M(r,. dt This is the reaction system in the Ricci Flow for the evolution of the curvature operator M when we compute (D t - ~)M and drop the Laplacian and replace D t - ~ by dl dt. Here the Lie algebra is the two-forms A2 which can be identified as the Lie algebra of the rigid rotations So(n), regarding M = Rabcd as an element of A2 ®. A2. The inner product used on A2 is the standard one. Now the Harnack quadratic can be regarded as an element of

and A2 EB AI, the space of pairs of a two-form and a one-form, is the Lie algebra of the group of rigid motions, which is a natural extension of the group of rigid rotations, with the group of translations as kernel. The Lie bracket on A2 = EBAI is given by

[U EB W, V EB X]

= [U, V] EB (UJX -

VJWl:

We can also introduce a degenerate inner product on A2 EB AI by letting (U EB W, V EB X) = (U, V)

ignoring the Al factor. Now if we form the ODE on the Lie algebra A2 EB Al according to the rules above for a quadratic Z, we get exactly the reaction system for (D t - ~)Z as given above! The geometry would seem to suggest that the Harnack inequality is some sort of jet extension of positive curvature operator on some bundle including translation as well as rotation, and this 'is

THE FORMATION OF SINGULARITIES

49

somehow all related to solitons where the solution moves by translation. It would be very helpful to have a proper understanding of this suggestion. At any rate, we can see why the Harnack expression stays positive. Write the Harnack quadratic Z as a sum of squares of linear functions (eigenvalues) weighed by constants (eigenvectors) Z

= LAM «(VM,U) + (XM' W})2. M

Then the previous formula yields

(Dt -

~)Z = ILAM «(VM,U) + (XM' W})VMr + LAMAN«[VM,VN],U) MN

+ (VMJXN -

VNJXM, W})2 .

This gives the identification of (D t - ~)Z in terms of the Lie algebra. Now if all AM ~ 0 then clearly (D t - ~)Z ~ 0, which is all we need to prove the Theorem. H.D. Cao([6]) has shown that the same conclusion holds if instead of a Riemannian metric with weakly positive curvature operator we have a Kahler metric with weakly positive holomorphic bisectional curvature (a weaker hypothesis in the Kahler case). This suggests trying to prove a Harnack inequality with other curvature hypotheses. For example, does there exist a Harnack inequality on three-manifolds with positive Ricci curvature? In two dimensions we can rewrite the Harnack inequality using the identification of two-forms with scalars and the rotation by 90° on the tangent space using a local orientation.

THEOREM 14.2. If we have a solution for t > 0 to the Ricci Flow on a surface which is compact, or complete with bounded curvature, and if the curvature R is weakly positive, and if we let

and define the quadratic

then Z

~

0 for all vectors Xa and all scalars V for all t

Proof. We substitute

> O.

RICHARD S. HAMILTON

50

in the original formula where J.l.ab is the volume 2-form in a local orientation. Note the choice of orientation disappears when we square. 0 In all cases we can trace the Harnack inequality by writing U = V A W and summing over an orthonormal basis of W to get the trace Harnack inequality

aR

1

at + tR+2DaR. Va + Rab Va Vb for all vectors Va for all t

~0

> O. This has the consequence, letting V

aR

= 0, that

1

-+-R>O at t or :t(tR)

~0

which implies that tR is increasing at each point! This is very useful if we combine it with the local derivative estimate of Shi.

COROLLARY 14.3. Suppose we have a solution to the Ricci Flow for t > 0 which is compact or complete with bounded curvature, and has weakly positive curvature operator or is Kahler with weakly positive holomorphic bisectional curvature. Suppose moreover that at some time t > 0 we have the scalar curvature R ~ M for some constant M in the ball of radius r around some point P. Then the derivatives of the curvature at P at time t satisfy a bound

IDRm(P,tW

~ CM 2 C~ + ~ + M)

for some constant C depending only on the dimension. Proof. Since tR increases, we get a bound R ~ 2M in the given region for times between t/2 and t. The positive curvature hypotheses each imply a bound on all the curvatures IRml from a bound on the trace R. The result'il,ow follows from the standard estimate. Likewise we get bounds on higher derivatives. Such instantaneous derivative estimates are more like what we expect for solutions of elliptic equations. We will use them subsequently in a variety of ways. 0

15 The Little Loop LeInIna. The following result gives a bound on the injectivity radius at a point in terms of a local bound on the curvature. LITTLE Loop LEMMA 15.1. There exists a constant f3 > 0 such that for any initial metric go on a compact manifold which either has positive curvature operator or is Kahler with positive holomorphic bisectional curvature, we can find a constant 'Y > 0 depending on go with the following property. If 9t is the

THE FORMATION OF SINGULARITIES

51

subsequent solution of the Ricci Flow with initial value 90 and if P is a point where

R:5 f3/W 2 in the ball around P of mdius W at time t, then the injectivity mdius of the metric gt at P at time t satisfies inj(P, t) ?

,/W.

Proof. Since the injectivity radius at P can be estimated in terms of the maximum curvature in a ball around P and the length of the shortest closed geodesic loop starting and ending at P, it suffices to get a lower bound on the length of the loop. The Lemma then follows from the following statement, which is what we actually prove. 0

THEOREM 15.2. There exists a constant f3 > 0 such that for an'll initial metric go on a compact manifold which either has positive curvature opemtor or is Kahler with positive biholomorphic sectional curvature, we can find a constant B < 00 with the following property. If gt is the subsequent solution of the Ricci flow and if P is a point where

R :5 f3/(W - s)2 in the ball of mdius W around P at time t where s is the distance of a point in the ball from P, then an'll geodesic loop starting and ending at P at time t has length L with W/L:5 B. Proof. H go has either positive curvature operator or is Kiihler with positive holomorphic bisectional curvature, then the subsequent solution gt does also, and hence gt has positive Ricci curvature. Moreover from [24] or from [6] we know that g satisfies a trace Harnack inequality oR R at + t + 2DR(V) + Rc(V, V) ?

0

for all vectors V at any time t > O. Any solution of the Ricci flow satisfying the trace Harnack inequality will also satisfy the Little Loop Lemma, as this is all we use in the proof. Since the Ricci curvature is always positive, distances always shrink as time increases. This makes it easier to control the geometry. Moreover since all the Ricci curvatures are positive, we have all the Ricci curvatures bounded by the scalar curvature, so

0:5 Rc(V, V) :5 Rg(V, V), and we can control the rate at which any distance shrinks by controlling R from above.

0

RICHARD S. HAMILTON

52

The first step is to check that we can find a constant Bl which works in the Theorem up to some time T > O. The reason is that the control on R from the Harnack estimate is not so good for small t.

LEMMA 15.3. For any f3 > 0 and any initial metric go as above, we can find T > 0 and a constant Bl with the following property. If at some subsequent time t with 0 ~ t ~ T we have

R ~ f3/(W - 8)2 in the ball of radius W around some point P, then any geodesic loop at P at time t has length L with

Proof. Let Mo be the maximum curvature at t maximum curvature up to time t. Since

=

0, and let M t be the

it follows from the maximum principle that

for some constant C (in the ·sense of the lim sup of forward difference quotients) and hence if we take T = c/Mo for some small constant c > 0 then ~

Mt

Since R > 0, we can let mo the maximum principle

2Mo

for 0

~ t ~ T.

> 0 be the minimum value of R at

t = OJ then by

R~mo'

everywhere for all t

~

O. Since at the center point P at time t

we see that gives an upper bound on W. Suppose now that there is a short loop at P of length L with

Then

o

THE FORMATION OF SINGULARITIES

53

and if Bl is large enough we can make

for any e

> 0 we like. Now as long as

for an appropriately small constant c > 0, the standard existence theory for geodesics tells us that in any nearby metric there will exist a geodesic loop starting and ending at P close to the original one; this is just an application of the inverse function theorem together with the observation that for L2 Mo :5 c there are no nonvanishing Jacobi fields on the loop which vanish at the end points. Thus we get a family of geodesic loops parametrized by t and varying smoothly, at least for some time backward. Under the Ricci flow the length of the loop varies by dL dt

=-

/ Rc(V, V)ds

where we integrate the Ricci curvature in the tangent direction V with respect to the arc length over the loop. (Since the loop is kept geodesic, the first variation in L from the motion of the loop is zero, and we only get the contribution from the change in the metric.) This gives an estimate dL

-dt > -CMoL showing the loop does not shrink too fast. In fact the length L t at t is related to the length L9 at () for () :5 t by

and hence in time 0 :5 t :5 T with T = c/Mo for a suitably small c, if the loop ends with length L :5 e /../MO it is never more than twice as large for as far back in time as we can continue it as a perturbation. But then we can do this all the way to t = 0 taking e > 0 small. Hence then must have been a geodesic loop at t = 0 oflength at most 2e/../MO. Now for any go we can take e so small this is false. Then making Bl large compared to e gives us a contradiction if W/L ~ B 1 • Thus W/L:5 B l , and we have established the Lemma. This Lemma has one very useful consequence. It is a Corollary of the trace Harnack inequality that for a solution of the Ricci Flow for t ~ 0 the quantity tR is pointwise increasing in t. Now we only have to worry for t ~ T with T > o. Moreover we can find a time T depending on go (in fact r = C /mo for some constant C, since by the maximum principle the minimum mt of R at time t grows by a rate

RICHARD S. HAMILTON

54

for some constant c > 0) such that the solution cannot exist longer than time T. Then for any time tl and t2 with

o < T ~ tl

~ t2 ~

T

and any point X we have

for the constant C = TIT. The next step is to find a constant B2 which works if W is not too small. THEOREM 15.4. For any f3 > 0 and any initial metric go as above and any Wo > 0 we can find a constant B2 with the following property. If at some subsequent time t ~ 0 we have

R ~ f31(W - 8)2 in the ball of radius W around some point P with W loop at P at time t has length L with WI L ~ B2. Proof. If we take B2 ~ Bl, we can assume t ~ if B2 is large we can make

T.

~

W o, then any geodesic

Suppose WIL ~ B 2 ; then

L~eW

for any e > 0 we like. Since distances shrink, if a point X has distance 8 at most WI2 from P at some earlier time 0 ~ t, it also has distance 8 at most WI2 from P at the later time t. By assumption R ~ f31(W - 8)2

and hence R(X, t) S, 4f3IW2.

Now for

T ~

0

~ t ~

T we have R(X, 0) ~ 4f3TITW 2

Putting C = 4{3TIT we get R(X, 0) ~ CIW 2

on the ball of radius W /2 around P at times 0 in

T

S, 9 S,

t.

o

Now from the existence of a short loop at P at time t we can deduce the existence of a short loop at earlier times 9, just as before. As long as the loop at P has length L S, W, it must stay in the ball ofradius W/2 around P where we have a curvature estimate R ~ C /W 2 • Then again the loop shrinks at a rate dL = dt Rc(V, V)ds ~ -CL/W 2

J

THE FORMATION OF SINGULARITIES

55

and for T ~ (J ~ t the length L t of the loop at time t is related to the length of the loop L(J at time (J by for the constant since t - (J ~ T - T and W ~ Wo. H c ~ 1/A and L t ~ cW then L9 ~ W and we can continue backward all the way to time T. Now at time T we have LT ~ cAW and we still have R ~ C/W 2 in the ball around P of radius W /2. Letting W = aW for an appropriate a > 0 gives

R ~ (3/(W - S)2 in the ball of radius W around P at time (J. Let L = LT be the length of the loop at P at time continuation.. Then

w/L

~ aW/cAW = a/cA

T

we constructed by

> Bl

if c < a/ABI' This contradicts our first estimate, which proves W/L is chosen small compared to B 2 •

~

B2 if c

COROLLARY 15.5. For any (3 > 0 and for any initial metric 90 as above and any 0: > 0 there exists a constant B3 with the following property. If R ~ (3/(W - s)2 in a ball of radius W around some point P at some time t with W2 ~ o:t, then any geodesic loop at P at time t has length L with W/L ~ B 3 .

Proof. Choose T > 0 from Lemma 15.3 and let wg = O:T in Theorem 15.4. Then take B3 to be the larger of Bl or B 2. H t ~ T then 15.3 gives the result; while if t ~ T and W 2 ~ o:t then W ~ Wo and 15.4 gives the result. 0 Now we come to the important case where W2 ~ o:t.

LEMMA 15.6. For any constant B ~ B 3 , if there exists a loop of length L at the center of a ball of radius W with R ~ (3/(W - S)2 and W/L ~ B, then there exists a first such time t. > 0, and at t. there is a point p. with a loop of length L. and a ball at p. of radius S. as above with W./ L. = B. Moreover

W~ ~ o:t.

Proof. Pick a decreasing sequence of times tj, and points Pj with loops of length L j at Pj and balls of radius Wj with R ~ (3/(Wj - 8j)2 on the ball, where Sj is the distance to Pj at time tj, such that tj converges to the greatest lower bound t. of all such t. For a subsequence, Pj -+ p. and 8j -+ S., the distance from p. at time t •. Since B ~ B 3 , we know tj ~ T > 0 so t. ~ T > O. Also Wj ::; Wo so a subsequence Wj -+ W. with W. ::; W 0 Now for t ::; tl 0

56

RICHARD S. HAMILTON

there is some 6> 0 such that every geodesic loop has length L ~ 6; so Lj ~ 6. This makes Wj ~ oB > 0, so W .. > o. If Sj/ Lj = Bj with Bi ~ B, we have Bj ~ w/6, and a subsequence Bj -+ B .. with B. ~ B. Thus Lj -+ L. where L. = W./B",. 0 Choose a subsequence so that the initial unit velocity vectors V; of the loop at Xi at time tj of length Lj converge to a vector V. ; then V. is the initial unit velocity vector of a loop at X. at time t. of length L •. Moreover in the ball of radius W. at X. at time t. we have R ~ {3/(W. _s.)2 by continuity. This gives a loop of length L. in a ball of radius W. at time t. with W./ L. = B •. Since B. is large enough, there is still a loop of almost the same length at X. at a slightly earlier time in a ball of radius almost as large where R ~ (3/(W - s). This would contradict the minimality of t. unless B. = B. Finally W~ ~ at. follows from Corollary 15.5. Now in reality we always have W/L < B above. To see this, we suppose not, pick the first time t. when W. / L. = B, and get a contradiction. The contradiction will come from demonstrating a loop and a ball as above at p. just a little before t. with W / L ~ B. First we show there will be a loop L at P", at earlier times which is not much longer. Since R ~ (3/(W. - s .. )2 in the ball of radius W. around p. at time t. where s. is the distance to p. at time t .. , we can bound R near p. at earlier times t ~ t. using the Harnack inequality. Recall that tR is pointwise increasing, so that tR(X, t) ~ t.R(X, t .. ) for t ~ t •. Now if seX, Y, t) denotes the distance from X to Y at time t, since lengths shrink we have

s = seX, P., t)

~

sex, p .. , t .. ) = s.

and W. - s ~ W. - s. and

This makes

t· {3 - t (W.. - s)2 in the ball of radius W. around X. at times t ~ t ... R<-·=-~

Thus R stays small compared to L~ in a ball of radius large compared to L. at times a little earlier than t .. , so by the theory of geodesics there will exist unique loop at X. near the original one for times t a little less than t •. Moreover the length L of this loop varies by dL dt

=-

f

Rc(V, V)ds

integrating over the loop. As long as L ~ W. the loop will stay in the ball of radius W./2, and as long as t ~ t .. /2 we have t .. /t :::; 2. Then on the loop

Rc(V, V) ~ R ~ 8{3 /W~

THE FORMATION OF SINGULARITIES

57

and

dL/dt ~ -8/3L/W~ from estimating the integral. IT L does not shrink fast and ends at L., it was not much larger than L. a little before t.. In fact d

dt log L ~ -8/3 /W~ and

L

< - L

.

e8{3(t.-t)/w~

for t a little earlier than t •. Finally, we want to show that at a time t a little before t. the curvature is small enough in a ball of radius W around p. with W appreciably larger than W., so that W / L > B. This will finish the proof.

LEMMA 15.7. At each time t ~ t. there is a largest W such that if s is the distance to p. then R(W - S)2 ~ f3

on the ball of radius W around p. at time t. Moreover there is at least one point X where the equality is attained with 0 =< s < W. Proof. Since the manifold is compact, the function s + J/3/R attains its infinimum W at some point X. (Even if it were not compact but complete, this would hold since s -+ 00 as X -+ 00.) Clearly W > 0 and s < W. Since s is conelike at p. but R is smooth, the minimum is not at P., so s > o. Now W is a function of t. D Choose a minimal geodesic 'Y from P. to X at time t, and let Y be its unit tangent vector at X pointing away from p.. The distance function s along the geodesic 'Y is realized by the arc length, so Ds(Y) = 1.

Now on 'Y R(W - s)2 ~ /3

and equality is attained at the end X, so DR(Y)::::

:~s.

The Harnack Estimate [24] in section 14 tells us that for all V 8R

R

7ft + t + 2DR(V) + Rc(V, V)::::

0

58

RICHARD S. HAMILTON

and since Rc(V, V)

~

RIV12 we have oR

R

at + T + 2DR(V) + RIV12

~

o.

Choose V = >.Y where Y is the unit tangent vector at the end of the geodesic above. Then oR + R + >.~ + >.2 R > 0 at t W-8 for all >.. Choose>. = -2/(W - 8); then

aR

R>

at+T -

4R (W-S)2

Now (W - s)2 ~ W 2, and we can choose a so that if W; ~ at. and t is near t. and W near W. then W 2 ~ 2t (as long as a < 2). This gives

oR

2R

-ot > ==::--~;:- (W-s)2. This inequality holds at any time t a little before t. at any point X where R(W - 8)2 = {J, and there is at least one such point. Now the distance 8 from X. must decrease as t increases. Then W must decrease fast enough to keep R(W - 8)2 ~ {J at the point X above. The function W may not be differentiable, so we proceed carefully. We know

W~8+J{J/R at each point and time with equality at X at time t, and W depends only on t while 8 decreases. Then at X at time t . . f W (t + h) - W (t) 1 1ImlD <-h.j.O h -W holds for all t a little before t •. Since we end up with W. at

t., the usual argument gives us that

W2 ~ W;

+ 2(t. -

t)

for all t a little before t.. Combining this with our previous estimate

L ~ L.e8{3(t.-t)/w~ shows that for small {J we get W / L series to get

W ~ W.

+

> W. / L.. To see this, expand in power

t - t

~. +O(t. _t)2

and

L < L + 8{J(t. - t)L. + OCt _ t)2

_.

and

W;

*

t - t WI L 2: W./ L. + (1 - 8{J) ;. W. + OCt. - t)2

showing we need {J

< 1/8. This completes the proof.

THE FORMATION OF SINGULARITIES

59

16 Limits of Solutions to the Ricci Flow. Given a sequence of manifolds Xj with origin OJ, frames :Fj at OJ and Riemannian metrics gj, we say that the sequence (Xj,Oj,:Fj,gj) converges to the limit (X,O,:F,g) if there exists a sequence of compact set K j exhausting X and a sequence of diffeomorphisms 'Pj of K j in X to Xj such that 'Pj takes 0 to OJ and :F to :Fj, and the pull-back metrics 'Pjgj converge to 9 uniformly on compact sets together with all their derivatives. This is the topology of Coo convergence on compact sets. If the limit exists, it is unique up to a unique isometry preserving the origin and frame. If (Xj, OJ,:Fj,gj) converges to (X, O,:F, g), then we clearly have the following properties: (a) for every radius s and every integer k there exists a constant B(s, k) independent of j such that the kth covariant derivative of the curvature Rmj of the metric gj satisfies a bound IDk Rmjl ~ B(s, k)

on the ball ofradius s around OJ in Xj in the metric gj; and (b) there exists a constant b > 0 independent of j such that the injectivity radii Pj of Xj at OJ in the metric gj satisfy the bound Pj ~ b.

Conversely we have the following existence result.

THEOREM 16.1. Given any sequence of manifolds (Xj , OJ,:Fj, gj) satisfying the bounds that IDk Rmjl ~ B(s, k) on balls of radi'US sand Pj ~ b> 0, there exists a subsequence which converges in the Coo topology on compact sets to a manifold (X,O,:F,g).

Proof. This is slightly more general even than what we did in [26], but follows again from an easy modification of the argument in Greene and Wu[19]. The only essential new feature is to bound the injectivity radius below at points at a large distance s from OJ in terms of the bounds on the curvature in a slightly larger ball. A lot of the subtlety of getting convergence using only bounds on curvature Rm and not its derivatives DRm is entirely unnecessary for solutions to parabolic equations which are automatically smoothing, such as the Ricci Flow. We have already seen how estimates on Rm give estimates on

Wm.

0

Now if we have a sequence of solutions to the Ricci Flow on some time interval, we can take a limit (if we have the appropriate bounds) and get another solution to the Ricci Flow. At each time t the metric in the limit solution is the limit of the metrics at the same time in each solution in the sequence. To extract the limit we only need bounds on the curvature at each point at each time, and bounds on the injectivity radius at the origins at time 0 (see [26]).

RICHARD

60

S.

HAMILTON

Consider a maximal solution g to the Ricci Flow on a manifold X for 0 :$

t < T, where either X is compact or at each time t the metric g is complete with bounded curvature, and either T = 00 or IRml is unbounded as t We let M(t) denote the maximum curvature at time t, i.e.,

M(t)

-4

T.

= sup{IRm(P, t)I}.

We need to assume a bound on the injectivity radius in terms of the maximum curvature. Let p(t) denote the infimum of the injectivity radii at all points at time t. DEFINITION 16.2. The solution satisfies an injectivity radius bound if there exists a constant c > 0 such that

p(t) ~ c/VM(t) at every time t. We classify maximal solutions into three types; every maximal solution is clearly of one and only one of the following three types: Type I: T < 00 and sup(T - t)M(t) < 00. Type II(a): T < 00 but sup(T - t)M(t) = 00. Type II(b): T = 00 but suptM(t) = 00. Type III: T = 00 and suptM(t) < 00. For each type of solution we get a different type of limit singularity model.

DEFINITION 16.3. A solution to the Ricci Flow, where either the manifold is compact or at each time t the metric g is complete with bounded curvature, is called a singularity model if it is not flat and of one of the following three types: Type I: The solution exists for -00 < t < 0 for some 0 with 0< 0 < +00 and

IRml :$ 0/(0 - t) Type II: Type III:

everywhere with equality somewhere at t = O. The solution exists for -00 < t < +00 and IRml :$ 1 everywhere with equality somewhere at t = O. The solution exists for -A < t < 00 for some constant A with 0 < A < 00 and

IRml :$ A/(A + t) with equality somewhere at t = O. We always take the equality to hold at some origin 0 at time O.

THEOREM 16.4. For any maximal solution to the Ricci Flow which sati~fies an injectivity radius estimate of the type above, of Type I, II, or III, there exists a

THE FORMATION OF SINGULARITIES

61

sequence of dilations of the solution which converges in the limit to a singularity model of the corresponding type. Proof. For Type I, let

There is some e

°

0= limsup{T - t)M{t)

> such that we always have 0

< 00. ~ cj

for M{t) satisfies an ODE

dM
For each of these solutions, let Pj be the origin 0, translate in time so that tj becomes 0, dilate in space by a factor oX so that R{Pj, tj) becomes 1 at the origin at t = 0, and dilate in time by oX 2 so it is still a solution to the Ricci Flow. The dilated solutions exist on a time interval

where and

Aj

= tjOj/{T -

tj) -t

00.

Moreover, they satisfy curvature bounds. For any e < T such that for T ~ t < T we have

>

°

we can find a time

T

IRml

~

(O + e)/(T - t)

by assumption, before dilation. After dilation this becomes a curvature bound

IRml for times -OJ

~

(O + e)/(o.j

~

-

t)

t < OJ where OJ

= (tj -

T)o.j/(T - tj) -t

Consequently the limit exists on the time interval

IRml

~

00.

-00

< t < 0. and satisfies

0./(0 - t)

everywhere, while IRm(O, 0)1 = 1. For Type II(a), we have to be a little more subtle. We start by picking a sequence Tj < T < 00 with Tj -t T. If the manifold is compact we can pick points Pj and tj where (Tj

-

tj)IRm(Pj , tj)1 =

sup (Tj - t)IRm(P, t)1 P,t
62

RICHARD S. HAMILTON

as the latter goes to zero as t /" Tj. If the manifold is not compact, we can take 'Yj /" 1 and find Pj and tj so that at least

(Tj - tj)IRm(Pj , tj)1 ~ 'Yj sup (Tj - t)IRm(P, t)l. P,t$,.T;

Now pick Pj to be the origin OJ, translate in time so tj becomes zero, dilate in space by a factor .x so R(Pj, tj) becomes 1 at the origin at t 0, and dilate time by .x2 so that we still have a solution to the Ricci Flow. The dilated solution exists for Aj ~ t < OJ where

=

= (Tj -

OJ and

A·3

tj)IRm(Pj , tj)1 ~

t· = Tj _'_0· _ tj 3 = t·IRm(P· 3 "

also. To see OJ --+ 00 for Type II(a) where T are chosen maximally and

00

t')I--+ 3

00

< 00, note that Tj /" T, Pj and tj

limsup(T - t)IRm(P, t)1

= 00.

To see Aj ~ 00 for Type II(a), use the fact that OJ ~ 00 forces tj ~ T and IRm(Pj , tj)1 --+ 00. We also get a bound on curvature. We have for 0 ~ t ~ Tj (Tj - t)IRm(P, t)1 ~ rj(Tj - tj)IRm(Pj, tj)1 where rj = l/'Yj -Aj ~ t ~ OJ

~

1 also, before dilation. After dilation this becomes for

(OJ - t)IRm(P, t)1

~

rjOj .

Write this as IRm(P, t)1 ~ rjOj/(Oj - t). As j ~ 00, rj --+ 1 and OJ/(Oj - t) --+ 1 for any fixed t. Hence the limit exists and satisfies IRm(P, t)1 ~ 1

=

everywhere for -00 < t < +00, while IRm(O, 0)1 1. For type II(b), we again choose a sequence Tj /" T Pj and tj so that tj(Tj - tj)IRm(Pj , tj)1 ~ 'Yj

sup

= 00, but now we pick

t(Tj - t)IRm(P, t)1

P,t$,.T;

where again 'Yj /,,1. Pick Pj to be the origin OJ, translate in time so tj becomes zero, dilate in space by a factor .x so that R(Pj, tj) becomes 1 at the origin at t = 0, and dilate time by ,X2 so it is still a solution of the Ricci Flow. Suppose T j dilates to OJ and 0 dilates to -A j • The solution now exists after dilation on a time interval -Aj < t < OJ where by dilation invariance AjOj

_ tj(Tj - tj)

--'<.......:.- -

Aj

+ OJ

Tj

IRm(Pj, tj)1

~ 00

THE FORMATION OF SINGULARITIES

63

since by assumption lim sup tIRm(P, t) I = 00. This forces Aj --t

00

and OJ --t

00

as well since

xy 1 = x+y l/x + l/y' ~-----,,-

Before dilation we have an estimate for 0

where again

rj =

~ t ~

Tj

l/'yj --t 1. After dilation this becomes for -Aj ~ t ~ OJ

Write this as

IRm (P,t)1 < -

rjAjO j (t+Aj)(Oj-t),

As j --t 00, rj --t 1 and AjOj/(t + Aj)(Oj - t) --t 1 also for any fixed t. Hence the limit exists and satisfies IRm(P, t)1 ~ 1

everywhere for -00 < t < 00 while IRm(O, 0) I = 1. Finally we come to Type III, where T = 00 and A = lim sup tM(t)

< 00.

First we claim A > O. Indeed if tIRm(P, t)1 ~ e for large t then the diameter L satisfies an estimate dL < GeL dt for a constant G independent of e and L. This makes L grow at most like tC~ while IRml falls off at least like l/t. If Ge < 1/2 we see that L2 M --t 0, which means that after rescaling the curvature collapses with bounded diameter. By a well-known result of Gromov the manifold is nilpotent; more to the point the injectivity radius bound we assumed would fail. Thus A > O. Now pick a sequence of points Pj and times tj so that tj --t 00 and

Choose Pj to be the origin OJ, translate in time so tj becomes OJ, dilate in space by a factor .x so that IRm(Pj , tj)1 becomes 1 at the origin at time t = 0, and dilate in time by a factor .x 2 so we still have a solution of the Ricci Flow. After dilation the solution will exist for times -Aj $ t < 00 where time 0 dilates to

64

RICHARD S. HAMILTON

Moreover for any

g

°

> we can find a time T < 00 such that for t tIRm(P, t)1 ~ A

~ T

+g

by hypothesis, before dilation. After dilation this becomes a bound

(t + A)IRm(P, t)1 for time t

since tj satisfies

~

~

A

+g

OJ = (tj - T)Aj/tj

-t

A

-OJ where

-t 00

and Aj

-t

A

°

> while T is fixed. Hence we get a limit which IRm(P, t)1 ~ A/(t + A)

on -A

< t < 00 while IRm(O,O)1

= 1. This completes the proof of the Theo-

0

~.

In the case of manifolds with positive curvature operator, or Kahler metries with positive holomorphie bisectional curvature, there is a small modification which is quite useful for Type II and Type III. Because we have positive curvature, we can bound the Riemannian curvature tensor just by the scalar curvature, with a bound IRml~CR

for a constant C depending only on the dimension. Then if we repeat the previous argument we get the following result. Note that we do not need to assume an injectivity radius bound this time; if the solution is compact the injectivity radius bound follows from the Little Loop Lemma in section 15, while if the manifold is complete but not compact and has positive sectional curvature the injectivity radius bound follows from the argument of Gronmoll + Meyer (see [9]) in the real case. THEOREM 16.5. For any maximal solution to the Ricci Flow with strictly positive sectional curvature on a compact manifold, or with a metric which is complete with bounded curvature at each time with strictly positive sectional curvature, or on a compact K iihler manifold with strictly positive holomorphic bisectional curvature, there exists a sequence of dilations which converges to a singularity model. For Type I solutions the limit exists for -00 < t < 0 and has R(P, t) ~ 0/(0 - t) with R(O,O) = 1, for Type II the limit exists for -00 < t < 00 with R ~ 1 and R(O,O) = 1, and for Type III the limit exists for -A < t < 00 with R ~ A/(t + A) and R(O, 0) = 1. These limits will have weakly positive curvature operator, or weakly positive holomorphic bisectional curvature.

COROLLARY

16.6.

In the real case such a Type II limit must be a Ricci

soliton with Rc= D2f.

THE FORMATION OF SINGULARITIES

65

Proof. This follows from the result in [25] on eternal solutions have weakly positive curvature operator and where the scalar curvature assumes its maximum, which happens by our construction at the origin at time zero. The proof is by applying the strong maximum principle to the Harnack inequality. D CONJECTURE 16.7. In the Kahler case such a Type II limit must be a Ricci-Kahler soliton with Rc = 8af and 88f = o. Proof. Try to use the strong maximum principle on Cao's Harnack inequal-

D

~

CONJECTURE 16.8 . In the real case such a Type III singularity must be a homothetically expanding Ricci soliton with Rc = D2 f + pg for some constant p> O. In the Kahler case such a Type III singularity must also be an expanding Ricci-Kahler soliton with Rc = 8af + pg and 88f = o. Proof. Apply the strong maximum principle to the Harnack inequality for solutions on t > 0 with the extra term (1/2t)Rc. We haven't checked the details, but it must work. D

Unfortunately we don't have injectivity radius bounds available in many cases; in fact in many cases we expect them to fail, particularly as t -+ 00 for example on a nilmanifold or two hyperbolic manifolds of finite volume joined along their cusps. However, recent work of Cheeger, Gromov and Futake ([11]) suggests that we should be able to get some kind of limit anyway. The manifolds will collapse to a lower dimensional manifold (or orbifold). However the solution to the rucci Flow on the limiting manifolds may not converge to a solution to the rucci Flow on the lower dimensional limit manifold (or orbifold). Rather there will be some extra information in the fibres that collapse, which should be represented by some information in a bundle over the lower dimensional limit manifold (or orbifold), and there should be a system for the joint evolution of the metric on the base and the information in the fibre reflecting the rucci Flow in the limiting manifolds.

17 Bounds on Changing Distances. It is useful to see how the actual geometry changes under the rucci Flow. For this purpose we need to control the change in the distance d(P, Q, t) between two points P and Q at time t when P and Q are fixed but t increases. The basic obvious estimate is the following. THEOREM 17.1. There exists a constant C depending only on the dimension, such that if the curvature Rm is bounded by a constant M

IRml~M

then for any points P and Q and any times tl and t2. There is a more subtle bound on how fast distances can shrink which is much better when the distance is large compared to the curvature.

RICHARD S. HAMILTON

66

THEOREM

17.2. There exists a constant C depending only on the dimension

such that if IRml~M

then d(P, Q, t2) ~ d(P, Q, tt} - C../M(t2 - tt} for any points P and Q and any times tl

~

t 2.

The second estimate says that the rate at which a distance shrinks can be bounded independently of how large it is. It is due to the fact that on a long minimal geodesic there cannot be too much positive curvature along its middle or it would be unstable. Both theorems are proved by the following observation. For any path 'Y its length L changes at a rate dL dt

1

= - -r Rc(T, T)ds

where T is the unit tangent vector to the path 'Y and we integrate along the path with respect to the arc length s. The function d(P, Q, t) is the least length L of all paths. In general it will not be smooth in t for fixed P and Q, but at least it will be Lipschitz continuous. Hence we can estimate its derivative above and below, in the sense of giving an upper bound on the lim sup of all forward difference quotients and a lower bound on the lim inf of all forward difference quotients.

LEMMA

17.3. The distance d(P, Q, t) satisfies the estimate

1

- sup Rc(T, T)ds -rEr -r

~

dd d(P, Q, t) t

~ 0 inf

rRc(T, T)ds

-rEr J-r

where the sup and inf are taken over the compact set r of all geodesics 'Y from P to Q realizing the distance as a minimal length. Proof. We can restrict our attention to the compact set of geodesics of some large but finite length and apply the argument in [21]. 0 For the first theorem we apply the bound -CMd(P,Q,t) to conclude

~

i

Rc(T,T)ds

~ CMd(P,Q,t)

d

-CM $ dt logd(P, Q, t) ~ CM and integrate and exponentiate to get the result. For the second theorem w~ apply the following result, which is an integral version of Meyer's Theorem.

THE FORMATION OF SINGULARITIES

67

THEOREM 17.4. On a Riemannian manifold suppose we have a geodesic from P to Q of length L with arc length s and unit tangent vector T. For 0< 0' '5 L/2 (a) if Rc(T, T) ~ 0 along 'Y then

l

L

-

u

Re(T, T)ds '5 2(n - 1)

u

0'

(b) if Rc(T, T) ~ (n - l)p2 then

l

L

u

-

u

Rc(T, T)ds '5 2(n - l)p tan po'

(c) if Re(T, T) ~ -en - l)p2 then

l

L

u

-

u

Rc(T, T)ds '5 2(n - l)p tan hpu

We give the proof shortly for convenience, but first we finish the proof of Theorem 17.2. We can bound the integral over the whole path 'Y in three pieces

fL

10 Rc(T, T)ds '5

10r Re(T, T)ds +

l

u

L- u

Rc(T, T)ds

+ l~u Rc(T, T)ds. We bound the first and third piece using the maximum of the curvature

1 u

Rc(T, T)ds '5 eMu

fL Rc(T,T)ds'5 eMu.

and

1L - u

We bound the middle piece using Theorem 17.4

fL Re(T, T)ds <

1L-u IT we take

0'

-

ev'M

tan h( v'M0')

= 1/ v'M both bounds are the same and we get

lL

Rc(T, T)ds '5

ev'M.

Using this bound in Lemma 17.3 gives the result in Theorem 17.2. Now we prove Theorem 17.4. Consider a geodesic from P to Q of length L with arc length s and unit tangent vector T. Choose an orthonormal frame F o, F I , ••. , F n - l at P with Fo = T, and extend it along the geodesic by parallel translation so that d

- Fa = 0 ds

for

0 ~ a ~ n - 1.

RICHARD S. HAMILTON

68

Then Fo continues to be T and the frame continues to be orthonormal. Jacobi's equation for a normal vector field V to the geodesic representing an infinitesimal geodesic perturbation is

(:2 w) + V,

R(T, V, T, W)

for all normal vectors W. Choose a basis Vl vanishing at P by choosing

Va

=0

and

d

ds Va

= Fa

at

=0

, ..• , Vn - l

P

for

for the Jacobi's fields

1 ~ a ~ n - 1.

In terms of the parallel frame we can write

Va

= V! F{3

for

1 ~ a, /3 ~ n - 1.

Then Jacobi's equation becomes ~

..,

fa.., ds 2 V{3

.., + Roao.., V{3 = 0

for the functions V;(s) for 0 ~ s ~ L, with initial conditions

where

Roao..,

= R(Fo, Fa, Fo, F..,)

so that Roaoa is the sectional curvature of the plane spanned by the tangent to the geodesic and the a th normal frame vector. Of course Roao.., is symmetric in a and 'Y.

LEMMA.

The matrix

is symmetric. Proof. We compute dS _ 1 d V'" d v: 8 ds a{3 - ..,8 ds a ' ds {3

-

8 Ro..,08 V..,v: a {3

using Jacobi's equation. This shows the derivative of Sa{3 is symmetric. But Sa{3 = 0 at s = 0, so Sa{3 is always symmetric. 0 Now by definition if the geodesic has no conjugate points to P before Q, then any Jacobi field vanishing at P does not vanish again before Q. Consequently

THE FORMATION OF SINGULARITIES

the matrix the matrix

69

VJ is invertible on 0 < s < L with an inverse we call W;. Define

Since ZQI3

= WJWgS'Y6

we see that ZQI3 is symmetric also. The formula for the derivative of the inverse of a matrix is

and we can easily compute

using Jacobi's equation. The trace

Z

d = [ 13 ZOllO"= -log det V 13 ds 01

01

represents the rate of growth of the transversal area along the geodesic. The function Z is defined and smooth on the interior 0 < s < L, while Z -+ +00 as s -+ 0, and Z -+ -00 as s -+ L also if and only if Q is a conjugate point to P. The usual inequality gives

with equality when ZQI3 is a multiple of the identity. Taking the trace of the equation above gives the inequality d

-d Z s

1

+ --lZ2 + Rc(T,T):::; 0 n-

where [ 01 13 RoQol3 = Rc(T, T) is the Ricci curvature in the direction T tangent to the geodesic. The only fact we use for the following estimate is that there is some smooth function Z finite on 0 < s < L for which this inequality holds. Since Z2 ~ 0 we always have dZ

ds

and hence

for any a in 0

+ Rc(T, T)

:::; 0

lL-tT Rc(T, T)ds :::; Z(a) - Z(L - a) < a :::; L/2.

If Rc(T, T) ~ 0 along the geodesic then

d _Z+ _1_ Z2 <0

ds

n-l

-

70

RICHARD S. HAMILTON

and we find that

n-l Z(O') ~ - 0' so we get

l

L

-

CT

CT

and

n-l Z(L - 0') ~ - - 0'

Rc(T, T)ds $ 2(n - 1) . 0'

If Rc(T, T) ~ (n - l)p2 then

~Z + _1_Z2 + (n - l)p2 ds

n-l

$ 0

and we find that

Z(O') $ (n - l)p tan pO' so we get

l

L

-

CT

CT

and

Z(L _ 0') ~ _ (n - l)p tan pO'

Rc(T, T)ds $ 2(n - l)p. tan pO'

Finally if Rc(T, T) ~ -(n - l)p2 then

~Z + _1_Z2 ds

n-l

(n - l)p2 $ 0

and we find that

Z(O') < (n - l)p - tanh pO' so we get

l

L

CT

-

CT

and

Z(L _ 0') > _ (n - l)p tanh pO'

Rc(T, T)ds $ 2(n - l)p. tanh pO'

This completes the proof. 18 Geometry of Complete Manifolds at Infinity. Given a complete Riemannian manifold, we define its aperture in the following way. Pick an origin 0, and let 8(s) be the sphere if radius s around the origin, the set of points whose distance to the origin 0 is exactly s. Its diameter diam 8. is the maximum distance between two points in the sphere. The aperture a of the manifold is defined as a = lim sup diam 8./2s . • -+00

Clearly a is invariant under dilation. We note that the aperture is independent of the choice of the origin. To see this, suppose 0 and 0' are two origins. Choose points P and Q on the sphere 8 8 around 0 with s very large compared to the distance r between 0 and 0', and so that d(P, Q) is nearly as where a is the

THE FORMATION OF SINGULARITIES

71

aperture at O. Then P and Q are nearly at distance as from 0', and by making one shorter we can make the distances equal, and at least s - r. For s large we can make as/(s - r) as close to a as we like. Then the aperture a' at 0' is at least the aperture a at O. By symmetry a = a'. Note the aperture of the paraboloid is 0, the aperture of a convex cone is between 0 and 1, the aperture of Euclidean space is 1, and the aperture of hyperbolic space is 00. In much the same way we can prove the following result. For a solution to the Ricci Flow the aperture a = a(t) is defined for each t.

THEOREM 18.1. For a complete solution to the Ricci flow with bounded curvature and weakly positive Ricci curvature the aperture a is constant.

Proof. Suppose Rc ~ 0 and IRml ~ M. In time ~t ~ 0 the distance between two points shrinks but not by more than e.JM~t. Let a be the aperture at time t. For any a < a and any (j < 00 we can find s ~ (j and two points P and Q such that d(O,P, t) = s

Then s-

d(O,Q,t)

=s

d(P, Q, t) ~

and

e../M~t ~ d(O, P, t ± ~t)

~ s + e.JM~t

s - e../M~t ~ d(O,Q,t ± ~t) ~ s

and

as.

+ e../M~t

as - e../M~t ~ d(P, Q, t + ~t) ~ as + e../M~t.

Now depending on which is further from 0, we can more P or Q back toward and Q from 0 equal again, without reducing the distance between P and Q by more than e.JM~t. Since

o by no more than e../M~t and make the distances of P

as- e../M~t . r>:I

s+evM~t

we see that the aperture at time t ± is constant.

_

---+ a ~t

as

s -+

00

is at least a also. Hence the aperture

0

THEOREM 18.2. Suppose we have a solution to the Ricci Flow on a complete manifold with bounded curvature. If IRml -+ 0 as s -+ 00 at t = 0, this remains true for t ~ O.

Proof. Suppose IRml ::; M for some constant M. For every e > 0 we can find (j < 00 such that IRml ::; e for s ~ (j. The curvature tensor evolves by a formula DtRm = ~Rm+Rm*Rm

RICHARD S. HAMILTON

72

which gives a formula

and an estimate

Zt IRml2 ~ AIRml

2

+ CI Rm l3

for some constant C depending only on the dimension. For any ,5

> 0 choose

and choose the continuous function 8 ~

a,

a::; 8 8 ~

~

p,

p.

where s is the distance from some origin at t = o. Then 1/; is Lipschitz continuous since s is, and since IDsl ~ 1 almost everywhere we also have ID1/;1 ~ ~ almost everywhere. Now we can smooth 'IjJ locally and patch together with a partition of unity to get a function ;p which is smooth and has _E;2

~

;p ~ M2 + E;2

ID;PI $

and

and

;p ~ M2 - E;2 for Lastly take cp = 'IjJ + 2E;2. Then E;2

~ cp $

M2

s~

+ 3E;2

(1

2~

and

;p ~ E;2

IDcpl

~ 2~

and

everywhere if

8

~ p.

everywhere

and

cp ~

M2

Now define cp for t

~

if s ~

(1

and

cp ~

3E;2

if s ~ p.

0 by solving the scalar heat equation

in the Laplacian of the metric evolving by the Ricci Flow. By the maximum principle we still have E;2 $ cp $ M2 everywhere for t ~ o. The derivative DaCP evolves in an evolving orthonormal frame by the formula

and hence

:t

IDcpl2 = AIDcpl2 - 21D2cp12.

Note this formula does not involve the curvature. Hence IDcpl $ 28 everywhere for t ~ 0 by the maximum principle.

THE FORMATION OF SINGULARITIES

73

The second derivative DaDb'() evolves by the formula

and hence

which gives an estimate

for some constant 0 depending only on the dimension. Let us put

and compute

a;; : ; ~F

- (1- OMt)ID2cpI2.

Then if t ::; e/M where e = 1/0 depends only on the dimension, we have

0/ <~F

ot -

and the maximum of F decreases. But

at t

= 0, and hence for t ~ 0 also.

Thus

Since l~cpl2 ::; nlD2cpl2 and cp solves the heat equation,

I~~ I::; 08Vt

for

0 < t ::; e/M

where this constant 0 = 2v1n depends only on the dimension n. Now an improper integral which is 20 which is finite, so for all P

Icp(P, t) - cp(P, 0) I ::; 208Vt for

1/0 has

0 < t ::; e/ M.

Since 8 > 0 is arbitrarily small, we can take

so that 2080 :S c 2 for t :S elM. Then cp :S 4c 2 at times t ~ elM on the set where s ~ p at t = O. Now distances can expand, but only at an exponential

RICHARD S. HAMILTON

74

=

rate governed by M. In particular if s s(P, 0, t) is the distance between a point P and the origin 0 at time t, we have

as < CMs at and

s(t) ~ s(O)eCMt . This gives us a constant C depending only on the dimension such that if s ~ C P at P at time t ~ clM then s ~ p at P at t = 0, and cp ~ 4€2 at P at time t. Now at t = 0 we have

and IRml2 ~ so IRml2 ~ cp everywhere at t

= O.

E2 ~

cp if s ~

0-

Since

we have

while

a

at (eCMtcp)

= ~ (eCMtcp) + CM (eCMtcp)

so IRml2 ~ eCMtcp by the maximum principle. For t ~ clM this gives IRml2 ~ Ccp for some other constant C depending only on the dimension. Hence at time t we have IRml2 ~ CE 2 for s ~ Cp where these constants C depend only on the dimension and are independent of Thus IRml -+ 0 for t ~ elM also as s -+ 00. Since the time interVal can always be advanced by clM as long as IRml ~ M, we get the result untillRml becomes unbounded or t -+ 00. 0

E.

Next we define the asymptotic volume ratio. Again let s denote the distance to an origin 0 in a complete manifold of dimension n, let BB denote the ball of radius s around the origin, and let V(Bs) be its volume. If the manifold has weakly positive Ricci curvature, then the standard volume comparison theorem tells us that V(Bs)/sn is monotone decreasing in s. We define the asymptotic volume ratio In Euclidean space II is the volume JJ of the unit ball, otherwise II ~ JJ. For all s, V(Bs) ~ vs n . In the same way as for a before, the value of II is independent

THE FORMATION OF SINGULARITIES

75

of the choice of the origin. (We omit the details.) Hence the lower bound holds on any ball around any point P

Often a volume bound can substitute for an injectivity radius bound. Of course we also have

THEOREM 18.3. Suppose we have a complete solution to the Ricci Flow with bounded curvature and weakly positive Ricci curvature, where IRml -+ 0 as s -+ 00 (a condition preserved by the flow). Then the asymptotic volume ratio 11 is constant.

Proof. Let 'Y be a small constant we shall choose soon, and consider the annulus

Since we have

V(Nu ) = V(Bu) - V(B-yu).

IT the asymptotic curvature ratio is at least

When "{ is small, the annulus.

11 - "(nil

is nearly

11

11,

then

and most of the volume of the ball is in

0

The volume of the annulus changes at a rate

For every e and every 'Y we can find This makes

IT Vi (Nu ) is the volume at time

t1

0"0

so that if 0"

~

0"0

then

IRml ~ e on N u •

and V:!(Nu ) is the volume at time t2 we have

V:!(Nu ) ~ e-~lt2-tlIVi(Nu). Let 111 be the asymptotic volume ratio at time t1 and Then

Vi (Nu ) for all

0"

and all "( >

o.

112

the ratio at time

~ (111 - "{nll)O"n

If V2 (Bo-) is the volume of Bu at time t2 then

t2.

RICHARD S. HAMILTON

76

Together these make

V2 (B u ) ~ e-f:lt2-ttl(Vi - ,..ni/)an . Fix,..

> 0 and let a ~ O. V2

Since this is true for all,.. and V is constant.

Then c ~ 0 and = lim V2 (B u )/a 2 ~ u-+oo

> 0, V2

~ Vi.

Vi -

,..ni/.

But we can switch

ti

and

t2,

so

Vi

= V2

19 Ancient Solutions. There is one other geometric invariant we shall consider. Let 0 be an origin, s the distance to the origin, and R the scalar curvature. We define the asymptotic scalar curvature ratio A

= lim sup RS2. 8-+00

Again the definition is independent of the choice of an origin and invariant under dilation. This is particularly useful on manifolds of positive curvature where R bounds IRml. On Euclidean space A = 0, on a manifold which opens like a cone 0 < A < 00, and on a manifold which opens like a paraboloid A = 00. Eschenberg, Shrader and Strake ([18]) have shown that on a complete odd-dimensional manifold of strictly positive sectional curvature A > OJ it is unknown whether this is true in even dimensions.

THEOREM 19.1. For a complete solution to the Ricci Flow with bounded curvature which is ancient (defined for -00 < t < T), and either with weakly positive curvature operator or Kiihler with weakly positive holomorphic bisectional curvature, the asymptotic scalar curvature ratio A is constant.

Proof. In either positive curvature case the Harnack estimate holds, and we conclude that the scalar curvature R is pointwise increasing. If the asymptotic curvature ratio is A at time t then for any finite A < A and any 8 we can find a point P at distance s ~ 8 from 0 at time t where RS2 ~ A. At a later time t + 6.t with 6.t ~ 0 the scalar curvature R at P is at least as big, while if M is a bound on the curvature everywhere the distance s of P from 0 will not have shrunk by more than CVM6.t. Since s - CVM6.t ~ 1 as s

s~oo

we see that the asymptotic scalar curvature ratio is at least A still at time t+6.t. Hence A does not decrease. To see A does not increase either, first suppose at some time t that A is finite. Then for any A> A we can find 8 ~ l/VM so that Rs2 ~ A for s ~ 8

THE FORMATION OF SINGULARITIES

77

at time t. Moreover for any 8 and any A < A we can again pick a point P at time t with R8 2 ~ A and s ~ 28. Consider any point Q at distance

d(P, Q, T) ~ s/2 for any

T ~

t. Since Rc

~

0, distances shrink and

d(P, Q, t) ~ 8/2 also. Then ~

d(Q,O,t) and by our choice of s

8/2 ~ S

R(Q, t) ~ 4A/8 2

and since R increases pointwise

also. Our interior derivative estimates allow us to bound DR and also D2 R, and hence oR/at. Recall from section 13 that if IRml :S M at all points at distance at most r from P for all times between T - r2 and T with M r2 :S I then

with a constant C depending only on the dimension. We can bound and take M = Ci/8 2 • When A :S I/C we can take r = 8/2; when s/2. In the first case we find that

oR

A~ ~

Ft(P,T) :S CA/s

I/C we can take r

IRml by R

= 1/2VM :S

4

and in the second case we find

for some constant C depending only on n, at all case. Pick Llt ~ 0. Then

T

:S t. Use

R(P, t - Llt) ~ R(P, t) - C(A + .A2)Llt/84 •

Also ~

d(P, 0, t - Llt)

Taking

8

very big compared to At and R(P, t) ~ A/8 2 ~

d(P, 0, t)

= 8.

A and A so that C(A + ~)At/s4

A + A2

for either

RICHARD S. HAMILTON

78

we have

R(P, t as

~t)S(P, 0, t - ~t)2 ~ S2 [ ~-

..... _ C(A ~4AA2 )~t

1-+ A

S -+ 00. Hence lim sup Rs2 ~ A at time t - ~t as well. In the case where A = 00 at time t, so that

limsupR(Q,t)d(Q,0,t)2

= 00

8-+00

we have to be more careful. For any

A < 00 choose the largest s so that

sup{R(Q, t)d(Q, 0, t)2 : d(Q, 0, t) :5 s} :5

A.

That a largest s exists is clear since if Q is any point at distance s we can find Qj at distance Sj with Sj /' sand Qj -+ Q. Moreover since the sphere ofradius 8 is compact, there must exist a Q with

d(Q,O,t)

=8

and or else 8 would not be maximal. Now choose P so that

d(P,O,t) and

R(P, t)

~

8

~ ~ sup {R(Q, t) : d(Q, 0, t) ~ s}

which is possible since R is bounded. Since

R(P, t)

Q is a

possible choice

~ ~R(Q, t)

and then 2

1-

R(P, t) d(P, 0, t) ~ 2" A. IT Q is any point with 1

d(P, Q, r) :5 2"d(P, 0, t) at some time r :5 t, then since distances shrink 1

d(P, Q, t) :5 2d(P, 0, t) as well, and 1 3 2d(P, 0, t) ::; d(Q, 0, t) ::; 2d(P, 0, t).

THE FORMATION OF SINGULARITIES

79

Either d(Q,O,t)$s

in which case R(Q, t)d(Q, 0, t)2 $ A

by our choice of s, and R(Q,t) $ 2A/d(P,0,t)2 $ 4R(P,t)j

or else d(Q,O,t) ~ s

in which case R(Q, t) $ 2R(P, t)

by our choice of Pj and so in either case R(Q, t) $ 4R(P, t).

Since R increases pointwise, R(Q,r) $ 4R(P,t)

for r $ t whenever

1 d(P, Q, r) $ 2d(P, 0, t).

Now we can use the interior derivative estimate again, for

A ~ 1 we get

and as before 1R(P, 0, t - Llt)d(P, 0, t - Llt)2 ~ 2A - CA 2Llt/d(P, 0, t)2

where d(P, 0, t) is large compared to Llt and lim sup RS2

A.

As d(P, 0, t)

-t 00

we see that

= 00

a time t - Llt as well. This finishes the proof of the Theorem.

o

Now we prove several results that show an ancient solution with positive curvature operator whose scalar curvature R falls off rapidly in space and time behaves like a cone at infinity.

THEOREM 19.2. Suppose we have a solution to the Ricci Flow on an ancient time interval-oo < t < T, complete with bounded curvature and strictly positive curvature operator. Assume

lim sup(T - t)R < t-+-oo

00

RICHARD S. HAMILTON

80

(as happens in Type I) and assume the asymptotic scalar curvature ratio (which we saw is constant in time) is finite

A

= lim sup Rs2 < 00. 8-+00

Then we get the following results: (a) The asymptotic volume ratio (which we saw is constant in time) is strictly positive v = lim V(Bs)/sn > OJ and 8-+00

(b) for any origin 0 and any time t there exists a constant ¢(O, t) that at all points at the time t

>0

such

RS2 ~ ¢(O, t).

Proof. We begin with a good estimate giving an upper bound on the curvature at all pairs of points and all time. D

LEMMA. There exists a constant C such that for all points P and Q at all times t ::; 0 we have

min[R(P, t), R(Q, t)]d(P, Q, t)2 ::; C

where d(P, Q, t) is the distance from P to Q at time t. Proof. Since A

< 00, some constant Co

works at t

= 0, so

min[R(P,0),R(Q,0)]d(P,Q,0)2::; Co

for all P and Q. Since R increases pointwise, R(P, t) ::; R(P, 0)

and

R( Q, t) ::; R( Q, 0)

for t ::; O. Since R::; C/(T - t), we can use Theorem 1.72 to get d(P, Q, t) ::; d(P, Q, 0)

+ C../T -

t .

This makes d(P, Q, t)2 ::; 2d(P, Q, 0)2

+ C(T -

t).

Thus min[R(P, t), R(Q, t)]d(P, Q, t)2 ~ 2min[R(P,0),R(Q,O)]d(P,Q,O)2

+Cmin[R(P,t),R(Q,t)](T-t) ~ C

for some constant C using the bound on the first term at t = 0 and the bound R ~ C/(T - t) everywhere. 0

THE FORMATION OF SINGULARITIES

LEMMA. There exists a constant c > 0 such that for every t a point Pt where R(Pt , t) ~ c/(T - t).

81

~

0 we can find

Proof. The maximum Rmax of R satisfies the ordinary differential inequality d 2 dt Rmax ~ C Rmax

for some constant C, by applying the maximum principle to the evolution of R. If Rmax(t) were even smaller than c/(T - t) for c small, it could not make it up to Rmax(O) in time. 0 Now fix an origin 0 and let s = d(P, t) = d(P, 0, t) be the distance of P to the origin at time t.

LEMMA.

There exists a constant C· so that

RS2

~ C· for all t ~

O.

Proof. Since R(Pt , 0) ~ R(Pt , t) ~ c/(T - t)

while min[R(Pt , 0), R(O, O)]d(Pt, 0, 0)2 ~ Co

we get an estimate d(Pt , 0, 0) ~ C../T - t

(where the case R(Pt , 0) ~ R(O,O) can be handled separately because R(O, 0) C /T anyway while T - t ~ T). Then using our distance shrinking bound d(Pt , 0, t) ~ C";T - t

for a larger constant C. For any P d(P, 0, t) ~ d(P, Pt , t)

+ d(Pt , 0, t)

by the triangle inequality. We already have min[R(P, t), R(Pt , t)]d(P, Pt , t)2 ~ C

for some constant C independent of t. If R(P, t) ~ R(Pt , t) ~ c/(T - t)

then the same argument that worked for P t proves that d(P, 0, t) ~ C";T - t

~

82

RICHARD S. HAMILTON

and since R :::; C I(T - t), Rs2 :::; C* for some C*. The other case when

R(P, t) :::; R(Pt , t) gives

R(P, t)d(P, Pt , t)2 :::; C in the estimate above, and since

d(P, 0, t) :::; d(P, Pt , t)

+ cv'T -

t

and R(P,t) :::; C/v'T - t, we get

R(P, t)d(P, 0, t)2 :::; C*

D

also for some C*. This proves this Lemma. Now we turn to the volume estimate. It is useful first to look at annuli.

LEMMA.

There exists a constant c

NiT

> 0 such that the annulus at t

=0

= {O' :::; 8 :::; 30'}

has volume

Proof. Let e

> 0 be a small constant we can choose later. Look at time

at the annulus

NiT

= {20':::; S :::; 30'}.

Since distances shrink as t increases from r to 0, the outer sphere of NiT surely lies inside the outer sphere of NiT' But we have seen

d(P, 0, r) ~ d(P, 0, 0) - Cv'T - r and so if 0' is large (which is our only concern), in particular 0' ~ T - r :::; 21rl :::; 2€,0'2 and

d(P, 0, r) ~ d(P, 0, 0) Choose

€'

vTle,

then

cv'2i 0'.

so small that c.,f2e :::; 1. Then

d(P, 0, r)

~

d(P, 0, 0) - 0'

so no distance from the origin shrinks by more than 0'. Hence the inner sphere of NiT lies outside the inner sphere of NiT, and NiT ~ N u • (Of course we don't need these to be topological annuli, we only estimate distances.) 0

THE FORMATION OF SINGULARITIES

83

Next we claim we can find d > 0 (depending on the e we choose) so that has volume V(Na) ~ dan at time

T

= -ca 2 • Since the curvature (for a ~

R~

G/ITI

Na

JT /c again) satisfies a bound

~ G/ca 2 ,

this remark follows from the following result by dilation, with d = (c/G)n/2. LEMMA. For every p > 0 there exists a ( > 0 so that if a complete manifold with positive sectional curvature has 0 < R ~ 1, then the annulus

Np = {2p ~ 8

~ 3p}

has volume

Proof. Since the manifold is complete with positive curvature but not compact, we can bound the injectivity radius by some apriori constant c> 0 below. The annulus contains a minimal geodesic of length p, as we see by intersecting it with a ray to infinity. IT p ~ c/2 the result is easy using geodesic coordinates at the origin, while if p > c/2 we can put a ball of radius c/2 inside the annulus. (In fact for large p we see the area is at least a constant times p. This is the best we can do if the manifold opens like a cylinder.) 0

Now we want to see that V(Na) still has a large area at t = o. At each time ~ t ~ 0 we still have all of Na outside the ball ofradius a, where R ~ G* /a 2 • Therefore we can estimate the rate at which the volume shrinks by

T

This makes Since

T

= ca 2 we get V(Na)

It=o ~ cV(Na)lt=r ~ cda

n.

But at t = 0, V(Ba) ~ V(Na) ~ V(Na) so 19.2(a) is done. Next we look at 19.2(b). Given a point P at distance a = d(P, 0, 0) from the origin at time t = 0, we let T = _ca 2 as before and find Pr where

R(Pn T) ;::: c/(T - T)

and d(Pn 0, T) ~ cv'T - T.

The Harnack inequality on a manifold with positive curvature operator in its integrated form (see [29]) gives R(P,O) ;::: R(Pr •T)e-Cd(p... P.r)2/lrl

RICHARD S. HAMILTON

84

for some constant C. The triangle inequality gives

d(PTlP,r) ~ d(PTlO,r)

+ d(P,O,r)

and

d(P, 0, r) ~ d(P, 0, 0)

= u.

Then

d(PTlP,r) ~ u + C"';T - r. Again if u ~ v'T/e we have T - r ~

21rl

and

d(PTlP,r) ~ Cu for some constant C, making

for some other constant C depending on e. This yields R(P, 0) ~ c/u 2 as desired. For u ~ v'T/e some constant c > works because R > O. Hence the Lemma is proved. A similar bound can be derived at any time.

°

20 Ricci Solitons. We will now examine the structure of a steady Ricci soliton of the sort we frequently get as a limit.

THEOREM 20.1. curvature, so that

Suppose we have a complete Ricci soliton with bounded

D2f = Rc for some function

f. Assume the Ricci curvature is weakly positive Rc~O

and assume the scalar curvature attains its maximum M at an origin. Then the function f i$ weakly convex and attains its minimum at the origin, and furthermore

IDfI2+R=M everywhere on the soliton. The soliton is not compact unless Rc = O. Proof. We show the equality first. Since

we have and

THE FORMATION OF SINGULARITIES

85

and so DiRik - Di~k = ~iklDd· Taking a trace on j and k, and using the contracted second Bianchi identity DiRii

1

= '2Di R

we get that Then Di(ID 112 + R) = 2Dil(DiDil - ~i) = 0 so ID 112 + R is constant. Call it M*. If M* = M, then DI = 0 at the origin. Since DiDil = Rij ~ 0, along any geodesic through the origin xi = xi(s) parameterized by arc length s we have dl dx i dB =Dd· ds

and

~I

= DiD I . dx i dxi > 0 ds 2 3 dB dB so I is convex and hence least at the origin. Since any point can be joined to the origin by a geodesic, we are done in this case. If M* > M, consider a gradient path of I through the origin xi = xi(n) parametrized by the parameter u with xi at the origin at u = 0 and 0

dx i _ iiDol du - 9 J .

Now ID 112 = M* - R so ID 112 ~ M* - M smallest at the origin. But we compute :UID112

> 0 everywhere, while ID 112 is

= 2gikgilRiiDdDd ~ 0

since Rii ~ 0 and ID 112 ~ o. Then ID 112 isn't smaller at the origin, and we have a contradiction. If the solution is compact then

AI = R ~ 0 implies I is constant, so Rc = D2 I = o.

o

THEOREM 20.2. For a complete Ricci soliton with bounded cUnJature and strictly positive sectional curvature 01 dimension n ~ 3 where the scalar cUnJature assumes its maximum at an origin, the asymtotic scalar curvature ratio is infinite; A = limsupRs2 = 00 0

8-+00

RICHARD S. HAMILTON

86

where

8

is the distance to the origin.

Proof. Suppose RS2 ~ C. The solution to the Ricci Flow corresponding to the soliton exists for -00 < t < 00 and is obtained by flowing along the gradient of I. We will show that the limit 9iJ'(X)

=

lim gij(X,t)

t~-oo

exists for x '" 0 on the manifold X and is a flat metric on X - {OJ which is complete. Since X has positive curvature operator it is diffeomorphic to R n , and X - {OJ to sn-l X Rl. For n ~ 3 there is no flat metric on this space, and this will finish the proof. 0 To see the limit metric exists, note that unless RS2 -+ 00 as S -+ 00, surely R -+ 0 as X -+ 00 so ID 112 -+ M as X -+ 00, at least at t = O. The function 1 itself can be taken to evolve with time, using the definition al at

= -ID112 = 6.1 -

M

which pulls 1 back by the flow along the gradient of I. Then we continue to have DiDjl = ~j for all time, and IDI12 -+ Mass -+ oo-for each time. When we go backwards in time, this is equivalent to flowing outwards along the gradient of I, and our speed approaches v'lJ. If 8 is the distance from 0, then s / It I -+ v'lJ. Since RS2 :::; C for some constant C, R :::; C /8 2, and starting outside of any neighborhood of 0 we have R:::; C/Mlt1 2 and hence

~g > -2RgIJ.. at 'J.. = -2R-· I, gives

If V

a

2C

o ~ atgij ~ - Mltl 2 gij' is a tangent vector and Wit denotes its length at time t, so

then

o~

d

I 12t

dt V

so

~

-

d

2C I 12 Mltl 2 V t 2

2C

o :::; dltl log Wit :::; Mltl 2 with t

< 0 decreasing and It I increasing. This makes d dltl

(

2

log IVlt +

so that log

2C ) Mltl :::; 0

IVI~ + !~I

WI~

increasing in

It I with

THE FORMATION OF SINGULARITIES

87

is actually decreasing. This shows Wit has a limit as t -+ -00. Since the metrics are all essentially the same, it always takes an infinite length to get out to 00. On the other hand, any point X other than 0 will eventually be arbitrarily far from 0, so the metric in the limit is also complete away from oin X - {OJ. Using the derivative estimates of W.-X. Shi [43] on the curvature it is straightforward to see that the 9ij(X, t) converge in Coo to a smooth limit metric 9ij(X) as t -+ -00. Since R ~ C/S2 and s ~ VMt we have the result that the limit metric is flat. This proves the theorem. 21 Bumps of Curvature. We shall show an interesting fact in this section about the influence of a bump of strictly positive curvature in a complete manifold of weakly positive curvature. Namely, minimal geodesic paths that go past the bump have to avoid it. As a consequence we get a bound on the number of bumps of curvature. This principle will be important for studying the behavior of singularity models at infinity when we do a dimension reduction argument. We begin by reviewing Toponogov's Theorem as given in Cheeger and Ebin [9]. Let M be a complete Riemannian manifold with all sectional curvatures K bounded below by a constant H. Suppose we have a geodesic triangle /:). in M with sides of lengths a, b, and c, and let 0: be the angle opposite the side of length a.

M

~ c anJ(~H

We make the following assumptions (1) the geodesics of lengths a and b are minimal (2) c ~ a + b (surely true if the geodesic of length c is also minimal) and (3) c ~ 7r/..fH if H > O. THEOREM 21.1. There exists a traingle /:). in the space M with constant curvature H whose sides hatle length a, band c, such that the angle Ii in /:). opposite the side of length a satisfies Ii ~ 0:. THEOREM

and angle

0:,

21.2. There exists a unique triangle /:). in M with sides band c such that the length a of the side opposite 0: satisfies a ~ a.

REMARK. . It

is not necessary to have sectional curvatures

K,

~

H in all of

RICHARD S. HAMILTON

88

M; it suffices to have this hold in the ball of radius a + b around any point in the triangle; because the construction only uses It on minimal geodesics joining two points on ~, and these all lie in such a ball. To see this, consider a geodesic triangle with sides a, b, and c $ a + b. If we join a point on the side a to a point on the side b with a minimal geodesic of length l, clearly l $ a + b. If we join a point on the side a to a point on the side c with a minimal geodesic of length i, and if the first point divides the side a into pieces a = al a2, and likewise the second point divides the side c into pieces c = Cl + C2, then

i

$ al

+ Cl

c

and l $ a2

+ b + C2

and by averaging

1 l$ 2(a+b+c) $a+b

as claimed. LEMMA 21.3. For every c > 0 there exist A < 00 and 15 > 0 such that if M is complete with K ~ 0, P is a point in M and K ~ c/r 2 everywhere in B 2r (P), if d(P, P') = r and if d(P, Q) ~ Ar, if P P', PQ and P' Q are minimal geodesics and if LP'PQ < ~ +15 - 2 then d(P', Q) < d(P, Q).

Proof. Pick a point Q' on the geodesic PQ at distance r from P, and choose a minimal geodesic from P' to Q'. 0 P'

Q

ct Let h = IP'Q'I,u LPQ' P' and /3' = 11' -

= IP'QI

and v = IQQ'I and let Q = LP'PQ and /3 = /3. We make three applications of Toponogov's Theorems.

89

THE FORMATION OF SINGULARITIES

(1) First note for every e > 0 we can find 5 > 0 and TJ > 0 such that if + 5 then h ~ (v'2 - 'I) r. This is because K ~ e/r 2 in B2r(P) and we can compare the triangle pI PQ' to the triangle with two sides equal to r and angle a in the sphere of curvature H = e/r2 using T2. All the sides are minimal, and we only need to check that

a ~ ~

h ~ 2r ~ Tr/m if e < 1 < {Tr/2)2. Hence the comparison can be made. Now on the sphere of radius 1, take an isosceles triangle of equal sides l ~ 1 with angle a ~ ~ + 5 between them and call the length of the third side k. In an isosceles right triangle k is strictly less than the Euclidean value of v'2 l, and hence depending on l we can find 5 > 0 and 'I > 0 such that if a ~ ~ + 5 then still k ~ (v'2 - TJ )l. If we scale the result to a sphere of radius r /.,fi with curvature H = e/r 2 , then taking l = .,fi gives the desired result. (2) Now we just use K ~ O. We compare the triangle P'Q' P with two sides equal to r and one equal to h ~ (v'2 - TJ)r to the Euclidean triangle with the same three sides using Tl. again all the sides are minimal, and we can do the comparison. We find that there exists a () > 0 depending on 'I only so that {3 ~ ~ + (). By scaling it suffices to observe that an isosceles Euclidean triangle with two equal sides 1 and the third side less than v'2 - TJ has the equal angles at least ~ + 6. (3) Finally we use T2 again to compare the triangle P'Q'Q to the Euclidean triangle with sides h and v and angle {3' ~ Again all the sides are minimal, and we find

3; - ().

Now h ~ v'2r while 1 +( cos{3, > - -

-

v'2

for some ( > 0 depending only on () > O. Therefore u 2 ~ (v + r)2 + r[r - 2V2(v] and for every (

> 0 we can choose A < 00 so that if

IPQI ~ Ar Thus IP'QI = u < v + r

v +r =

then v ~ (A - l)r and 2v'2(v > r. Now we prove an important repulsion principle.

=

IPQI

as desired.

THEOREM 21.4. For every e > 0 we can find A < 00 such that if M is a complete Riemannian manifold with K 2: 0, if P is a point i;'l M such that K 2: e/r 2 everywhere in B 3r (P), if S 2: rand Q1 and Q2 lie outside B)..B(P) and'Y is a minimal geodesic from Q1 to Q2, then'Y stays outside B.(P).

Proof. Let X be the closest point on QIQ2 to P. Draw a minimal geodesic from X to P and let its length be 0'. Extend the geodesic X P an equal length

RICHARD S. HAMILTON

90

beyond P, ending at a point Y. Draw minimal geodesics Q1Y and Q2Y. We claim IQ1 Y I < IQ1 X I and IQ2YI < IQ2XI

0'

which will show Q1Q2 is not minimal, provided

0'

:5 s.

y

x Since both halves of the argument are the same, we drop the subscripts 1 and 2. Consider the geodesic triangle QXY with P the midpoint of XY, where QX and QY and PX are minimal and LQXP = 7r/2.

Y

Q

x Choose the point Z at distance r from P towards X, and draw minimal geodesics QP and QZ. Let a = LQZX and a' = 7r-a, while "1 = LQPY and "1' = 7r-'Y. Again we make several applications of Toponogov's Theorems. First note that IQPI ~ >.s and IPXI:5 8 so

IQZI

~

IQPI-IPZI

~

(>. -1)8

and

IQXI :5 IQPI + IPXI :5 (>. + l)s . . Therefore comparing the triangle QZX to the Euclidean one with the same three sides, we find by T2 that for every 8 > 0 there exists a >. < 00 such that a ~ ~ - 8, as is easily seen by first comparing the Euclidean triangle to one of sides proportional to >. + 1, >. - 1, and 1 with a more extreme angle a, and observing a -+ 7r/2 as >. -+ 00. Consequently a' :5 ~ + lS. Now choosing 8 small and >'large compared to e, and noting that if K ~ e/r 2 in B 3r (P) then B 3r (P) ~ B 2r (Z), we see that Lemma 1 implies IQPI < IQZI. Now if we also had "1' :5 ~ + 8 we would also have QZ < QP by Lemma 3, and we cannot have both .. Hence 'Y' 2: ~ + lS and this gives 'Y' :5 ~ - lS.

THE FORMATION OF SINGULARITIES

91

Now we apply Toponogov's Theorem 21.2 to the triangle QPY to compare it to the Euclidean triangle of sides IQPI and IPYI and angle 'Y. We do not know if PY is minimal, but QP and QY are by construction, and

IPYI = u ~ 8 while IQPI

~ A8

and hence IPYI ~ IQPI + IQYI, which is all we need. Then by the law of cosines

IQYI 2 ~ IQPI 2 + IPYI 2

-

2IQPI·IPYI· cOS'Y.

But we also have

IQPI 2 ~

IQXI 2 + IPXI 2

by Tl on the triangle of sides QX and PX and angle 7r/2. Then

IQYI 2 ~ Use

IQXI 2 + IPXI 2 + IPYI 2 - 2IQPI·IPYI· cos'y.

IPXI = IPYI = u ~ 8 IQYI 2 ~

and

IQPI

~ A8 and 'Y ~ ~ - &to get

IQXI 2 + 2u 2 [1- Acos

(i - &)] .

Picking A large compared to &, we get Acos and IQYI

(i - 8) = Asin& > 1

< IQXI as desired. This proves the theorem.

D

We apply the previous repulsion theorem to prove a result on remote curvature bumps in complete manifolds of positive curvature. DEFINITION 21.5 . A ball Br(P) of radius r around P is a cUnJature {3-bump if K ~ (3/r 2 at all points in the ball. The ball is A-remote from the origin 0 if d(P, 0) ~ Ar.

°

THEOREM 21.6. For every {3 > there exists A < 00 such that in any complete manifold of positive cUnJature there are at most a finite number of disjoint balls which are A-remote cUnJature (3-bumps.

Proof. If the ball Br(P) is a A-remote curvature {3-bump, and if Q is any point such that d(O, Q} ~ 2d(0, P}

then if we take minimal geodesics OP and OQ. we claim that for any {3 can find A < 00 and (J > 0 such that LPOQ

> 0 we

~ (J.

To see this, let X be the point on OQ with OX = OP. Since K ~ 0 everywhere and !:l.OPQ has minimal sides, if the angle LPOQ is < 9,then for every>. < 00

92

RICHARD S. HAMILTON

we can find () theorem.

> 0 such that PX < OP/>... But this contradicts the repulsion ~,

"

.. --- ....... , p

I

,

'.

I

I

o

Q

x Note there is a curvature ,B-bump at P,OP> >"P X and PQ

~

OQ - OP

~

OP

so the theorem applies (with c = ,B/9 to get K ~ c/ p2 on the ball of radius p = r/3). Now pick any sequence Pi of curvature ,B-bumps with

and we find for j

< k the angle

for a fixed () > O. This is impossible. Hence there cannot be an infinite sequence of >"-remote disjoint curvature ,B-bumpsj for since K is bounded on any compact set and r ~ ..;c / K on each bump, we can only get a finite number of disjoint bumps into any compact set, and this lets us find PiH with d(PiH' 0) ~ 2d(Pi , 0). Thus the theorem is proved. 0

22 DiInension Reduction. There is a general principle of dimension reduction which has proved useful in minimal surface theory and also the theory of Harmonic maps. The idea is that having first taken a limit of a sequence of dilations to model a singularity, we should study this limit by next taking a sequence of origins going out to infinity and shrinking back down to get a new limit of lower dimension. On a complete manifold the idea is that in dimension at least three, as we go out to infinity the radial curvatures will fall off faster than the meridian curvature, so the new limit of the contractions will be flat in the radial direction. We will illustrate this idea by proving a result on solutions with positive curvature operator, where the Little Loop Lemma gives injectivity radius controlj but the same idea will work in any other case where we can control the injectivity radius. THEOREM 22.1. Suppose we have a solution to the Ricci Flow on a compact manifold Mm of dimension m with weakly positive curvature operator for a

THE FORMATION OF SINGULARITIES

93

maximal time interval 0 :::; t < T. Then we can find a sequence of dilations which converge to a complete solution of the Ricci Flow with curvature bounded at each time on an ancient time interval -00 < t < 0 with scalar curvature R bounded by R:::; 0/(0 - t) everywhere and R = 1 at some origin 0 at time t = 0, which again has weakly positive curvature operator. Moreover the limit splits as a quotient of a product NR x Rk with m = n + k flat in the directions Rk with k ~ 0, and where the interesting factor Nn either is compact or has finite asymptotic curvature ratio lim Rs2 8-+00

= A < 00.

Moreover the limit factor NR will still satisfy a local injectivity radius estimate.

sn

Of course we conjecture the only possible limit is the round sphere or a quotient of it shrinking to a point. In dimension 3 or 4 we have pinching estimates that keep the curvature operator strictly positive if it starts strictly positive, that prevent limits NR x Rk with k > o. We do not know any examples of complete non compact ancient solutions of positive curvature operator with RS2 < 00 and Rltl < 00, and we conjecture none exist, since the curvature has had plenty of space and time to dissipate. Proof. The Little Loop Lemma gives us a bound on the injectivity radius in terms of the local maximum of the curvature; if R :::; 1/r2 in the ball ofradius r around a point P, then the injectivity radius at P is at least fJr for some fJ > O. This allows us to take limits by dilating to make the maximum curvature 1. From the results in section 16 we get a limit solution of Type I or Type II . Any such limit will split as a product NR x Rk with k ~ 0 as large as possible, and where N R has strictly positive sectional curvature; for any zero sectional curvature is a zero eigenvector of the curvature operator, producing a reduction of the holonomy to the nilgroup O(n) ~ O(m). Among all possible Type I or II limits choose one where k is maximal. We shall then get a contradiction unless NR has finite asymptotic scalar curvature ratio A < 00. We have seen in Corollary 16.6 that a Type II limit with weakly positive curvature operator must be a Ricci soliton, and in Theorem 19.2 we have seen that in dimension n ~ 3 such a Ricci soliton must have A = 00. In dimension n = 2 the only Ricci soliton is the cigar 2: 2 (see [22]) which does not satisfy the local injectivity radius bound, since R goes to zero exponentially in the distance s from the origin, while the circumference of the circle at distance 8 approaches 1 as it opens like a cylinder. Thus if we prove NR is compact or has A < 00, it must be Type I. Suppose therefore that N R is not compact and A = 00, and we shall contradict k maximal. We shall pick a sequence of dilations of NR which converges to a limit with a flat factor. We need the following result. LEMMA 22.2. Given a complete noncompact solution to the Ricci Flow on an ancient time interval - 00 < t < T with T > 0 with curvature bounded at

94

RICHARD S. HAMILTON

each time and with asymptotic scalar cUnJature ratio

A = limsupRs2 =

00

8-+00

we can find a sequence of points Pj -+ 00 at time t = 0, a sequence of radii and a sequence of numbers 8j -+ such that (a) R(P, 0) ~ (1 + 8j )R(Pj, 0) for all P in the ball Br(Pj , 0) of radius r j around Pj at time t = (b) rJR(Pj,O) -+ 00 (c) if Sj = d(Pj,O,O) is the distance of Pj from some origin at time t = 0, then Aj = SjlTj -+ 00 (d) the balls Brj (Pj, 0) are disjoint.

°

rj

°

°

Proof. Pick a sequence ej -+ 0, then choose Aj -+ 00 so that AjeJ -+ well. As in Theorem 18.2, let Uj be the largest number such that sup{R(Q,0)d(Q,0,0)2: d(Q,O,O) ~

Uj}

00

as

~ Aj .

Then

R(P,0)d(Q,0,0)2 ~ Aj if d(P,O,O) ~

Uj

while there exists some Qj with

R(Qj,0)d(Q,0,0)2

= Aj

and d(Q,O,O)

= Uj

( or else U j would not be maximal). Now pick Pj so that d(Pj, 0, 0) ~

Uj

and

1

R(Pj,O) ~ 1 +e. sup{R(Q,O): d(Q,O,O) ~

Uj}

J

which is possible since even on a noncom pact set we can come as close to the sup as we wish. Finally pick rj = ejuj. First we check (a). IT P is in the ball of radius rj around Pj at time t = 0, either d(P, 0,0) ~ Uj or d(P, 0,0) ~ Uj. In the first case we have from the choice of Pj R(P,O) ~ (1 +ej)R(Pj,O) which satisfies condition (a) with OJ = ej. In the second case, we have from the . choice of U j R(P,O) ~ Aj/d(P,0,O)2 and

d(P, 0, 0) ~ d(Pj, 0, 0) - d(P, Pj, 0) ~ so

Uj -

1 A· R(P,O) ~ (1-e;)2' uJ'

On the other hand, from the choice of Q; ARCQ;, 0) = u~ J

rj

= (1 -

ej)uj

THE FORMATION OF SINGULARITIES

95

and from the choice of Pj 1 R(Pj,O) ~ -1-R(Qj,O) +Cj

since Qj is a possible choice of Q, then R(P· 0) 3'

and

1

AO'J

> - - . ---1...

-l+cj

1 +Cj R(P, 0) ~ (1_cj)2R(Pj,0)

which satisfies condition (a) with

and in either case 8j -+ 0 as Cj -+ O. Next we check condition (b). We have from our previous estimate 2

2 Cj

rjR(Pj,O) ~ -l--Aj -+

00

+Cj

by our choice of Aj . To check condition (c) note Sj ~ O'j so that Aj ~ 1/Cj -+ 00. Finally note that (a), (b) and (c) continue to hold if we pass to a subsequence. Any point Pin Br;(Pj,O) has distance from the origin at time 0 d(P,O, 0) ~ d(Pj , 0, 0) - d(P, Pj, 0) ~ (1 -

Cj)O'j

and since Aj -+ 00 we must have O'j -+ 00. Thus any fixed compact set does not meet the balls Brj (Pj, 0) for large enough j. IT we pass to a subsequence, the balls will all avoid each other. This proves the Lemma. 0 The next step is to take a sequence of dilations of the limit factor Nn around a sequence of points Pj which we take as our new origins OJ, only now we shrink down instead of expanding to make R(Pj , 0) dilate to R(Oj, 0) = 1. The points Pj are chosen at time t = 0 according to the previous Lemma. The balls Br; (Pj, 0) dilate to balls of radius fj -+ 00 by condition 4(b). Condition (a) gives good bounds on the curvature in these balls at time t = 0, while the same bounds for t ~ 0 follow from the Harnack inequality, which has as a Corollary that R is pointwise increasing on an ancient solution with weakly positive curvature operator. The Little Loop Lemma provides a bound on the injectivity radius at a point in terms of the maximum curvature in a ball around the point, in a form invariant under dilation. Hence this local injectivity radius estimate survives into the limit Nn, and gives an injectivity radius estimate at each Pj from the estimate on R in the ball of radius r j. We now have everything we need to take a limit of the dilations of the Ricci Flow around the (Pj,O), dilating time like distance squared and keeping t = 0 in N n as t = 0 in the

96

RICHARD S. HAMILTON

new limit, which we call "'if". This new limit will be a complete solution to the Ricci Flow on an ancient time interval -00 < t ~ with bounded curvature and weakly positive curvature operator. (Note our bounds on R do not hold for t> 0. Once we have "'if" we could extend it for t > by Shi's existence result [42].) Moreover "'if" has an origin 0 and R(O,O) = 1, while R ~ 1 everywhere for t ~ since OJ -+ 00. We claim a cover of "'if" splits as a product with a flat factor. To show this, it suffices to show that "'if" has a zero sectional curvature at (0,0). Suppose it does not. Then we have some lower bound 'Y > on the sectional curvatures at (0,0). This means that there will be a uniform lower bound 'Y' (say'Y' = 'Y /2) so that we have a lower bound K ~ 'Y'R(Pj , 0) on the sectional curvatures at the (Pj , 0) for all large enough j. The bounds on R in the balls Brj(Pj , 0) give bounds on R backwards in time by the Harnack inequality (as we mentioned), and now since R bounds IRml the interior derivative estimates give bounds on the first derivatives IDRml in smaller balls. Since these bounds are dilation invariant, we find that the sectional curvatures all have a uniform lower bound 'Y" (say 'Y' /2) so that we have a lower bound K ~ 'Y" R{Pj , 0) in balls around the Pj at time t = of radii

° °

°

°

°

Pj = c/VR(Pj,O)

for some constant c > 0 depending only on the dimension. Thus there exists a f3 > 0 such that for large j every Pj at t = 0 is the center of a f3-bump, and these bumps are all disjoint. Moreover since PJR(Pj , 0) = c2

and rJR(Pj , 0) -+

00

we see rj/pj -+ 00; and also sj/rj -+ 00 where Sj is the distance of Pj from the origin 0 in Nn at time t = 0, so for any A < 00 the f3-bumps at Pj are A-remote for large j. But this contradicts Theorem 21.6. Hence a cover of "'if" splits as a product and with p + q = n and q > 0, and r is a group of isometries. (Is r = O?) The limit factor NP may not be yet of Type I or II because we did not choose it in the usual way. What we can do is to take a further limit of dilations of . NP, also by shrinking, to get yet another limit 4 NP which will be of Type I or II. We get a Type I limit when the backwards limit is

n=

lim sup It IsupR(P, t) t~-oo

P

< 00

and Type II when this limit is infinite. To extract the Type I limit we choose a sequence of points Pj tj -+ -00 so that the lim sup is attained

= and times

and then make P j the new origin OJ, translate in time so tj becomes 0, dilate in space so R(Pj, tj) becomes 1 and dilate time like distance squared. To extract

THE FORMATION OF SINGULARITIES

97

the Type II limit we choose a sequence nj -+ 00, pick Tj with ITj I as large as possible so that sup{ltIR(P, t) : Tj ~ t ~ O} ~ nj and pick

Pj

and

tj ~ Tj

so that

where ej -+ 0, and dilate the same way. In both cases we have an injectivity radius estimate coming originally from the Little Loop Lemma on Mm and surviving all the dilating and limiting procedures. The rest of this argument proceeds as before. Now a sequence of dilations of Mm converges to N' x Rio, and a sequence of dilations of N' converges to NP x Rq, and a sequence of dilations of NP converges to .4 NP which is Type I or II. Thus a sequence of dilations of N' x Rio converges to NP x Rq+k, and a sequence of dilations of NP x R9+k converges to .4 NP x Rq+k. Now a dilation of a dilation is a dilation, and a limit of limits is a limit by picking an appropriate subsequence. Thus a limit of dilations of Mm converges to .4 NP x Rq+k where q + k > k. This contradicts the hypothesis that k is maximal, which proves the Theorem. 0 There is another case where the blow-down argument can be used.

THEOREM

22.3. Suppose we have a complete Ricci soliton solution

in odd dimension 2n + 1 with bounded curvature and strictly positive curvature operator. Then there exists a sequence of dilations around origins Pj at time o which converges to a limit which splits as a product of Rl with a solution of even dimension 2n which is ancient and complete with bounded curvature and weakly positive curvature operator. Proof. In section 19 we say that ID 112 approaches the maximum curvature M as 8 -+ 00 where 8 is the distance from some origin. Thus for every 8 > 0 we can find (J' < 00 so that for 8 ~ (J'

(VM which makes 1 comparable to

8.

8)8 ~

Hence on the level set

S.., the distance

8

1 ~ (VM + 8)8

= {/ = cp}

is nearly cp / YM for large r, in particular. Hence on the level set SI'={/=J.t}

98

RICHARD S. HAMILTON

the distance s is nearly J.L/ VM, in particular

IJ./(Vii + 6)

$ s $ J.L/(VM - 6)

o

for large 11:.

Now choose the point P j and radii rj as before and let R; = R(P;,O) and = I(Pj,O). Then the CUrvature R at any point P on any sphere 8/A at time = 0 with

J.Li

satisfies an estimate for large j, where again

Rjr] -+

00

and

ej

-+ O.

We can argue as before if we can control the injectivity radii Pj at (Pj, 0) with an estimate Pj ~ e/..jifj. We get this estimate in odd dimensions as follows. Each level set 8/A for large IJ. is a smooth submanifold which is strictly convex since I is convex. The second fundamental form II of 8/A is given by

II(X, Y) = D2 I(X, Y)/ID/I on vectors X and Y where

DI(X)

= DI(Y) = 0

makes them tangent to 8/A' Since IDII -+ .JM and D2 I = Re, we can control the second fundamental form on 8/A by the maximum of Re on 8/A' hence by R j • Thus

IIII $ CRj/Vii r;j.JM. Each 8/A has positive sectional curvature in the

on all 8/, with 1J.L-J.Ljl $ induced metric by the Gauss curvature equation, and each 8/, is orientable since the whole soliton is diffeomorphic to R 2n +l and the normal bundle is oriented by D I > O. H the dimension 2n + 1 of the soliton is odd, the dimension 2n of 8/A is even. Then by a theorem in [9] the injectivity radius of 8/A in the induced metric can be bounded ~ c/.jlfj. This gives a similar bound on the injectivity radius of the soliton at P; in the following way. Since the curvature is positive it is bounded below, and it suffice to show that a ball around Pj in the soliton of radius o/.jlfj has

J

volume ~ c/ R~n+l for some 0 > 0 and c > 0 independent of j. We do this by taking a coordinate chart inside the ball and estimating its volume. First go a distance of VJfj from P; in the direction of ±DI. This moves us out and back some comparable distance. Then take the exponential map of radius of VJfj

THE FORMATION OF SINGULARITIES

99

out from each point on this curve in the spheres S,.. in their induced metric. Start with a frame on the tangent space at Pj and parallelly translate it along the curve in the direction D f to get a frame at each point on this curve, and use it to refer the exponential map on a standard ball in R 2n into S,.. for each p.. Then this gives a coordinate chart in a neighborhood of Pj on the soliton. Since each curvature in the soliton and each second fundamental form on the hypersurfaces S,.. can be controlled by Rj, for a suitable small a the coordinate chart will inject with derivative close to an isometry. This shows the image

J

has volume ~ cl RJn+l. The rest of the proof proceeds just as before, up to taking the first limit. Unfortunately we cannot do the backward limit in time without more injectivity radius control. 23 An Isoperimetric Ratio Bound in Dimension Three. In this section we ·shall prove an isoperimetric ratio bound for solutions to the Ricci Flow in dimension three in the special case of a Type I singularity where we have a solution for 0 ~ t < T < 00 with

IRml(T - t) ~

n

for some constant n < 00, and where we also assume a bound below on the total volume Vet) of the form

v ~ aCT -

t)3/2

for some constant a > O. The first assumption is special; but the second is not so important, since if IRml(T - t) ~ n < 00 but V I(T - t)3/2 -+ 0 (at least for a subsequence of times) then IRmIV 2 / 3 -+ 0, and since IRml controls all the curvatures, the curvature collapes with bounded volume; and it follows from the work of Cheeger and Gromov [10] the manifold has an F-structure, and hence its topology is understood already. .

THEOREM 23.1. For every (3 > 0, p < 00, T < 00, n < 00 and a > 0 we can find a constant'Y = 'Y((3, p, T, 0, a) with the following property. If an initial metric 90 has the property that every surface which bounds a volume at least V on each side has area A ~ (3V 2/ 3, and the initial metric has scalar curvature R ~ -p, and if the subsequent solution if the Ricci Flow exists for 0 ~ t < T with IRml(T - t) ~ 0 and V ~ aCT - t)3/2,

then at any time t any surface which bounds a volume at least V on each side has area A ~ 'Ymin(T - t, V 2/ 3). Proof. Let G(V, t) be the function defined for 0 :5 t < T and 0 :5 V :5 Vet) which for 0 < V < Vet) is the infimum of the areas of surfaces of any type

100

RICHARD S. HAMILTON

=

which divide the manifold into regions of volumes V and V - V, with G 0 if V = 0 or V = V. Then so much is known about the theory of minimal surfaces (see Almgren [1]) that we know G is continuous in V and t, and for any t in o :5 t < T and any V in 0 < V < V the infimum is attained on a smooth surface of constant mean curvature H. Moreover if f3 < f3E where f3E is its Euclidean value f3E = (3611")1/3 then for any metric on a compact manifold we can find ~ > 0 depending on the metric so that any surface bounding a volume V :5 8 has area A ~ f3V 2 / 3 • This 0 means we do not need to concern ourselves with very small volumes. We shall prove a lower bound of the form G function F(V, t) is chosen of the form 1 F

2pt

= ea

> F for 0 < V < V where the

{Q B B} T _ t + V2/3 + [Vet) - Vj2/3

for some suitably large constants Q and B which we are free to choose later. Since ept :5 epT we can find 'Y > 0 in terms of epT , A and B, which will prove the Theorem. IT B is large enough, then by the previous remark we do not have to worry when V or V - V is very small. IT this estimate fails, there will be a first time t* and a volume V* with o < V* < [aCT - t)]3/2 when G = F, and G(V*, t*) will be attained by the area of a smooth surface E* of constant mean curvature H. Consider a oneparameter family of smooth surfaces E(r) for r near 0 by taking the parallel surface to E* at distance'lr, with E(r) inside the part with volume V* for r < 0 and outside for r > o. Note that E(O) = E*. Define the smooth functions A(r, t) and VCr, t) for r near 0 and t near t* by letting A(r, t) be the area of E(r) at time t, and letting VCr, t) be the volume enclosed by E(r) at time t on the side of the part with volume V*. Note that A(O, t*) is the area of E* which is G(V*,t*), while V(O,t*) = V*. It is clear we have the inequality A(r,t*)

~

G(V(r,t*),t*)

since G is the least area among all surfaces enclosing the given volume at the given time. But G ~ F up to time t*, so A(r, t*)

~

F(V(r, t*), t*)

for all r near 0, and equality is attained at r = 0 where G A and F are both smooth, at r = 0 and t = t* we get 8A

8r

8F8V

= 8V&

and 8 2 A > 8 2F (8V)2 8r2 - 8V2 8r

8F8 2 V.

+ 8V 8r2

=F

at time t*. Since

THE FORMATION OF SINGULARITIES

In addition, it is also clear that we have the inequality

A(O, t) ;::: G(V(O, t), t) for t $ t*, and since G ;::: F up to time t* we get

A(O, t) ;::: F(V(O, t), t) for t $ t*, with equality at t

= t*,

Thus at r

8A 8F at $ at Now at r

= 0 and t = t* ,

= 0 and t = t* ,

8F8V

+ 8Vat'

8V -=A=F 8r

aqd

8 2V 8r2

= 8A = H A = H F 8r

where H is the constant mean curvature, Then the equality

8A 8F8V =-8r 8V 8r makes

8F 8V=H,

From this we get

8 2V 8r2

8F

= F 8V'

Now our inequality on 8 2 A/8r 2 becomes

The volume V shrinks at a rate

8V= 8t

!

Rdv
(since the inequality R ;::: -p, which we assume at t = 0, is preserved by the Ricci Flow), Since both F and G are symmetric in V -t V(t) - V, it is no loss to assume V* $ V(t)/2; and this makes 8F/8V = H ~ 0 at r = 0 and t = t*, Then our inequality on aA/8t becomes

aA < aF vaF at - at +p av' These are the inequalities we need,

101

RICHARD S. HAMILTON

102

The remaining fact we use comes from section 12, where we showed that for a family of parallel surfaces I:;(r) we have

8A

82 A

at = 8r2

-471"X

where X is the Euler class of I:; * . We claim that for a suitably large constant Q in the definition of F, which makes F ~ (T - t)/Q,

we can make X ~ 0, so that I:;* is not a sphere or a projective plane. What is required for this? We have 271"X = [ [Rm(P)

lEo

+ K]da

where Rm(P) is the ambient curvature of the tangent plane P to I:; * , and K is the determinant of the second fundamental form. We have Rm(P) ~ CO/(T - t)

for some constant C. Then

Lo Rm(P)da ~ CO/Q which is as small as we like for Q large. Also K ~ H 2/4 and H [

lEO

K da

= 8F/8V so

< !F (8F) 2

-

4

8V

Now we claim that by making B large we can make F(8F/8V)2 as small as we like. Recall 1 F

=e

£pt S

{Q B B} T _ t + V2/3 + [Vet) - V]2/3 .

Differentiate implicitly to get 8F _ ~ fpt 2 8V - 3 Be F

Since we have assumed V

~

{_1_ _[Vet) _1 } V5/3

V(t)/2,

o <- av 8F < ~BefPt F 2 /v 5 / 3 - 3 and This makes

V]5/3

.

103

THE FORMATION OF SINGULARITIES

which is indeed as small as we like when B is large. Thus X < 1, and since it is an integer we must have X ~ O. This makes

8A> 8 2 A 8t - 8r 2 '

Now we can combine our inequalities to include that

at V = V* and t = t*. However, we claim if Q and B are large enough the opposite inequality holds everywhere. This contradiction implies G > F for all t < T, which will prove the Theorem. 0

LEMMA

23.2. If Q and B are large enough then the function F defined by

.!.. _

F - e

fpt

{-.!L + ~ + T- t

V2/3

B } [V(t) - V]2/3

satisfies 8F

28 2F

8F

~ F 8V2 + F

(8F)2 8V

Proof. We look for a function F in the form F

= I/H.

at + pV 8V for 0 ~ t


and 0


~

V(t)/2.

Then we need

IT H takes the form then K must satisfy e

fPtK3 [8K 8t

V 8K )] 3 (8K)2 > K8 2K + p (~K 3 + 8V + 8V - 8V2 .

Our K has the form K

= T Q_ t + B

[1

V2/3

+ (V _

1] V)2/3

.

We compute

8K 8t

(2

8K)

+ '32 B

2 pQ

Q

+ p '3 K + V 8V = (T 1

. (V _ V)5/3

t)2

(

+ '3 T

pV -

- t

8V) at .

RICHARD S. HAMILTON

104

Since R

~

-

p makes

8V

<

V

8t - p

we have 8K

at + P

(2a

K

+V

8K) 8V

Q ~ (T - t)2'

Since e~pt ~ 1 it is sufficient to verify Q K3 (T _ t)2

+3

(8K)2 8V

~

and now we can forget about p. We consider two cases. The first is where V constant g we shall choose shortly. We have 8K 8V

~

2[1

= -a B

and

8 2K K 8V2

gV for some small absolute

1]

V5/3 - (V - V)5/3

10 [V8/31 + (V - 1] V)8/3 .

82 K

= 9'B

8V2

Then 3 (8K)2 > 12 B2 [1- 5/3]2. _1_ 8V - 9 g VlO/3

and

82 K

10 [ 8/3] 8V2 ~ 9'B 1 + g

•

1 V8/3'

Also 1 Q B . V2/3 ~ K ~ T _ t

+ B [1 + g 2/3]

Therefore our inequality will hold if we choose 12 9'

g

•

1 V2/3'

> 0 so small that

[1 - g5/3 ] 2 ~ 9' 10 [1 + g8/3 ]

+ '19

and if in addition we have 3 1 B Q. (T _ t)2V6/3

Since

g

1 2

+ '9 B

1 10 Q [ 8/3] . VlO/3 ~ 9'B 1+g

•

1 1 T - t . V8/3'

is small we can take

and then this inequality holds if B3 ~ 3Q. Thus we have the estimate for V:::; g(V - V) by making B large compared to Q.

THE FORMATION OF SINGULARITIES

Consider the other case where V compare everything to V and

~

105

c(V - V). This is easier because we can

V 2 / 3 ~ a(T - t)

for some a > 0 by our hypothesis. With various constants C < 00 and c > 0 independent of B, Q, and a (but depending on c > 0 which is fixed) we have

82 K

1

8V2 ~ CB . V8/3

and

1 K ~ cB· V2/3.

Our estimate holds if Q 8 2 K
which holds if V ~ a(T - t)3/2 and BQ ~ C/a 4 / 3

for some constant C as above. This is easily arranged also, and the Theorem is established. 0 COROLLARY 23.3. If IRml ~ O(T - t) and V ~ a(T - t)2/3 as before then the injectivity radius r satisfies an estimate

for some

(J

>0

depending on {3, p, T, 0, a as before.

Proof. IT the injectivity radius is very small compared to the maximum curvature then the isoperimetric ratio A/V2 !3 will also be very small for a torus of area A enclosing a volume V very small compared to the maximum curvature. 0

24 Curvature Pinching in Three Dimensions. In three dimensions we can extract more information from the explicit form of the curvature reaction. Recall from 5(c) that when the curvature operator matrix M is diagonal

where M(X, Y)

= Rg(X, Y) -

2Rc(X, Y)

106

RICHARD S. HAMILTON

and the trace of M is the scalar curvature R the reaction ODE system becomes

Any closed convex set of curvature operator matrices M which is SO(3) invariant (and hence invariant under parallel translation) and preserved by the reaction ODE is also preserved by the Ricci Flow. Since the system of ODEs is homogeneous, it is natural to first study the radial motion, and then examine the solution curves projectivly. The radius p is given by and we compute

which shows the radius p increases for positive scalar curvature R = A+J.&+V > 0, and decreases for negative scalar curvature R = A+J.&+V < O. Next note that if a vector VERn evolves by a system of ODE 8

~

=F(V)

then this system and the associated system

~ = a(V)F(V) -

b(V)V

have the same oriented family of solution curves in the projective sphere sn-1 = Rn - {O}/R+, for any scalar valued functions a(V) and beY). We take V = (A,J.&, v) and a = (A2 + J.&2 + v 2 ) b = A3 + J.&3 + v 3 + 3AJ.&V • Then the associated system keeps A2 + J.& 2 v 2 constant, so we can restrict our attention to the unit sphere p = 1. It has the explicit form dA dt = A2( J.& - v) 2 - J.& 3( A - v ) - v 3(A - J.& ) •

Clearly it has fixed points A = J.& = v and A = J.& = 0, A = v = 0, J.& = v = 0. This gives eight fixed points on the sphere p = 1. It is easiest to display the flow on the front of the sphere R > and on the back R < 0. We denote the circles A = 0, J.& = 0, v = with solid lines, and the circles >. + J.& = 0, >. + v = 0, J.& + v = with dotted lines. In the hemisphere

°

°

°

THE FORMATION OF SINGULARITIES

107

R > 0 the region of positive sectional curvature lies inside the solid triangle, the region of positive Ricci curvature inside the dotted one; similarly for negative sectional and Ricci curvature on the other. RO

The center point A = J1. = II > 0 represents the sphere 8 3 , and the center point A = J1. = II < 0 represents the hyperbolic space H3. Note 8 3 is attractive while H3 is repulsive. The three vertices A > 0, J1. = II = 0 and J1. > 0, A = II = 0 and II > 0, A = J1. = 0 represent the cylinder 8 2 x Rl, while the three vertices A < 0, J1. = II = 0 and J1. < 0, A = II = 0 and II < 0, A = J1. = 0 represent H2 x Rl. These are degenerate fixed points which all attract in one direction from one side, and repel in the opposite direction on the other side. Of course the picture on the back R < 0 is the reverse of the picture on the front. We can examine the degenerate fixed point at the cylinder 8 2 x Rl where A = J1. = 0 more precisely by taking instead the associated system with a=

and b =

II

112

+ AJ1.

which preserves the planes where II is constant. Restricting to associated system dA = J1. - A + A2 - A2 J1. { dt dJ1. = A - J1. + J1.2 - AJ1.2

dt

with a degenerate fixed point at >.

= J1. = O.

II

= 1 gives the

IT we substitute J1.=X-Y

A=X+Y

we get the system dx { dt

= X2 + y2 _

~~ = -

x(x2 _ y2)

[2(1 - x)

+ x2 _

When we are close to the origin x increases and y2 +3x

=0

y2] y.

IYI

decreases. On the parabola

RICHARD S. HAMILTON

108

we have

d

dt (y2

+ 3x) = 3x (1 + x 2) ~ 0

so if we start inside this parabola we must stay inside, and if we start close to the origin we must appraoch the origin. But on the parabola y2 +4x -e

we have

=0

d dt (y2+4x-c) =4x 2(I+x)+2c(c-4x-x 2) ~o

when -1 ~ x ~ 0, so if we start outside this parabola but close to the origin with x < 0 we must stay outside until x > 0, after which x becomes large before y reaches o. The envelope of all the solution curves attracted to the origin will again be a solution curve between the parabolas y2 + 3x = 0 and y2 + 4x = 0, so this separatrix has a vertical tangent near the orgin. On the other hand, near the origin dx 2 - ~ x + y2 dt

to a good approximation. If x dx dt which gives solution curves

~

~

x

and

0 and 0 2

and

~

dy - ~ -2y dt y « Ixl then

dy dt

~

-2y

which keep y « Ixl and approach the x-axis very fast. We expect the solution curves inside the separatrix to look like these. In fact we expect the solution curves of the original system and the simple approximation are conjugate by a diffeomorphism. This gives the following picture for the solution curves near S2 x R where A = 0, I' = 0, II = 1, projected radially onto the plane II = 1. (Recall x and yare rotated 90 0 from A and 1'.)

:;,--------:, , , ,," , , " , " , ,,"

Note that a sizable region in A, 1', II space is attracted into the fixed radial line

THE FORMATION OF SINGULARITIES

109

A = I-' = 0 while the rest flows past it, on towards 8 3 along the fixed radial line A=I-'=v. Tom Ivey has used a computer to produce a picture of the solution curves for the associated system obtained by projecting radially on the plane A+I-' + v = 1. The picture looks like this

RICHARD S. HAMILTON

110

Hopefully some geometric insight into the following pinching results.

THEOREM

24.1.

For any

g

in 0

~ g ~

1/3, the pinching condition R

~

0

and Rc(x,y)

~

gRg(x,y)

is preserved by the Ricci Flow in dimension three. Proof. H the curvature operator M has eigenvalues A ~ IJ conditions become IJ + II ~ 0 and

~ II,

the pinching

=

with 6 2g / (1 - 2g). Since A is a convex function of M while J.I. + II is a concave function, the inequalities define a convex set of matrices, so we only have to check that this set is preserved by the ODE system. So we must check

or

JL2

+ All + 112 + AJL ~ 6 (A2 + JLII)

on the boundary where

JL + II

= 6A ~ o.

This is equivalent (solving for 6) to

A (JL2

+ All + 112 + AJL)

~ (JL + II) (A2

+ JLII)

which reduces to

A2(JL + II) ~ JLII(JL + II) which clearly holds if JL

+ II ~ 0 and

A ~ JL

~

o

II.

THEOREM 24.2. For any {J > 0, B < 00, and'Y > 0 we can find a constant C < 00 depending on {J, Band 'Y with the following property. If a solution to the Ricci Flow in dimension three has

{Jg(x,y) ~ Rc(x,y) ~ Bg(x,y) at the beginning t

= 0,

then for all subsequent times t

~

0 we have

IRC - ~Rgi ~ 'YR + C as a bound on the trace-free part of the Ricci tensor. Proof. Depending on {J and B we can choose d > 0 so that

_£ • . ,

rUKMATION OF SINGULARITIES

111

at t = 0, and hence for t ;:: 0 by the proof of the previous theorem. Choose the constant A so that the inequality

A - v :5 A(JL + v)I-6 holds at t

= 0, which is possible since

at t = O. We claim this inequality is also preserved by the rucci Flow. Clearly it defines a convex set of matrices M with eigenvalues A ;:: I' ;:: v and I' + v ;:: o. So we only must check that the inequality is preserved by the ODE system. Now so

d

-In(A - v) dt while

d

dt (I'

+ v)

= 1'2

so

= A - I' + v

+ AV + v 2 + AI' ;:: A(JL + v)

d

dt In(JL + v) ;:: A. Then

d dt In [(A - v)/(JL + v)I-6] :5 <SA - (I' + v) :5 0

so the ratio (A-II)/(JL+II)1-6 decreases. Hit is less than A to start, it remains so. We can estimate

for some constant C, and I'

+ v:5 CR

for some the constant C. For any ( > 0 we can find yet another constant C (() with R I - 6 :5 (R + C(() for all R ;:: O. Then we get IRe -

~Rgl :5 C(R + C(()

and we only need to take ( :5 'Y / C to finish the proof.

o

COROLLARY 24.3. For any (3 > 0, B < 00 and (J > 0 we can find a constant C < 00 with the following property. If a solution to the Ricci Flow in dimension three has (3g(x, y) :5 Rc(x, y) :5 Bg(x, y)

112

RICHARD S. HAMILTON

at the beginning t

= 0,

then for any subsequent T

~

0

max maxIDRm(P, t)1 ~ (}max maxIRm(P, t)1 3 / 2 t~T

t~T

P

P

+ C.

Proof. We can recover this result by a limiting procedure; an explicit estimate using the maximum principle is given in [20]. Suppose the estimate fails for all C. Pick a sequence Cj -+ 00, and pick points Pj and times Tj such that IDRm(Pj , Tj)1 ~ (}max max IRm(P, t)13/2 t~T;

P

+ Cj .

Choose the Pj to be the origin, and pull the metric back to a small ball of radius r j proportional to the reciprocal of the square root of the maximum curvature

up to time metrics so

T.

Clearly these go to infinity by our derivative bounds. Dilate the max max IRm(P, t) I t~T;

P

becomes 1 and translate so time Tj becomes time O. Then Cj dilates to zero, but in the limit metric IDRm(O, 0)1 ~

e.

However the limit metric has 1

RC-'3 Rg =O by the previous theorem. But then it has constant curvature, which is a contradiction. This proves the corollary. 0 We can now see that the solution to the Ricci Flow on a compact threemanifold with positive Ricci curvature becomes round. Since ~IN > 0, ~AX goes to infinity in a finite time. Pick a sequence of points Pj and times Tj where the curvature at Pj is as large as it has been anywhere for 0 ~ t ~ Tj. Since IDRml is very small conpared to R(Pj , tj) and IRc - ~Rgi is also, the curvature is nearly constant and positive in a large ball around Pj • But then Myer's Theorem tells us this is the whole manifold. Our next result is even more interesting, because it applies to any threemanifold regardless of the sign of the curvature tensor. It was also observed independently by Ivey [30]. Consider the function y=J(x) =xlogx-x

for 1 ~ x < 00, where it is increasing and convex with range -1 ~ y < 00. We let f- 1 (y) = x be the inverse function, which is also increasing but concave and satisfies lim J-1(y)/y = O. y-+oo

THEOREM

24.4. Suppose we have a solution to the Ricci flow

a at gij = - 2Rij

THE FORMATION OF SINGULARITIES

113

on a compact three-manifold which satisfies the inequalities R

at t

= O.

Then it will continue to satisfy them for t

~

~

-1 and

O.

Note that since f-l(y) ~ +1 always, any matrix with eigenvalues at least -1 and trace at least -1 satisfies the inequalities. For any metric we can achieve this by dilation. Then the inequalities will continue to hold under the Ricci flow. Then if the curvatures go to infinity, the most negative will be small compared to the most positive.

LEMMA.

The set P of matrices

P: {

Mab

defined by the inequalities

..\ + J.' + v ~ -1 v+I- 1 (..\+J,t+v) ~ 0

is closed, convex and preserved by the ODE.

Proof. P is closed because 1-1 is continuous. The function ..\+ J.'+v is just the trace, which is a linear function. Therefore the first inequality defines a linear half-space, which is convex. The function v is concave, and 1-1 is concave and increasing, so the second inequality defines a convex set as well. 0 Under the ODE d

.~+J.'+0=~+~+~+~+~+~ and this quadratic can be written as

so it is clearly non-negative. Thus the first inequality is preserved. The second inequality can be written as

which becomes

..\ + J.' ~ (-v) log( -v). It is easier to keep track of the signs if we let n

= -v, and write it as

..\ + J.' ~ nlogn.

,-1 (. \

_,-1 (. \

To show the inequality is preserved we only need to look at points on the boundary of the set. If v + + J.' + v) = 0 then v = + J.' + v) ::; -1 since ,-I(y) ~ 1 for all y. This makes n ~ 1, so nlogn ~ 0 and ..\ + J.' ~ O. Since ..\ ~ IJ we must at least have ..\ ~ O. But IJ may have either sign.

RICHARD S. HAMILTON

114

We deal first with the case where 11

~

O. Then we need to verify

dA d~ dn dt + dt ~ (logn + 1) dt when A +

~

= n log n. Solving for A+~

logn= - n and substituting above, we must show

+~ A2 - ~n + ~2 - An ~ (A -n-

+ 1)

(-n 2-

A~ )

which reduces to

and since A, ~ and n are all positive or zero we are done here. In the other case where 11 ~ 0 we again change the sign by letting 11 = -m. Then the inequality becomes A ~ m+nlogn. To show the inequality is preserved we must verify that

dA dm dn + (logn+ 1)dt-dt dt

- >-

when A = m + nlogn. Solving for

A-m logn= - n and substituting above, we must show

when A showing

~

0 and 0

~

m

~

n (and n

A2 n + Am2 which is equivalent to

which must hold because

Hence the proof is complete.

~

1). This simplifies algebraically to

+ m 2 n + n 3 ~ A2m + Amn

THE FORMATION OF SINGULARITIES

115

COROLLARY 24.5. For any constants B < 00 and 6 > 0 there exists a constant C < 00 with the following property. If any solution to the Ricci Flow on a complete three-manifold with bounded cUnJature satisfies IRml ~ B at t = 0, then for t ~ 0 it satisfies the estimate

M(X, Y)

~

-(6R + C)g(X, Y)

on the cUnJature operator M. Hence when the cUnJature R is big, any negative cUnJature is very small in comparison. The following refinement of these techniques gives a curvature pinching result useful for classifying Type I singularities on a three-manifold. THEOREM 24.6. Suppose we have a solution to the Ricci Flow on a compact three-manifold on a maximal time intenJal 0 ~ t < T which is Type I, so

lim sup(T - t) IMI

< 00

t-+T

and suppose the manifold never acquires positive sectional cUnJature everywhere. Then there exists a () > 0 such that for every T < T and every 6 > 0 we can find a time t in T ~ t < T and a point P where (T - t)IMI 2: () and a frame at P in which 1M - REI ~ 61MI where the scalar cUnJature R = tr M is the trace of the cUnJature operator M and E is the curvature operator matrix of a round cylinder 8 2 X RI given by

E= ( \ ) .

COROLLARY 24.7. The limit of dilations of the solution around these points and times gives an ancient solution with bounded non-negative sectional cUnJature whose holonomy reduces. Consequently it splits as a product of a surface with RI.

Proof. Since the minimum of R increases, we can choose a constant p 2: 0 so that

R+p>O for all t ~ O. The pinching estimates imply that for large IMI any negative eigenvalues M may have are not nearly as great in absolute value as some positive one; and hence there is some constant A < 00 so that

IMI :5 A(R + p) for all t

~

O.

o

RICHARD S. HAMILTON

116

We shall prove the converse of the Theorem. Suppose that for every (J > 0 there exist T < T and 6 > 0 such that at every point and in every frame at any time t with T ~ t < T we always have (T -

t)IMI

~ (J

or else 1M - REI ~

61MI.

We shall then show the manifold shrinks to a point and becomes round. We shall let C < 00 and c > 0 denote various constants which may depend on A and p (as well as the dimension n = 3) but which for now are independent of the parameters (J, T, 6, TI, e which we will choose as follows. We pick (J small enough to start, choose T and 6 depending on (J from the new hypothesis, pick TI depending on 6, and finally choose e depending on ,. The exact choices of (J, T, 6, TI, e will be explained as the proof evolves. Using R + P > 0, consider the function

F = (T -

WI ~ MI2 f(R + p)2-e

where

1

o

-+M=M- -RI 3 is the trace-free part of M when I is the identity matrix in an orthonormal frame. The matrix M evolves by DtM=~M+M'

where if

in an appropriate frame then

M'

= (A2 + P.1I p.2 + All

) . 112

+ Ap.

The trace R evolves by

DtR= ~R+R' where R' is the trace of M', and the trace-free part ~ M evolves by Dt ~ M=~ ~M+ ~ M' where ~ M' is the trace-free part of M'. Using the identity

~

[I ~ MI2 f(R + p)2-e] [I

+(2 - c) [DR/(R + p)] D ~ MI2 /(R + p)2-e] = 2 ...; M· ~ ...; M/(R + p)2-E - (2 - e)1 ...; MI2~R/(R + p)3-E

+ {eID"'; MI2 + (2 - c)

ID"'; M-"'; ~~~12} (R + p)E-2

THE FORMATION OF SINGULARITIES

117

and discarding the last term in braces which is clearly positive, we can compute the evolution of F as

DtF = ~F + (2 - e)[DRj(R + p)] . DF + F' where F' is computed from the ODE's as

F'

= 2(T - W ~ M· ~

+ p)2-~ - e(T - t)~-l / ~ M/2 f(R + p)2-e - (2 - e)(T - tY/ M' f(R ~

M/2R'j(R+ p)3-e.

We can regroup this as

F'

= (T -

W[X - 2Y]j(R + p)3-e

where

x = 2p ~ M· ~ M' +e/ and

~ M/2R' -e(R+p)/ ~ M/ 2j(T-t)

= / ~ M/ 2R' -R ~ M· ~ M'. is the only term we would have if e = 0 and p = 0.) y

(Note Y we compute explicitly

Using the ODE's

and note Y = 0 on the symmetric spaces 8 3 , 8 2 X Rl, R 3, H2 X Rl, and H3 where>. = ,." = v or >. = ,." = 0 or >. = v = 0 or ,." = v = 0, while Y > 0 elsewhere. We can estimate X from above as follows. The matrix ~ M has diagonal entries like 1 -[(>' - ,.,,) + (>. - v)] 3 so I ~

MI is comparable to (>. - ,.,,)

+ (>. -

v)

+ (,." -

v)

up to a constant factor above and below. The matrix ~ M' has diagonal entries like 1 3[(>' - ,.,,)(>' + ,." - v) + (>. - v)(>. + v - ,.,,)] so I ~ M'I ~ CIMII ~ first term in X

MI

for some constant C. This gives a bound on the

We also have a bound IM'I~CIMI2

and R' is the trace of M', so we get a bound on the second term in X

118

RICHARD S. HAMILTON

Finally

IMI $

A(R + p) so we get a bound on the third term f:(R + p)1 ~ MI2/(T - t) ~

cclMIl ~ MI2 /(T - t).

This gives a bound

on the quantity X. We can also estimate Y from below.

LEMMA

24.8. For every fJ

> 0 there exists an ( > 0 such that if the matrix

M satisfies 1M - REI ~

fJlMI

in every frame then

Proof. We saw Y > 0 if we avoid the lines where M Hence by homogeneity Y ~ (IMI 4 for some ( > 0 if 1M - REI

~ fJlMI

and

1M -

= RE or M = lRI.

~RII ~ fJIMI·

1M -

If lRII $ fJlMI for fJ small, we surely have all the eigenvalues of the same sign with comparable magnitudes, and

>..2(JJ _11)2 +JJ2(>.. - 11)2 + 112(>.. - JJ)2 ~ (>..2 + JJ2 + 112)[(>.. - JJ)2 + (>.. - 11)2 + (JJ - 11)2] for some ( > O. Hence in either case we are done. Given ( as above, choose f: > 0 so small that Of: S ( for the constant 0 in the bound on X. If 1M - REI ~ fJlMI then

S OplM11 ~ MI2 - (IM121 ~ M12. On the other hand, if (T - t)IMI S (J then neglecting Y ~ 0 we have X - 2Y

X - 2Y $ OplM11

~ MI2 - (~- 0) IMI21 ~ MI2

and if we pick (J > 0 at the beginning with (J S c/(C Since f: is small compared to (, we have

in either case. As a consequence

+ 1)

then c/(J - 0

~ 1-

THE FORMATION OF SINGULARITIES

119

Having come this far, since e is now chosen we loose nothing to let our constants C and c depend on e from now on . Then we can write this as

o We summarize our argument so far. LEMMA

24.9. There exist constants p ~ 0, A < 00, C + p > 0 and IMI $ A(R + p), and if

< 00, c > 0 and e > 0

such that R

F

= (T- WI ~ MI 2/(R+ p)2-t:

then DtF

= fl.F + V . D F + F'

where V

= (2 -

e)DR/(R + p)

and

COROLLARY

24.10.

We have F -+ 0 as t -+ T.

Proof. Choose any A > O. When

(T-t)IMI $ A since I ~

MI $

IMI and IMI $ A(R + p) we have

which is as small as we like if A is small enough. But when

(T-t)IMI

~

A

we have IMI quite large for t near T, so

and

F' $ On the other hand, now that

-~cIMI2F/(R+P)'

IMI

R+p $

is large

V3IMI +p $

21M I

120

RICHARD S. HAMILTON

so

Using

IMI ~ >'/(T -

t) we get

F'

~ -~C>'F/(T -

t)

for t near T. Thus when the maximum F MAX of F exceeds ,..!2- e )..E it must decrease at a rate-

where p

= i'c>. > O.

This implies

so if (T - t)-P F MAX = B at some time T close enough to T for the above estimates to hold then subsequently FMAX

~

B(T - t)P

and so when t is even closer to T the second holds. But >. > 0 is arbitrary, so F-t O. Now we can show that the manifold shrinks to a point and becomes round. By assumption

(T-t)IMI ~

n

for some constant n. On the other hand there exists a constant w that at each t we have

>

0 such

(T-t)IMI ~w somewhere, or else

IMI could not go to infinity as t -t T

because

would not allow such rapid growth. Hence the maximum of IMI is always proportional to l/(T - t). The quantity F is dilation-invariant, so when we form the Type I limit (which must exist by our injectivity radius estimate which we proved in Corollary 23.3) we have F = 0 on the l:'mit. Hence the limit metric has ~ M = 0, and hence has M = ~RI. But this implies the curvature is constant, (as we have had occasion to observe before from the contracted second Bianchi identity). Since the curvature is positive, the limit is a sphere 8 3 or a quotient space-form 8 3 /r. This proves the theorem. 0

THE FORMATION OF SINGULARITIES

121

25 Limits with Strictly Positive Curvature Operator. Given a sequence of complete solutions to the Ricci Flow with uniformly bounded curvature on some time interval, we can extract a convergent subsequence by the result in [26] provided we can control the injectivity radius at the origin points. In general this may be hard, but there is one important case where we get it for free. This is based on the observation that for a complete non-compact manifold with strictly positive sectional curvature we can bound the injectivity radius by the maximum of the curvature. The situation we consider here is not quite that simple, but with some work it is also possible to estimate the injectivity radius. We have a sequence of solutions to the Ricci flow where the sectional curvatures are bounded, where the lower bound is negative but increases to zero (as we have seen always happens after dilation if n = 3), and where the sectional curvatures at the origin points are uniformly bounded positive away from zero, and where the diameters go to infinity. In this case when we are far out in the sequence the curvature stays positive a long way out, and is never very negative. This is enough to produce a neighborhood of the origin which is convex and contains a ball of enough size to give a good lower bound on the injectivity radius. We now make this precise. THEOREM 25.1. Suppose we have a sequence of solutions to the Ricci Flow given by metrics G j on manifolds M j with origins OJ and frames:Fj for times 0: < t < w (with 0: < 0 < w) which are all complete, and such that for some

p>O (a) all the sectional curvatures of all the metrics G j are at most 1/p2 (b) there is a sequence dj -+ 0 such that all the eigenvalues of the curvature opemtor Rmj of the metric Gj are at least -dj / p2 (c) there is an e > 0 such that all the eigenvalues of the curvature opemtor Rmj of G j at the origin OJ are at least e/ p2 (d) the diameters dj of the metrics G j go to 00. Then there is a subsequence of the metrics such that all the injectivity mdii at the origins are at least this p > O. Hence a subsequence converges to a solution Goo of the Ricci Flow on 0: < t < w. Proof. The first step is to extract a subsequence which would want to converge if we could control the injectivity radii. To do this we introduce the notion of a geodesic tube in a manifold M with origin. Given a frame:F = (F}, F 2 , ••• , Fn) at the origin 0 and a length L, we begin by constructing the geodesic of length L out of 0 in the direction Fl and its opposite. Then we parallelly translate the frame :F along this geodesic, and take the geodesic out of each point in the direction F2 and its opposite of length p. Parallelly translate :F along these also, and take the geodesic out of each of these points in the direction F3 and its opposite of length p, and so on. Notice that only in the first direction do we go a long way L, while in the other directions we don't go farther than p. The curvature satisfies IKI :$ p, so this construction gives a local diffeomorphism of (-L,L) x (-p,p) x··· x (-p,p) ---+ M.

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RICHARD S. HAMILTON

Consider the pull-back metrics. For the Ricci Flow a bound on the curvature gives a bound on all the derivatives of the curvature. Then by ordinary differential equations we get bounds on the pull-back metric and all its derivatives with respect to the tube coordinates. (Here we omit the details.) H we consider a fixed reference frame :Fj at the origin in each Mj and take an element A of the orthogonal group, then A:Fj is a frame at the origin in M j , and we can take the pull-back metric for the geodesic tube on A:Fj. For a fixed A and a fixed L, we can always find a convergent subsequence of the pull-back metrics. By choosing a countable dense set of A's and a sequence of L's going to infinity, and by a diagonalization argument, we can find a subsequence of metrics so that the pull-back metrics to the tube on the frames A:Fj of length L converge for every A and every L. In this case we say the metrics preconverge along geodesic tubes. (Note any convergent sequence would be preconvergent.) The advantage of preconvergence is that we do not need to control the injectivity radius to get it. Form now on we only deal with such a preconvergent sequence. We can strengthen the notion of preconvergence to compare one tube with another. For any two vectors X and Y in Rn (which we identify with the tangent spaces at the origins OJ in the Mj with the frames :Fj) we can consider the sequence of distances dj

= dj(exPjX,exPj Y)

~ IXI

+ IYI

in Mj i by picking a subsequence we can assume the dj converge. H we do this by diagonalization for a countable dense set of pairs (Xo<, Yo<) then in fact dj will converge for every pair (X, Y). To see this take any c > O. Choose a sequence of pairs with Xo< -+ X and Yo< -+ Y. Since we have preconvergence in geodesic tubes in the directions X and Y, the metrics G j converge to a limit G! in the tube on X, and to a limit G~ in the tube on Y. We can find a constant c > 0 depending only on the dimension so that for any ( > 0 small enough, if IX - XI ~ c(p then in the metric G!

and likewise if Iii' -

Given (

YI

~ c(p then in the metric G~

> 0, choose 0 so large that IXo< -

XI

~ c(p

and

IYo< -

YI

~ c(p.

Then and

d~ (exp~ yo<,exp~ Y) ~ (.p.

Now choose j large enough depending on X, Y, 0, 1], and p so that Idj(exPjXa,exPjX) -d~ (exp!Xo<,exp!X)1 ~ (.p

THE FORMATION OF SINGULARITIES

and

123

Idj(exPj Ya,exPj Y) - d! (exp~ Ya,exp~ Y)I ::5 (p.

Finally make j large enough also depending on X, Y, cr, ( and p so that dj(exPj Xa,exPj Ya ) ::5 (p since Xa and Ya are in the countable set for which the sequence is preconvergent in distances. Then dj(exp, X, exPj Y) ::5 S(p. Since , is arbitrary, the sequence is preconvergent in distances for all X and Y as claimed. In fact we can do a little better along the lines of [H]. Using the geodesic tube coordinates at t = 0, we can also consider the pull-back of the metric at earlier or later times, which we can bound using curvature bounds, since we know the metric evolves by the curvature under Ricci Flow. Then we can actually make the pull~backs of the Ricci Flow converge to a solution of the Ricci Flow in every geodesic tube. We can also keep the solutions preconvergent in the distances dj(exPm X, eXPj Y)(t) for all X and Y at every time t. 0 LEMMA 2S.2. For every length L we can find eeL) > 0 and J(L) > 00 such that all eigenvalues of the curvature operator on M j at points within distance L of the origin have It ~ eeL) when j ~ J(L).

Proof. Suppose not. Then we can find a sequence of points Xj = expo(ij V;) at distances ij ::5 L from the origin in some directions V; with IV; I = 1 such that some eigenvalues of the curvature operators at the Xj are not bounded away from 0 on the positive side. Since on Mj we have there eigenvalues ~ -6j / p2 with 6 -t 0, they in fact go to zero. 0

Find a convergent subsequence V; -t V and ij -t i and pick a geodesic tube in each M j starting in the direction V. By preconvergence we get a limit which solves the Ricci Flow in the tube and the limit will have some eigenvalue of the curvature operator equal to zero at the point iV with i ::5 L. But in the limit all the eigenvalues of the curvature operator are ~ 0, so by the strong maximum principle (see[29]j the argument works locally also) there must be a zero eigenvalue of the curvature operator everywhere in the tube at every time, in particular at the origin at t = O. But for the sequence we had the eigenvalues of the curvature operator at OJ ~ e, so this holds in the limit also. Since this is a contradiction, the Lemma is established. In a manifold M with origin 0, we define the function i(V) on unit tangent vectors V at with values in [0,00] to be the distance to the cut locus in the direction V. IT exp is the exponential map at the origin, then

°

i(V)

= max{ljd(explV,O) = l}.

It is well-known (see Cheeger and Ebin [9]) that the distance to the cut locus is a continuous function. Moreover if l = leV) then either the geodesic exp(sv)

124

RICHARD S. HAMILTON

for 0 ::; s ::; l has a non-zero Jacobi field vanishing at the ends, or there exists another W '" V write exp(lW) == exp(lV). The choice of frames Fi at the origins OJ in M j allows us to identify the tangent spaces at the origins with Rn. We define the set V of distinguished directions as those in which we can go off to infinity as j -t 00. To see this is well-defined, let lj (V) for a unit vector V in R n be the distance to the cut locus in M j in the direction V relative to the frame Fj.

LEMMA.

For any sequence Vj

-t

V, the limit loo(V) == .lim lj(Vj) exists J~OO

and depends only on V and is a continuous function of V, when the sequence of manifolds is preconvergent. We can always define Proof. First we show the limit exists. loo(V) ==. -t liminf lj(Vj). Choose a subsequence of j's for which the lim J~OO

inf is attained as a limit. IT lim inf == 00 we are done. Otherwise for each j, either there is a non-zero Jacobi field Jj or an alternate geodesic in the direction Wi' By passing to a subsequence, there is always either one or the other. [] IT there is always a Jacobi field Jj, we can take its derivative dJj/ds at the origin to be a unit tangent vector Xj. By choosing a subsequence we can make Xj converge to some unit tangent vector X. The metrics preconverge in the geodesic tube around V, so the limit metric has a non-zero Jacobi field J vanishing at 0 with dJ/ds = X, and J vanishes again at expoo(sV) with s == loo (V). This means that the index form

I(J, J) ==

f

[lDJI2 - R(T, J, T, J)] ds

on the geodesic expoo(sV) on 0 ::; s ::; loo(V) has a null space, and hence has a strictly negative direction on 0 ::; s ::; loo (V) + e for any e > O. Then it also has a negative direction on 0 ::; s ::; loo (V) + e in any metric G j when j is large enough, and thus lj(V) ::; loo(V) + e. Therefore lj(V) -t loo(V) for all j -t 00, not just for the subsequence. Otherwise we find a subsequence where eXPj(ljWj) == eXPj(ljVj) for some sequence Wj '" Vj with lj == lj(V). By taking a subsequence we can assume Wj -t W. IT W = V, then the limit metric in a geodesic tube in the direction V again has a non-zero Jacobi field on expoo(sV) vanishing at s = loo(V), and we are done. This Jacobi field J can be bound by taking J = 0 a~d dJ/ds = X at the origin 0 where for some subsequence · X = 11m

j~oo

W·-V3 3 IWj - V,I

THE FORMATION OF SINGULARITIES

125

Since eXPj(sV) and eXPj(sWj ) are geodesics in the metrics Gj and Gj -+ G in the tube on V = W, we can check that J(s)

= lim j-+oo

exp/sWj) - eXPj(sV;) IWj - V;I

converges for the subsequence chosen above to the desired Jacobi field, with J = 0 again at s = ioo(V). If W # V, we take two geodesic tubes in the directions V and W. Then for our subsequence dj(exPj(ijV;),exPj(ijWj )) -t 0

and since i j on V

-t

i = ioo(V) and V; -t V and Wj -t W we also have in the tube

and in the tube on W

which makes dj(exPj(ijV;),exPj(iW))

-+ 0

for our subsequence. But this sequence is defined for all j, and the limit exists because we have made our metrics preconvergent in distance. Hence this sequence not just the subsequence, goes to zero for all j. Now consider the picture in the geodesic tube in the direction V for each Mj with j large. There is the geodesic out of V from the center, and close to it is the geodesic out of V;. At distance i out the tube there is another geodesic passing through the tube which came out of W, and at a distance i out of W it is close to the point at distance i out of V. The metrics converge in the tube, and the geodesics out of W will converge in the tube to a limit geodesic which we call 'Y. Now 'Y passes through the point P at distance i out along the geodesic 7 down the center which came out of V. But we claim 'Y cannot coincide with 7. For if it did, the corresponding 'Yj out of Wj and 7j out of V; in Mj for the subsequence of j would be close, and hence both in the tube in direction V, and their starting vectors V; and Wj would be as close as we like. But V; -+ V and Wj -+ W with W # V. Hence 'Y and 7 are distinct. Now the argument is a little subtle, because 'Y is only defined in the tube around 7. If we had a limit metric, then 'Y would be a geodesic out of W, and the distance to the crossing point P would be the same along 'Y and 7. In this case it would be a shorter path, once we are beyond P along 7, to go in a perpendicular from 7 over to 'Y and then follow 'Y back to the origin. For short distances beyond P, the savings in distance is on the order of a fraction given by the sine of the angle between 'Y and 7. (This would be exact for the flat metric.) Since we have a uniform curvature bound, for short distances beyond P we still save almost this much. Now if we take j large enough, since the metrics converge in the tube our savings in cutting over from the geodesic 7j out of Vi to the geodesic 7j out of W will still be almost this much. Thus

RICHARD S. HAMILTON

126

for every e > 0 we can find J(g) so that if j ~ J(g) then lj(V;) < loo(V) + g, since the geodesic out of V; does not minimize length at distance much past l = loo(V). This proves the assertion that

loo (V)

= 3-+00 .lim lj (V;)

always exists. It follows easily that loo (V) is independent of the choice of the sequence V; -t V. For if we have two different sequences, we can collate them to get a new sequence by odd and even j and the odd and even subsequences cannot have different limits. It also follows that loo (V) is continuous in V. For let Vie be any sequence which converges to V. For each k choose jle so large that in Mjle we have

Ilj. (Vie) -loo(VIe)1 ~ l/k if loo(Vle) < 00, otherwise we make lj. (Vie) ~ kif loo(VIe) subsequence jle we have

= 00.

Then for the

by the previous argument. Hence

also, and we are done proving the Lemma. Now we let V be the set of directions in which we can go off to infinity without hitting the cut locus in M j as j -t OOi specifically V

= {V E sn-l : loo(V) = 00 } •

Since the diameters of the M j go to infinity, the set V is not empty. To see this, pick a sequence V; with lj (V;) -t 00 and find a subsequence with V;1e -t V; then loo(V) = 00 so V E V. Moreover .lim inf lj (V) = 00. 3-+00 VEV

For if not, pick V; E V with lj (V;) ~ l < 00 for some subsequence j and some l < 00. For another subsequence V; -t V. But loo(V) = .lim loo(V;) since loo 3-+00

is continuous, and loo(V;) = 00 since V; E V, so loo(V) = 00 also and V E V. But lj(V;) -t loo(V) also, which is a contradiction. Now recall that all sectional curvatures on M j have K.j ~ 1/p2 for some p > 0 independent of j. We define the set Nj in Mj in the following way: Nj

=

{exPj(sW):

IWI = 1

and for s:5 lj(W); and for all V E V, s' :5 S and r:5 lj(V) we have dj exPj(s'W),exPj(rV) ~ r - pl.

THE FORMATION OF SINGULARITIES

127

First note that Nj is closed and not empty; for Nj is defined as an intersection of closed sets, and contains the ball of radius p around the origin OJ in Mj. LEMMA 25.3. There exists an L < 00 such that for all large enough j the set Nj lies in the ball of radius L around the origin OJ in M j •

Proof. If not, we could pass to a subsequence of j's and find a sequence Sj -+ 00 and Wj in the unit sphere with eXPj(sjWj) E N j .

For another subsequence we have Wjh -+ v for some V. Now Sj $lj(Wj ) and Sj -+ 00 so ioo(V) = 00 and V E V. However if we take any s' > p fixed then dj(exPj($'Wj),exPj(s'V)) -+ 0

and we get a contradiction, since we must have

once

Sj ~ S'.

o

Among all geodesic loops starting and ending at the same point and lying entirely in the compact set Nj there will be a shortest one. Call it 'Yj, and suppose j starts and ends at a point we call Pj • If 'Yj has length at least p for all j, we are done. When 'Yj is shorter than p we consider two cases (and rule them both out). The first case is when 'Yj makes an angle 7r with itself at Pj , hence forming a geodesic circle. For any r no matter how large and any V E D we can take j large enough to make ij (V) ~ r. Consider the point Xj = expj(rV), and find the point Y; on 'Yj closest to Xj. Let Y; = expj(sW) with IWI = 1 be an exponential representation of Y; in N j • Then taking s' = s we get d(Xj, Y;) ~ r - p. Now we can find e > 0 depending on L + P so that all sectional curvatures Itj on M j in the ball of radius L + P around the origin OJ have Itj ~ e/ p2 independent of j, by our previous Lemma. Take a shortest geodesic (j from Xj to Y;. Then along (j for a distance p from Y; we have allltj ~ e/r > O. Moreover by taking j large we can make all sectional curvatures Itj ~ -6j / p2 for 6j as small as we like, and we can make r as large as we like. In this case the standard computation shows the second variation of the arc length of the geodesic f]j fixing one endpoint at Xj and the other on 'Yj is strictly negative. Indeed let Zj be the unit tangent vector to 'Yi at Y; and extend Zj to (j by parallel translation. Choose a function cp to be identically 1 within distance p of Y; along f]j and then to drop linearly to zero. The second variation of arc length in the direction cpZj is

128

RICHARD S. HAMILTON

where K.j

= Rm(Tj,Zj,Tj,Zi)

is the sectional curvature of the plane spanned by the unit tangent vector T j to (i and by Zi' Considering the separate contributions from the part OfTJi within p of}j and the past beyond I(cpZi,cpZj) $ -c/p+ 1/r+8i r/p2.

First take r so large that

l/r $ c/3p and then take j so large that

and we still have so the second variation is strictly negative. But now we see }j is not the closest point on Ii to Xj, which is a contradiction. Thus Ij cannot be a geodesic circle. However if Ii makes an angle different from 7r at Pj , we are no better off. For now we can shorten the geodesic loop Ii' Since its length is no more than p, and all sectional curvatures satisfy K.i $ 1/p2, there will be a geodesic loop 7i close to Ii starting and ending at any point Pi close to Pi' If we take Pi to be along Ij itself then the loop 7j is shorter than Ij, since for angle less than 7r the first variation in arc length of this motion is strictly negative. Moreover Pj is still in N j . If the whole loop 7j is in N j then Ij wasn't the shortest. On the other hand if 7j doesn't stay in Nj there must be a point Qj on 7j lying outside of N j • Now if Pj is close to Pj, then Qj cannot lie far from Nj, so in particular its distance from the origin can be kept less than L+p. Let Qj = expj(sjWj) with IWjl = 1 and Sj $lj(Wj) be some exponential representation of Qj and let

OJ

= {exPj(sWj) : 0 $

s $ Sj}

be the corresponding geodesic from OJ to Qj. Since Qj is not in N j , we can find some Vj E V and some Tj $lj(Vj) and some sj $ Sj such that dj(exPj(sjWj),eXpj(TjVj))

< Tj

- p.

In fact we may as well take Tj = lj(Vj) since the inequality gets stronger as Tj increases. Choose g > 0 depending on L+2p so that all the sectional curvatures K.j on M j in the ball of radius L+ 2p around the origin OJ in mj are at least t/ p2 independent of j. The previous argument shows that the closest point on OJ to Xj = exp(TjV}) cannot be an interior point for large j. We only need observe that Tj is as large as we want when j is large by our previous observation.

THE FORMATION OF SINGULARITIES

129

Moreover the closest point is not the origin, since there the distance is rj while at exp;(sjWj) it is less than fj - p. Hence the closest point is at the end Qj, so

< fj

dj (Xj , Qj)

while surely dj(Xj,Pj

)

- p

~ fj

since Pj lies in N j • Thus the closest point Qj to Xj on 1; is not its end point Pj. But now the second variation of arc length from Xj to Qj will be negative, giving a contradiction as before. Hence the only possibility is that the shortest 100p'Y in N j has length at least p, and we have our injectivity radius estimate. 26 Singularities in Dimension Two and Three. The Ricci Flow on a compact surface cannot form any singularity except for the sphere or projective plane shrinking to a point and becoming round. One way to prove this now is to examine the possible singularities and see there are no others. We have an injectivity rdius estimate in terms of the maximum of the curvature valid for all 'time. So unless the solution exists for all time with curvature decaying like

/R/5C/t as t -t I first.

00,

we can form a singularity model of Type I or II. We examine Type

THEOREM 26.1. The only solutions to the Ricci Flow on a surface which are complete with bounded curvature on an ancient time interval -00 < t < T and where the curvature R has

limsup(T t-+-oo

t)IRI < 00

are the round sphere 8 2 and the fiat plane R2, and their quotients. Proof. Since IRI :$ C/(T - t) and the minimum of R increases, R ~ O. Moreover by the strong maximum principle R = 0 everywhere and it is fiat, or R > 0 everywhere. If the solution is compact with R > 0, either it is the sphere or it is the projective plane Rp2 = 8 2 / Z2 whose double cover is the sphere. Assume it is the sphere, and we shall see it is round. Then Rp2 must be round also since its cover is. We know from [22] that the sphere shrinks to a point at some future time which we can take to be T, when it becomes round. Its area A shrinks at a constant rate dA = -JR da = -871' dt so A = 87r(T - t). On an even dimensional oriented manifold the injectivity radius can be bounded by the maximum curvature. Since R'5,Cj(T-t)

130

RICHARD S. HAMILTON

by hypothesis, we must have injectivity radius p with

p? cJT - t for some c. Now the diameter L has

L 5, CA/p 5, CJT - t as a bound also. Hence the diameter, the injectivity radius and the maximum curvature all scale proportionally to the time to blow-up. The scaled entropy E =

f

R In[R(T - t)]da

is monotone decreasing in t. Since R(T - t) 5, C

and

f

R da

= 811'

we have an upper bound E 5, 811'lnC

so E- oo

=

lim E t

t~-oo

exists. Pick a sequence of points tj -t -00 and points Pj where the curvature is as big as anywhere at time tj. Then it was never larger anywhere at any earlier time, since an ancient solution with R > 0 has R pointwise increasing by the Harnack inequality. Make Pj the new origin and tj the new time 0 and T the same blow-up time by translation and dilation. We can then take a limit using the curvature and injectivity radius bounds. The backwards limit is still compact by the diameter bound. Moreover the scaled entropy is now constant at the value E- oo • But the only way this happens is on a shrinking soliton, and (except for orbifolds) the only one is the round sphere. Then E has its minimal value at t = -00, so it was constant all along. hence the sphere was round all along. There remains the case where the surface is complete but not compact. Since R > 0, the surface is diffeomorphic to the plane. We proceed to examine such a surface until we learn enough about it to get a contradiction. Recall first that the asymptotic scalar curvature ratio

A

= lim sup Rs2 B~OO

is constant on an ancient solution with weakly positive curvature operator by Theorem 18.3.

LEMMA

26.2. For our solution A

< 00.

Proof. Suppose A = 00. Then as in the dimension reduction argument of Lemma 22.2 and the following, we can choose a sequence of points Pi at t = 0

THE FORMATION OF SINGULARITIES

131

and radii rj which give Aj remote .8-bumps for a fixed .8 > 0 and Aj -+ 00. This works in dimension 2 only, because the only curvature is the scalar curvature, so when it is big every curvature is big. But now this contradicts Theorem 21.6. (Once the dimension is 2 we cannot reduce it further, since everything in 0 dimension 1 is intrinsically fiat.) Thus A < 00. From our previous results in the proof of Theorem 18.3 we know that an annulus NO' = {O' ~ S ~ 30'} has an area A (NO' ) ~

CO'2

for some constant c, and the scalar curvature at distance s falls off at most by

for some other constant c. Now we can explain how we get a contradiction. For a complete surface with R> 0 we have

by the Gauss-Bonnet Theorem since the surface is exhausted by discs bounded by convex circles with geodesic curvature k ~ 0 and on a disc

ff

= 411".

R da + 21a k ds

However on our surface we claim

II

Rda=oo.

This is because each annulus NO' makes a contribution

IL" R

da

~ :2 . ~ c 00

2

for some constants c > 0, using R ~ c/s 2 and A(NO') ~ 00 2 • But we can take an infinite sequence of disjoint annuli, and their contributions add up to 00. This finishes the proof of the Theorem. 0 Next we examine Type II limits. Since R assumes it maximum at an origin.

> 0 it must be a soliton which

THEOREM 26.3. The only complete Ricci soliton on a surface with bounded curvature which assumes its maximum 1 at an origin is the "cigar" soliton E2 with metric

RICHARD S. HAMILTON

132

Proof. The soliton moves along a vector field V = D f. Since the Ricci Flow preserves the conformal structure, which gives a complex structure J, the vector field V is holomorphic. Then it turns out that JV is a Killing vector field; this trick works on any Ricci-Kahler soliton. This gives a circle action on the soliton which makes it rotationally symmetric, and the Ricci soliton equation reduces to an ordinary differential equation which we can solve. We refer the reader to [22] for details. 0

In our paper [28] we prove the following isoperimetric estimate, which is similar to our study of minimal geodesic loops on a surface. Suppose a loop 'Y of length L divides the surface into two pieces of areas Al and A 2 • Define the isoperimetric ratio 1('Y) of the loop 'Y by

and let [=

inf [('Y) 'Y

be the infimum over all 'Y of any length or shape. THEOREM

26.4.

On the sphere 8 2 the isoperimetric ratio [ is increasing.

It follows that we cannot form the cigar as a limit on 8 2 , because the cigar opens like a cylinder. IT the surface develops a piece like a long thin cylinder it will have a short curve in the cylinder with a comparably large area on either side, and the ratio I will be close to 0 . IT we approach the cigar as a singularity forms, I must decrease to zero. But on the sphere [increases. The projective plane Rp2 can be treated by looking at its cover 8 2 • Other surfaces have Euler class X ::; 0 and can be treated directly (as in [22]) or as a special case of Kahler manifolds with [Rc] = p[w] with p $ 0 (as in [4]). The rescaled flow converges to a constant curvature metric. It is very interesting to see how much we can say about the formation of singularities in dimension three.

THEOREM 26.5. Suppose we have a solution to the Ricci Flow on a compact three- manifold, and suppose R becomes unbounded in some finite time T. Then there exists a sequence of dilations of the solution which converges to S3 or S2 x Rl or E2 x Rl {where E2 is the "cigar" soliton} or to a quotient of one of these solutions by a finite group of isometries acting freely {these quotients are the space forms S3 h, Rp2 X Rl, and Rp2 x Rl and S2 x S:, Rp2 x S: and E2 x S: for circles S: of any radius r}, except possibly for the case of a Type [ singularity where the injectivity radius times the square root of the maximum cUnJature goes to zero.

Proof. When we get an injectivity radius estimate valid for finite time we can always for a singularity model of Type I or II. First consider Type I. If

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133

the sectional curvature ever becomes positive everywhere, it becomes round and our limit is 8 3 or 8 3 h. Otherwise in Type I we get a limit which is an ancient solution with bounded non-negative sectional curvature which splits as a product of a surface with Rl. For the surface, if (T - t)R $ C it must be a round sphere or projective plane by Theorem 26.1. Otherwise we can take a backwards limit as t -t -00 to get a Type II limit, which must be the cigar ~2 • Since a limit of a limit is also a limit, we get ~2 x Rl or ~2 X 8 1 as a limit of the three-manifold solution. In order to take this backward limit we need an injectivity radius estimate on the surface in terms of the maximum curvature R at the current time. Since R > 0 this is easy. There are three cases. H the surface is compact and oriented, it is 8 2 and the result follows from a theorem for positive sectional curvature in even dimensions of Klingenberg ([9], 5.9). H it is compact but not oriented, it is Rp2 and the double cover can be handled as before. H it is not compact, it is diffeomorphic to R2 and we can use the estimate for complete noncompact manifolds of positive sectional curvature. H the limit is Type II, it must be a Ricci soliton of weakly positive sectional curvature from our pinching result in Theorem 24.4. H the sectional curvature is not strictly positive, it splits as a product of a surface soliton, which must be ~2, with a flat factor R2 or 8 1 (of any radius). Even if it does not split, we know the asymptotic curvature ratio is infinite

A

= limsupRs2 = 00 8--+00

by Theorem 20.2, and by Theorem 22.3 since the dimension n = 3 is odd, we can do dimension reduction to find a limit of a limit which splits as a product with Rl of an ancient solution with bounded positive curvature on a surface. Again a limit of a limit is a limit, and we can classify the surface as a round 8 2 (not RP2 because it is oriented) or ~2. This finishes the proof of the Theorem. D Of course 8 3 or 8 3 /'Y can actually occur as limits from the homothetically shrinking solutions, and we expect to get 8 2 x Rl from a neck pinch (or a degenerate neck pinch after dimension reduction). We even expect Rp2 x Rl as the limit from doing a neck pinch on 8 3 shaped like a dumb-bell and then quotienting by Z2. Some of the other quotients are harder to picture. For example if 8 2 x 8 1 has a product metric, the 8 2 factor shrinks but the 8 1 factor does not. Hence the limit of its dilations is 8 2 x Rl, not 8 2 x 8 1 • We conjecture 8 2 X 8 1 cannot form. More importantly, we conjecture ~2 x Rl and ~2 x 8 1 cannot form as limits of dilations of a compact solution. Here are the reasons for our belief. First, ~2 cannot form starting from a compact surface. Second, we could rule out ~2 x Rl on a three-manifold the same way we can rule out ~2 occuring as a factor in limits coming from compact manifolds with positive curvature operator, because ~2 violates the local injectivity radius estimate coming from the Little Loop Lemma. Moreover the Little Loop Lemma only depends on having some kind of backwards control on the scalar curvature R locally. This control came from the Harnack estimate, which uses positive curvature operator. But in three

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dimensions 'lur pinching estimates show that we only miss positive curvature by a little bit. This gives hope that we can get an approximate Harnack estimate giving some backwards control on R as desired. Backwards control means that R does not falloff too rapidly. This raises the following interesting problem. If we almost have a degenerate neck pinch, but at the last moment the little bubble on the end of the neck gets pulled through, leaving a little bump, how fast can the curvature of this little bump decay? REFERENCES

[1] [2] [3] [4] [5]

[6] [7]

l81 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

F. Almgren,_ _ A. Bach and W.-Y. Hsiang, Einstein metrics on Kervaire spheres, preprint. Cornell U., 1982. R. Bryant, Local existence of gradient Ricci solitons, preprint. Duke U., 1992. H.-D. Cao, Deformation of Kahler metrics to Kahler-Einstein metrics, Invent. Math. 81(1985), 359-372. _ _, Ricci-Kahler soliton, preprint, Texas A&M, 1992. _ _, The Harnack estimate for the Ricci-Kahler flow, preprint, Texas A&M, 1992. H.-D. Cao and B. Chow, Compact Kahler manifolds with nonnegative curvature operator, Invent. Math. 83(1986), 553-556. M. Carfora, J. Isenberg and M. Jackson, Convergence of the Ricci flow for a class of Riemannian memcs with indefinite Ricci curvature, J. Differential Geom. J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry. J. Cheeger, M. Gromov, Existence of F -structures. Cheeger and Gromov and Futake, H. Chen, Pointwise quarter-pinched 4 manifolds, Ann. Global Anal. Geom. 9(1991), 161-176. B. Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33(1991), 325-334. _ _, On the entropy estimate for the Ricci flow on compact 2-orbifolds, J. Differential Geom. 33(1991), 597-. B. Chow and L.-F. Wu, The Ricci flow on compact 2-orbifolds with curvature negative somewhere, Comm. Pure Appl. Math 44(1991), 275-286. D. De Turck, Short time existence for the Ricci flow, J. Differential Geom. J. Eells Jr. and J. Sampson, Harmonic Maps, Amer. J. Math. (1964). Eschenberg, Shrader and Strake, J. Differential Geom. (1989). R. Greene and Wu, Limits of Riemannian manifolds, Pacific J. Math. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ-

THE FORMATION OF SINGULARITIES

[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] "[36] [37]

[38] [39] [40]

[41] [42] [43]

135

ential Geom. 17(1982),255-306. _ _ , Four-manifolds with positive curvature operator, J. Differential Geom. 24(1986),153-179. _ _, The Ricci flow on surfaces, Contemp. Math. 71(1988),237-261. _ _, Lecture Notes on Heat Equations in Geometry, Honolulu, Hawaii, 1989. _ _ , The Harnack estimate for the Ricci flow, preprint 1991. _ _, Eternal solutions to the Ricci flow, preprint 1991. _ _, A compactness property for solutions of the Ricci flow, preprint 1991. _ _, A matrix Harnack estimate for the heat equation. _ _ , An isoperimetric estimate for the Ricci flow on surfaces, preprint VCSD,1992. G. Huisken, Ricci deformation of the metric on a riemannian manifold, J. Differential Geom. 21(1985). T. Ivey, Local existence of Ricci solitons in dimension three, preprint. _ _, Local existence of non-gradient Ricci solitons, preprint, Duke V., 1992. J. Jost and H. Karcher, Geom. Meth. zur gewinnung fUr harmonische Abbilding, Manuscripta Math. 40(1982),27-77. Koiso, _ __ P. R. A. Leviton and J. H. Rubinstein, Deforming Riemannian memcs on the 2-sphere, 10(1985). _ _, Deforming Riemannian metrics on Complex projective spaces, Centre for Math Analysis 12(1987), 86-95. C. Margerin, A sharp theorem for weakly pinched .. -manifolds, C.R. Acad. Sci. Paris Serie 117(1986), 303. - ' Pointwise pinched manifolds are space forms, Geometric Measure Theory Conference at Arcata-Proc. of Sympos. in Pure Math. Vol. 44 (1984). N. Mok, _ _ S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Proc. Sympos. Pure Math. 44(1986),343-352. _ _ , On deformation of Riemannian metrics and manifolds with positive curvature operator, Lecture Notes in Math. Vol. 1201, Springer, Berlin, 1986, 201-211. W. X. Shi, Complete noncompact three-manifolds with nonnegative Ricci curvature, J. Differential Geom. 29(1989),353-360. _ _, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30(1989), 223-301. _ _ , Ricci deformation of the metric on complete noncompact Rieman-

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[44]

nian manifolds, J. Differential Geom. 30(1989), 303-394. _ _ , Complete Kahler manifolds with positive holomorphic bisectional curvature, preprint 1993.

[45] [46] [47]

L.-F. Wu, The Ricci flow on B-orbifolds with positive curvature, J. Differential Geom. 33(1991),575-596. _ _ , A new result for the porous medium equation derived from the Ricci flow, preprint 1991. _ _, The Ricci flow on complete R2, preprint 1991.

SURVEYS IN DIFFERENTIAL GEOMEI.RY, 1995 Vol. 2 ©1995, International Press

Spaces of Algebraic Cycles H. BLAINE LAWSON, JR. Table of Contents

Chapter O. Introduction Chapter I. Algebraic Cycles 1. Algebraic subsets 2. Algebraic cycles 3. Symmetric products 4. Divisors 5. Curves on a 3-fold 6. The Euler-Poincare series of the Chow monoid 7. Functoriality 8. The homotopy relationship between Cp(X) and Zp(X). 9. Cycles and the Plateau Problem Chapter II. Suspension and Join 1. Algebraic suspension 2. Algebraic join and the cup product 3. The Algebraic Suspension Theorem 4. Some immediate applications 5. The relation to topological cycles 6. The ring 11". Z· (pO) 7. Suspension and symmetry Chapter III. Cycles on pH and Classifying Spaces 1. The total Chern class 2. Algebraic join and the cup product 3. Real cycles and the total Stiefel-Whitney class 4. A conjecture of G. Segal 5. Equivariant theories Chapter IV. The Functor L.H. 1. Definition and basic properties 2. The natural transformation to H.(ejZ) 3. Coefficients in Zm 4. Relative groups 5. Localization 6. Computations 7. A local-to-global spectral sequence 8. Intersection theory 9. Operations and filtrations 10. Mixed Hodge structures 11. Chern classes for higher algebraic K-theory 12. Relation to Bloch's higher Chow groups 13. The theory for varieties defined over fields of positive characteristic 14. New directions

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Chapter V. The Functor L* H* (Morphic Cohomology) 1. Effective algebraic co cycles 2. Morphic cohomology 3. Functorlality 4. Ring structure 5. The natural transformation to H* (e; Z) 6. Operations and filtrations 7. Computations at level 1. 8. Chern classes 9. An existence theorem 10. A Kronecker pairing with L*H* Chapter VI. Duality 1. Definition 2. The duality isomorphism: L* H*

~

L n-*H2n-*

SPACES OF ALGEBRAIC CYCLES

139

Introduction

In this article 1 we shall review a recent body of work which is concerned with the structure of the spaces of algebraic cycles on an algebraic variety. Before embarking on this survey we should offer some general motivation for such a study.

The fundamental objects of interest in algebraic geometry are the sets of solutions of polynomial equations in affine or projective space. Any profound understanding of such sets must be at least in part geometric. However, if the field in question is, say finite, in what sense can one speak of geometry? This geometry comes from the network of algebraic subsets. An algebraic variety has not only points, but also a family of "algebraic curves" , algebraic subsets of dimension 1. As in most geometries the distinguished curves give structure to the space. Of course here there are also distinguished algebraic surfaces, 3-folds, etc. It is the interlocking web of these subvarieties which endows an algebraic variety with a rich geometric structure. For an affine variety X c eN this picture translates faithfully into algebraic terms. Irreducible subvarieties of X correspond to prime ideals in the ring O(X) of polynomials restricted to X. The inclusion of subvarieties corresponds to the (reverse) inclusion of ideals. For general X Grothendieck took all this a step further. He taught us to consider the irreducible subvarieties to be "points" of the space. On this enhanced set of points he introduced a topology and a sheaf of rings - classical stuctures of geometry. In this spirit of purely elementary considerations, there is a related construction which also uses subvarieties and in fact predates Grothendieck. Fix a variety X, and for p ~ 0 let X (P) denote the set of "p-dimensional points" of X, i.e., the set of irreducible p-dimensional subvarieties. Then one defines the Chow monoid of X to be simply the free abelian monoid

generated by this set. The points c E Cp(X), which are expressed uniquely as finite formal sums c = E ni Vi with ni E Z + and Vi E X (P), are called effective algebraic p cycles on X. Now the surprizing fact - established by Chow and Van der Waerden in 1937 - is that when X C pN is projective, this monoid itself is an algebraic variety. Specifically, it can be written as a countable disjoint union (0.1)

II

QEH2p(X;Z) 1 Partially

supported by the NSF and I.H.E.S.

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H. BLAINE LAWSON, JR.

where each Cp,a(X) canonically carries the structure of a projective algebraic set. This gives us a constellation of geometric objects naturally associated to X. They can be thought of as compactifications of the moduli spaces of pdimensional algebraic subsets of X, and have considerable independent interest, particularly when X = lP'n. (For example, for varieties over C, Cp,a(X) has been shown to represent the space of solutions to the Plateau problem in the homology class 0:.) These Cp,a(X) fit together to form a monoid whose addition when restricted to the algebraic pieces is a morphism of varieties. There is of course also the free abelian group

of all algebraic p-cyc1es on X. It is functorially related to X, but appears at first to be just a huge, infinitely generated group. However, it carries a very interesting structure which comes from the Chow monoid

as follows. Note that Zp(X) can be written as a quotient

(0.2) where (a, b) '"" (a', b') <==> a + b' = a' + b. By (0.1) Cp(X) can be written as a monotone union VI C V2 C ... of projective algebraic sets. Therefore, Zp(X) carries an intrinsic filtration

(0.3) where Kl

=

U

Vi x Vi / '"" .

i+i9

Each Kl is the quotient of an algebraic set by a proper algebraic equivalence relation. Note that when X is defined over C, each Kl is a compact Hausdorff space. This induces a topology on Zp(X) in the standard way (by defining C C Zp(X) to be closed iff C n Kl is closed for alIt), making Zp(X) a topological abelian group. Its homotopy groups, as we shall see, constitute an interesting set of invariants. They characterize Zp(X) up to homotopy equivalence and reflect the algebraic structure of X. I have gotten somewhat ahead of myself. Let's return to elementary considerations. As mentioned above, spaces of cycles have considerable geometric interest, particularly when X = lP'n. Consider for example the set Cp,l (lP'n) of effective p-cycles of homology degree 1. This is exactly the Grassmannian of (p + I)-planes in en+! , a space of fundamental importance in geometry. One reason for its importance is that, as n goes to infinity Cp-l,l (lP'n) approximates

SPACES OF ALGEBRAIC CYCLES

141

the classifying space BUp for p-dimensional vector bundles; and as both nand p go to infinity, one obtains the classifying space BU for reduced K-theory. Despite the beauty and importance of the Grassmannians, until seven years ago surprizingly little was known about spaces of cycle of higher degree. In fact, the work surveyed here was motivated by a desire to understand these other components of 00

Cp(JP n)

=

II Cp,d(JPn). d=O

(Here Cp,d(JPn) denotes the effective p-cycles of homology degree d.) One could see straightforwardly that Cp,d(JPn) is always connected and simply-connected, and it seemed plausible to conjecture that 7r2Cp,d(JPn) ~ Z. This and much more turned out to be true. The first interesting discovery was that as d -+ 00 the sets Cp,d(lr n) "stabilize" to become classifying spaces for integral cohomology in even degrees. This says much about their structure. It also means that the Chow varieties are in fact fundamental objects in topology. This stablization result can be rigorously expressed by the assertion that there exists a homotopy equivalence

(0.4)

Zp(JP n)

~

K(Z,O) x K(Z,2) x K(Z,4) x··· x K(Z,2(n- p))

for all 0 ~ p ~ n, where K(Z, 2k) denotes the Eilenberg-MacLane space (See I.3 below). Since Zp(JPn) is a group, this is equivalent to the assertion that

Note the simplicity of these homotopy groups. By contrast the homology groups of Zp(JPn) are quite complicated. Note also that

(0.5) This equivalence is induced by an algebraic suspension mapping which is described in Chapter II. Now from the introduction of cycle groups into topology something new emerges. The first surprizing fact is that the simple inclusion Cp,1 (JPn) C Zp (JPn) canonically represents the total Chern class of the tautological (n - p)-plane bundle over the Grassmannian Cp,l (lrn). Furthermore, on projective algebraic cycles there exists an elementary binary operation, called the algebraic join. It is a direct generalization of the direct sum of linear spaces, which gives a pairing on Grassmannians and corresponds to addition in K-theory. It turns out to canonically represent the cup product in cohomology. Using this join

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H. BLAINE LAWSON, JR.

structure one has been able to answer some old questions in homotopy theory (cf. Chapter III). Now the homotopy groups of Zp(pn) turn out to be simple and to playa central role in certain universal constructions in topology. It seems reasonable to think therefore that the groups 1I"iZp(X) might be important for any projective variety X. 2 They are functorial. FUrthermore there is an Algebraic Suspension Theorem: Zp(X) ~ Zp+lCEX) generalizing (0.5) above, which gives 1I"iZp(X) an unexpected and useful structure. Consider some basic examples. Example 1.

1I"OZp(X)

=

=

Ap(X)

algebraic p-eycles on X modulo algebraic equivalence,

Example 2. By a classical theorem of Dold and Thorn, one has that for all k~O

1I"A:ZO(X) = HA:(Xj Z). This shows that the functor 1I".Z. (X) not only contains the integral homology of X but it also contains the groups A. (X) which are purely algebraic invariants. So this functor represents something new which should be of interest to algebraic geometers. On the other hand the groups 1I".Z.(X) have definite geometric interest since they tell us about the global structure of the Chow varieties Cp,d(X). (See Chapter 1.8.) For these reasons the groups 1I".Z.(X) have been systematically studied over the past few years. They turn out to have a rich internal structure and to be related to many of the standard invariants of algebraic geometry. For example P. Lima-Filho has shown that these groups can be defined for quasi-projective varieties, and they fit into localization exact sequences. This allows complete computations in many cases. He has also extended the definition from quasiprojective to general algebraic varieties. It was Eric Friedlander who laid the foundations for the study of these invariants. He realized the importance of Example 1 above and introduced methods of formal group completion into the theory. He made sense of the groups 1I".Z.(X) for varieties defined over any algebraically closed field and proved the suspension theorem in this context. In his fundamental paper he introduced the notation

2It may seem at first that homotopy groups, which involve continl10us mappings of spheres, are particularly non-algebraic in their construction. However, the homotopy of an abelian topological group Z has a beautiful realization as the homology of the simplicial group Sing.(Z).

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143

where Lp indicates that the algebraic level is p, i.e., there are p algebraic parameters, and where Hk indicates that the homology degree is k. Friedlander and Mazur have shown that the algebraic join of cycles leads to a natural transformation s : LpHk(X) --+ Lp-1Hk(X) which in turn induces filtrations on the groups H.(XjZ) and A.(X). These filtrations have alternative, purely algebraic interpretations and are subordinate to the filtrations of Grothendieck and of Bloch-Ogus. Grothendieck's standard Conjecture B actually implies that the filtrations coincide. The suspension theorem has been extended Oy Friedlander and Gabber to a general intersection product in Z.(X) which gives a graded ring structure to L.H.(X). There exists a local-to-global spectral sequence with an identifiable E 2 -term as in Bloch-Ogus theory. There are relations to algebraic K-theory and to Bloch's higher Chow groups. All this is discussed in Chapter IV. Now the groups L.H.(X) behave like a homology theory on the category of quasi-projective varieties and proper morphisms, and it is natural to ask whether there is an associated "cohomology" theory. In [FL 1,2] such a theory was introduced, based on a new concept of an effective algebraic co cycle on a variety X. Such a cocycle is defined as an algebraic family of affine subvarieties parameterized by X. The set of all such cocycles in degree-q is roughly speaking the monoid Cq(Xj en) = Mor(X, Cq(en)) with a natural topology. Taking homotopy groups of the group completion gives a contravariant functor L· H· (X) which enjoys a rich structure. There is a "cup product" induced by taking the pointwise join of cocycles, there is a natural transformation of rings

Lq Hk(X) --+ Hk(Xj Z), there are s-maps and filtrations, Chern classes, etc. All this is discussed in Chapter V. Although the functors L.H. and L· H· are quite differently defined, they are surprizingly related. There is for example a Kronecker pairing between them. However, much more interesting is the recently established fact that on smooth projective varieties they satisfy Poincare duality. In fact for any projective variety X of dimension n there is a naturally defined homomorphism

LqHk(X)

..!4 L n- qH2n-k(X)

for all q, k which under the natural transformation to singular theory becomes the Poincare duality map, i.e., there is a commutative diagram

LqHk(X) ~ Ln-qH2n-k(X)

1

Hk(Xj Z) ~

1 H 2n -k.{Xj Z)

H. BLAINE LAWSON, JR.

144

where V(a) = a n [X]. When X is smooth and projective, the map 15 is an isomorphism. This result, discussed in Chapter VI, has a broad range of consequences. I would like to express my sincere thanks to Eric Friedlander, Paulo LimaFilho and Pawel Gajer for having made a number of suggestions which greatly improved the original version of this manuscript.

Chapter I - Algebraic Cycles §1. Algebraic subsets. Let IP'n denote complex projective n-space, the space of all lines through the origin in en+!. Then there is a natural map 11' :

en+! - {O} -+ IP'n

which assigns to v the 1-dimensional subspace it generates. Definition 1.1. A subset V c IP'n is said to be algebraic if there exists a finite collection of homogeneous polynomials PI, ".,PN E C[zo, ZI, ... , zn] such that

An algebraic subset V is said to be irreducible if it cannot be written as a union V = VI U V2 of two algebraic subsets where VI i. V2 and V2 i. VI. Basic results in algebra tell us that every algebraic subset V c IP'n can be written uniquely as a finite union of irreducible ones, and each irreducible one has a well defined dimension (cf. [4], [43]). From a differential geometric point of view, irreducibility is nicely characterized as follows. For V C IP'n, let Reg(V) denote the set of manifold points of V, i.e., the set of points x E V for which there is an open neighborhood U and local holomorphic coordinates «(1,'" , (n) on U such that

From the Weierstrass Preparation Theorem one proves the following. If V is an algebraic subset, then so is Sing(V) ~f V - Reg(V). Furthermore, V is irreducible <==> Reg(V) is connected,

and the algebraic dimension of an irreducible V equals the complex dimension of Reg(V). For a general algebraic subset V, Reg(V) can be written as a finite disjoint union Reg(V) = RI II ... II RN of submanifolds, and the unique decomposition V VI U··· U VN is given by V; R j •

=

=

We now introduce some terminology. An irreducible algebraic subset V C pn is called a projective subvariety. The set theoretic difference V = VI - V2 of two projective subvarieties is called a quasi-projective subvariety. For any

SPACES OF ALGEBRAIC CYCLES

145

such V, let n{V) denote the field of rational functions on V (the restrictions of rational functions on pn whose polar divisor does not contain V). Then a morphism between quasi-projective subvarieties is a map f : VI -+- V2 such that rn(~) ~ n(l'I). By a projective or quasi-projective variety we mean an isomorphism class of such subvarieties. Of fundamental importance to us here is the fact that projective subvarieties V C pn determine "topological cycles". This can be seen, for example, from the following. Let EI = V - Reg(V), E2 = EI - Reg (E I ), etc. denote the singular strata of V. Then there exists a semi-algebraic triangulation of V for which the singular strata are subcomplexes. This triangulation is unique up to P L homeomorphism (see [85]). H W c V is also a subvariety, then this triangulation of V can be chosen so that W is a sub complex and the induced triangulation on W is as above. See [44] for an elementary proof. Now fix V with such a triangulation, and suppose p = dim{V). Let [V] be the chain consisting of all 2p-dimensional simplices oriented by the canonical orientation of Reg{V). Then a[V] lies in the 2p - 2 skeleton (since it is supported in V - Reg(V», and so a[V] = o. This is the fundamental cycle of V. It can be seen to generate H 2p {Vj Z) ~ Z. The cycle [V] also determines a class in H 2p (pnj Z) ~ Z PPPI where pp is a p-dimensional linear subspace. The integer d such that [V] is homologous to d [PP] is called the degree of V. One has that # (V n pn-p ) = d for almost all linear subspaces pn-p of co dimension p. Furthermore for almost all pn-p-l we have pn-p-l n V = 0, and the linear projection 11" : pn - pn-p-l -+- pp restricts to a map 11" : V -+- pp of degree d. There is another more intrinsic definition of the cycle [V] in terms of deRhamFederer Theory. Denote by £k{M) the space of smooth differential k-forms on a manifold M equipped with the COO topology (uniform convergence of derivatives on compacta). The topological dual space £k{M) ~f £k(M), is called the space of deRham currents of dimension k on M. Taking the adjoint of exterior differentation gives a complex (£.{M), d) whose homology is isomorphic to H.{Mj JR) (cf. [16]). Let now V C pn be a projective subvariety of dimension p. Then the Hausdorff 2p-measure of V is finite, and so V defines an element [V] E £21' (pn) by [V]( r,o)

{I. 1)

=(

r,o

iReg(V)

for all r,o

E

£2p

(PH). As a current we have that dry] = 0 i.e.,

d[V] (1jJ) ~f [V]( d1jJ) = 0

H. BLAINE LAWSON, JR.

146

for all ,p E £2p-l (lpn). (For proofs of these and of subsequent assertions about currents, see [42]). Of course we have [V] ~ d [PP] also in deRham cohomology. §2. Algebraic cycles. Let Xc pN be an n-dimensional projective subvariety and for each p, 0 ~ p ~ n consider the set X(p) of all p-dimensional subvarieties contained in X. In Grothendieck's picture these are the p-dimensional points . of X • the pth level of the web of points, curves surfaces etc, which encode the rigid algebro-geometric structure of X. It is natural to consider the following. Definition 2.1. The group of p-cycles on X is the free abelian group Zp(X) generated by X(P). The positive (or effective) p-cycles on X are the elements of the free abelian monoid Cp(X) C Zp(X) generated by X(P). We will call Cp(X) the Chow monoid of X. In other words Zp(X) consists of all finite formal sums

where ~ E X(P) and ni E Z for each i There is a group homomorphism

j

and we have

C

E Cp(X) if each ni ~

o.

deg: Zp(X) -+ Z given by deg(c)

= L:ni degree(l-'i).

Letting Cp,d(X) C Zp,d(X) denote the subset of cycles of degree d gives us a graded group and a graded submonoid : 00

(2.1)

Cp(X)

= II Cp,d(X) c d=O

00

Zp(X)

=

EB

Zp,d(X),

d=-oo

Now comes the magic. In 1937 Chow and van der Waerden discovered the following fundamental result (cf. [14], [70], [81]). Theorem 2.2. (Chow [13]) Each of the sets Cp,d(X) for d structure of a projective algebraic subset.

~

0 carnes the

When X = pn Chow's construction goes as follows. Let G n- p - l (pn) ~ G denote the Grassmannian of linear subspaces of co dimension p + 1 on pn. Holomorphic line bundles on G are in one-to-one correspondence with Z via the first Chern class CI. Suppose V C pn is a projective variety of dimension p and degree d. Set Dv = {L E G : L n Vi: OJ. Then Dv can be seen to be an algebraic subset of codimension one in G. Any such set is the divisor of a holomorphic section Uv of a holomorphic line bundle ld of Chern classs d on G. The section Uv is unique up to scalar multiples. Thus V determines a point

SPACES OF ALGEBRAIC CYCLES

147

[o"V] in P (HO (Gj 0 (la))). To a general positive cycle c = E ni Vi we associate the section (Fe = O'~: ® ... ® O'~:. This gives an embedding

A careful analysis involving resultants shows the image to be an algebraic subset. Furthermore it is proven that if X c pn is an algebraic subset, then Cp,a(X) C Cp,a (pn) is also an algebraic subset. Notice what this gives us. Our monoid Cp(X) is now equipped with a topology so that each piece Cp,a(X) is a compact Hausdorff space, in fact an algebraic set. The addition map Cp(X) x Cp(X) -t Cp(X) is easily seen to be an algebraic map on these components. Hence, Cp(X) is an algebraic abelian monoid - quite a nice object ! It is natural to wonder about the uniqueness of this canonical algebraic structure on Cp(X). For this we need the following.

Definition 2.3. A continuous algebraic map is a map 'P : V -t W between projective algebraic subsets whose graph is an algebraic subset of the product VxW. IT V is normal (in particular if V is smooth) every such map is a morphism. Note however that the inverse of the map C -t Y = {(z, w) E (!2 : Z2 = w3 } given by t 1-+ (t 3 , t 2 ), is continuous algebraic but not a morphism. We now have the following. Proposition 2.4. The canonical algebraic structure on Cp(X) is uniquely detennined up to algebraic homeomorphism by any projective embedding of X. Proof. (Sketch.) Let j : X ~ pn be the given embedding and suppose j' : X ~ pn' is another. Using the Veronese embedding (Le., the tensor product of homogeneous coordinates) we get an embedding j x j' : X x X ~ pn X pn, C pnn'+ n+' n. Define ~ : X ~ IP'nn ,+n+n, via the diagonal in X x X. The Veronese is linear on each factor, so our original embeddings are recaptured by restriction. Now we see above that if A c B C IP'N are projective varieties, then Cp,a(A) is an algebraic subset of Cp,a(B) for all d. Thus we have three algebraic embeddings

corresponding to j,j' and~. Since ~Cp(X) is the graph of the identity map on Cp(X), and it is also algebraic, we are done. 0 Note. In the above proof it is better to use the intrinsic grading of Cp(X) given by the map (2.2)

H. BLAINE LAWSON, JR.

148

From 2.4 we conclude that the topology on Cp(X) is intrinsically defined, i.e., it depends only on the isomorphism class of X. This makes it natural to pass the topology on to the group completion Zp(X). Note that

(2.3) where (Cl, C2) "" (ci, c~) ¢:::::> Cl classes gives a surjective map

+ c~ = C2 + ci.

Therefore, taking equivalence

(2.4) Now Cp(X) x Cp(X) is a monotone union of compact sets

K; =

(2.5)

II

Cp,d(X) X Cp,d ' (X)

d+dl~i

for i

~

O. The equivalence relation is closed and so the quotients Ki =

1I'K;

are compact Hausdorff spaces topologically embedded in one another: (2.6) with U Ki = Zp(X). Whenever one is in this situation, there is a natural topology induced on the space, called the topology associated to the family {Ki}. It is defined by declaring subset C to be closed if and only if en Ki is closed for all i. With this topology Zp(X) is a topological group. This group is characterized by the universal property that any continuous homomorphism h : Cp(X) -+ G into an abelian topological group G determines a continuous homomorphism h: Zp(X) -+ G so that

commutes. Remark 2.5. In a beautiful paper [54] P. Lima-FiIho recently established several equivalent formulations of the topology on Zp(X). One is engendered by considering flat families of cycles over smooth base spaces and is related to work of Rojtman. With this definition many properties, such as functoriality, are clear. Another definition involves "Chow envelopes" and is useful for establishing the existence of fibration sequences, etc. Lima-Filho shows that these definitions with all their properties extend to arbitrary algebraic varieties (not just quasi-projective ones), and that on this general category they coincide.

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At this point it could be useful to examine a number of examples. §3. Symmetric products. Note that for any projective variety X

CO,d(X)

= p::nixi : Xi E X = X x ... X X/Sd

and ni E Z+ with Eni

= d}

~f Spd(X) (the d-fold symmetric product)

where Sd is the symmetric group acting by permutation of the factors. Hence,

Co(X)

= II Spd(X) d~O

is the free abelian topological monoid generated by the space: its components are evidently varieties. A particularly nice case is that where X = ]pl. Lemma 3.1. As projective varieties we have that Spd (]PI)

Co

(]pI)

= ]pd.

Hence,

II ]pd.

=

d~O

Proof. Define the map ]pd -+ spd (]pI) by assigning to the point with homogeneous coordinates [aD : al : •.. : ad] the zeros of the homogeneous polynomial equation d ~

k d-k

~akZOzl

0 =.

k=O

The inverse to this map is given by expressing the coefficients of a polynomial as elementary symmetric functions of its roots; namely, if [6 : 7]1] , .•• , [~d : 7]d] are d-points in ]pI, then [aD : ... : ad] are the coefficients of the polynomial d

II (~iZI -

p(Zo, zt} =

7]iZO) •

0

i=1

Note that the additive structure in this monoid is given by the maps ]pd

x ]p'" -+

([a], [b]) where

Ck

=

L

aibj

for

]pd+d'

I-t

[c]

k = 0,··· ,d + rI.

i+j=k

The case where X is a non-singular curve of higher genus is even more interesting. Here we must use more sophisticated geometry. By an elementary construction (cf. [64] and §4 below) one associates to every positive O-cycle L niX. on the curve X, a holomorphic line bundle l of degree d = Ln.,

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150

and a holomorphic section u of l such that E niXi is the zero divisor of u. The pair (l,u) is unique up to scalar multiples of u. Now holomorphic line bundles on X correspond to elements in Hl(X,OX) where Ox is the sheaf of non-vanishing holomorphic functions on X. It sits in an exact sequence 0-+ Z -+ 0 ~ Ox -+ 0 which gives an exact sequence

where Cl is the degree or first Chern class of the bundle. Resolving 0 and using harmonic theory gives an isomorphism Hl(XjO) ~ Hl(X,IR), and (3.1) is of the form 0-+ 1R2g /Z2 g -+ Hl(Xj OX) -+ Z -+ O.

In particular the components of Hl(Xj OX) are all tori of dimension 2g where 9 is the genus of X. The map above gives us a monoid homomorphism (3.2)

The preimage of each point l is the projective space IP (HO (Xj O(l))) of all global holomorphic sections of i. Hence, component by component we get maps

For d sufficiently large, this is surjective. In fact it is a fibre bundle whose fibre is IP d - g (a non-obvious result even topologically). We now observe that for any topological space Y, the symmetric products Spd(y) = Y X •.• X Y/Sd and therefore the topological monoid Co(Y) are well defined. When Y is compact and Hausdorff we can also define the topological group Zo(Y) exactly as in §2 above. The spaces Spd(y) are beautiful and of fundamental importance in topology. This is due to the following classical result originally conjectured by Serre. It was proved by Dold and Thorn and, independently and simultaneously, by loan James.

Theorem 3.2. (Dold and Thorn [17], [18]). Let Y be a connected finite complex with base point Yo. Then under the embeddings Spd(y) <-+ Spd+1(y) given by c t-+ c + Yo, there is an isomorphism (3.3)

~

11'.

(Spd(y)) ~ H.(Yj Z).

d

Furthermore for any finite complex Y, there is an isomorphism (3.4)

1I'.Zo(Y)

~

H.(Yj Z).

The first statement can be rephrased by considering the limiting space SP(Y)

= lli:!l Spd(y) d

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151

with topology induced by the family of compact sets Ki = Spi(y) as in (2.6)forward. This space, called the infinite symmetric product of Y, has the property that

(3.5)

1I".sP(Y) =:! H.(YjZ).

One nice consequence of 3.2 is MacLane spaces. Recall that for Eilenberg-MacLane space K(r, n) lence in the category of countable

that it gives beautiful models of Eilenbergany finitely generated abelian group r, the is the space, unique up to homotopy equivaCW-complexes, such that if k = n otherwise.

(See [80] for this and what follows). These spaces are classifying spaces for the functor Hn (.j r) in the sense that for any finite complex Y, there is a natural isomorphism Hn(Yj r)

(3.6) where [Y, K(r, n)] = from Y to K(r,n).

11"0

~

[Y, K(r, n)]

Map (Y, K(r, n)) denotes homotopy classes of maps

Theorem 3.2 gives homotopy equivalences

(3.7) for all n. Hence, from (3.6) we see that for a connected finite complex Y, Hn(Yj Z) =:! [Y, SP (sn)] =:! ~d [Y, Spd (sn)]

(3.8)

= ~d 11"0 Map (Y, Spd (sn)) .

This is interesting since maps from Y to Spd (sn) are simply d-valued maps from Y to sn. They correspond, under graphing, to topological cycles in the product Y x sn which project d-to-l onto Y. We will return to this point later when we discuss algebraic cocycles.

s,. y

Equation (3.8) generalizes to higher homotoPY groups. One has that

(3.9) for all k

~

o.

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H. BLAINE LAWSON, JR.

There is a relative version of Theorem 3.2 which was important in the original proof.

Theorem 3.3. (Dold and Thom [17], [18]). Let A c Y be a pair of finite complexes (i.e., A is a subcomplex of Y). Then there is a natural isomorphism (3.10)

11'.

{Zo(Y)/Zo(A)}

Furthermore, given any integer m

> 0,

~

H.(X,Aj Z).

there is a natural isomorphism

(3.11)

Proving (3.10) involves proving that the homomorphism Zo(X) -t Zo(X)/Zo(Y) is a principal bundle. The long exact sequence for 11'. then results in the long exact sequence in homology for the pair. Note that algebraically we have that

Zo(X)/mZo(X)

= Zo(X) ®z Zm

is just the free Zm-module generated by the points of X. The topology on this is interesting to contemplate. It is an important fact that the results of Dold-Thom completely determine the homotopy type of these spaces. This is due to the following result.

Theorem 3.4 (John Moore [66]). Let A be a connected topological abelian monoid or a topological abelian group. Then A is homotopy equivalent to a product of Eilenberg-MacLane spaces. In other words the Postnikov k-invariants all vanish, and so Y is completely determined by its homotopy groups. Thus Theorem 3.2 implies that for any connected finite complex Y there are homotopy equivalences 00

(3.12)

Z x SP(Y) ~ Zo(Y) ~

II K (Hp(YjZ),p). p=o

There are analogous statements for Zo(Y)/Zo(A) and Zo(Y)/mZo(Y) corresponding to (3.10) and (3.11). Now for a projective variety X we see that Cp(X) and Zp(X) are natural generalizations of Co(X) and Zo(X) to the p-dimensional points of the space. It is certainly intriguing to speculate about the extent to which these gorgeous results can be generalized. §4. Divisors. Let us now examine cycles of codimension one. Suppose X is a non-singular projective variety of dimension n. The fundamental result

SPACES OF ALGEBRAIC CYCLES

153

here is that locally every effective cycle of co dimension one on X is the divisor of a holomorphic function, in fact a rational function which is regular in the neighborhood and is unique up to multiplication by non-vanishing functions. Thus given c E Cn-l(X), there is a family {(Ui, li)}~l' where {Ui}~l is an open covering of X, and Ii E 0 (Ui) has the property that

Clu,

= Div (Ii) .

The quotients gij = Io/!; : Ui n Uj ~ C - {O} define transition functions for a line bundle I.e on X for which the 10 determine a global holomorphic crosssection. (See [64] for more details). When X = pn there is exactly one holomorphic line bundle for each integer d, which is denoted Oed). The sections of Oed) for d > 0 are in natural one-to-one correspondence with homogeneous polynomials of degree din (n + I)-variables. This gives the following generalization of Lemma 3.1 above. Lemma 4.1.

Cn- l (pn)

= II p(n!d)_l. d~O

For a general X, the construction above gives a homomorphism (4.1) generalizing (3.2). The preimage of t E HI(Xj OX) under (4.1) is the projective space P (HO(X, OCt))) of holomorphic sections of t. One can prove that there is an exact sequence

o ~ HI(XjJR)/HI(XjZ) ~ HI(XjOX)

~ NS(X) ~ 0

where NS(X) = HI,I(Xj Z) ~ H2(Xj Z) is the set of classes whose image in H2(XjC) is represented by a (1, I)-form. NS(X) is called the Neron-Severi group of X. Thus HI (Xj OX) is a discrete group extended by a torus of dimension bl(X)

= rank(HI(XjJR».

By universality, the homomorphism (4.1) extends to a continuous homomorphism (4.2)

This extension'is explicitly given by extending the construction given at the beginning of this section to general (non-positive) cycles. Every line bundle admits a meromorphic (i.e., rational) section, and the quotient of two sections of the same bundle is a rational function. Hence, (4.2) expands to an exact sequence (4.3)

0 -+ IP (K X (X» ~ Zn_I(X) -+ HI (Xi OX) -+ 0

H. BLAINE LAWSON, JR.

154

where P (K,x (X)) is the projective group of the non-zero rational functions on X under multiplication. In more classical terms there is a tautologically defined exact sequence of groups o -+ p (K,x (X)) -+ Zn-1 (X) -+ Pic(X) -+ 0 where Pic(X) is the Picard group of divisors on X modulo rational equivalence. The above remarks constitute a computation of Pic(X) in terms of sheaf cohomology. What is of interest here is that with our given topology on Zn-l (X) the homomorphism (4.2) is continuous, and in fact (4.3) is a fibration. Setting poo = !i!!lpd we have the following result of E. Friedlander. d

Theorem 4.2 [22] Let X be any non-singular projective variety of dimension n. Then there is a homotopy equivalence

From this we see that certain classical invariants occur as homotopy groups of Zn-l (X) namely

7l'oZn-1(X) 7l'1 Zn-1(X) and also 7l'2Zn-l(X)

~

~

~

NS(X) Hl(XjZ)

~

H 2n - 1 (X,Z)

Z.

§5. Curves on a 3-fold. The next interesting case to examine is when p = 1 and n = 3. Here life can be quite complicated. Consider for example X = p3. Every cycle of degree 1 is linear, so

where {h (~) denotes the Grassmannian of 2-planes in ~. It is not difficult to see that C1,2 (p3) = Sp2(Q) U Q where SP2(Q) corresponds to pairs of lines in p3 and Q consists of plane quadrics, i.e., all quadratic curves lying in hyperplanes in p3. These two subsets of Cl,2 (p3) are algebraically irreducible and of dimension 8. The set S p2 (9) n Q consists of degenerate plane quadrics, i.e., pairs of lines which meet in p3. It has dimension 7. In degree 3 we have the decomposition

SPACES OF ALGEBRAIC CYCLES

155

where SP3(Q) consists of triples of lines, 9 + Q = {i + q : i E 9 and q E Q}, C consists of plane cubics, and N consists of "twisted cubics" and their limits. Each of these is irreducible and of dimension 12. The elements in N which do not belong to other components are exactly the images of maps ]pI1 -4 ]pI3 given by a full basis of homogeneous polynomials of degree 3, the so-called rational normal curves in ]pI3. One example is the map

[zo : zd

1-4

[zg : v'3z~zl : v'3zoz~ : z:]

(which has constant curvature 1/3 in the standard metric). All others are obtained from this one by a change of basis in C4, i.e., N is the closure of an orbit of PGL 4 (C) acting on CI ,3 (]pI3). It is interesting to examine the intersections of the various components of Cl,3 (]pI3). For example en N consists of those plane cubics which are rational. It has dimension 11 and generically fibres over (]pI3)·. Clearly as d increases the geometry of Cl,d (]pI3) becomes tremendously complicated. Each map CI,d' (]pI3) X Cl,d" -4 CI,d (]pI3) given by addition, where d' + d" = d, contributes a large number of irreducible components to Cl,d (]pI3). In addition to this there will always be new ones, for example plane curves of degree d. For further discussion see [74] for example. §6. The Euler-Poincare series of the Chow monoid. Despite their complicated structure, it is possible to compute the Euler characteristics of the Chow sets Cp,d (]pin). An intriguing consequence of the calculation is that the generating function associated to these numbers is rational. For fixed p :5 n consider the formal power series in one variable 00

(6.1)

wp,n(t)

= LX (Cp,d (]pin)) t d d=O

where X(Y) denotes the Euler characteristic of Y. Then in collaboration with Steven Yau the following was proved. Theorem 6.1. ([60]) For all 0 :5 p


(6.2)

When p = n - 1, i.e., in the case of divisors, this is a classical Hilbert polynomial calculation. When p = 0, it is a special case of the general MacDonald formula [61]

156

H. BLAINE LAWSON, JR.

which holds for any connected finite complex Y. Nevertheless, the rationality of q,p,n(t) for general p is somewhat surprising. Note that the exponent in (6.2) is just the Euler characteristic of the Grassmannian Cp,l (pn) of p-planes in pn. The result above has been generalized by J. Elizondo to general toric varieties. Note that for any projective variety X and any p < dim(X) we can define q,p

=

L

X (Cp,Q(X)) 0:

QE r 2p

where r 2p = H2p(Xj Z)mod torsion, by convention x(0) = O. Given a basis el,'" ,eN ofr2P we let tl,'" ,tN denote the linear coordinates on H2 p(XjJR) with respect to the dual basis. We then "rewrite" q,p as a formal sum (6.3)

q,p

=

L

X(Cp,n(X))t n .

nEZ N

Theorem 6.2. (J. Elizondo [19], [20]). If X is a non-singular toric variety, then q, p is an intrinsically defined mtional function on H2p (X j JR) which can be explicitly and canonically computed from the combinatorial data (the "fan") of the variety. This result has an elegant formulation in terms of equivariant cohomology suggested by E. Bifet (See [19], [20]). §7. Functoriality. Despite their complicated nature, the Chow monoid and its group completion do behave nicely under algebraic maps. Let f:X~Y

be a morphism of projective varieties and suppose V C X is a irreducible subvariety of co dimension p. Then f(V) C Y is a subvariety of Y, and we define if dim(f(V)) < p f.V = if dim(f(V)) = p

{~f(V)

where k is the degree of the map homomorphisms

f

: V

~

f(V).

This determines group

(7.1)

and (7.2)

Proposition 7.1. (Friedlander [22]) The homomorphisms (7.1) and (7.2) are continuous.

SPACES OF ALGEBRAIC CYCLES

157

These homomorphisms clearly have the property that (g

for morphisms

I : X -+ Y

0

I).

= g.

0

I.

and 9 : Y -+ Z.

For nice morphisms there are maps in the opposite direction. Proposition 7.2. (cf. [34], [22]) Suppose I : X -+ Y is a flat morphism of projective varieties with fibre dimension k. Then the flat pull-back of cycles gives continuous homomorphisms

1* : Cp(Y) -+ Cp+k(X) 1* : Zp(Y) -+ Zp+k(X), Examples of flat morphisms are proper submersions and branched coverings. For a rigorous definition of flatness see [43] §8. The homotopy relationship between Cp(X) and Zp(X). In our discussion there are two objects of interest. The first is the Chow monoid Cp(X), a geometric object. The components of Cp(X) are algebraic spaces whose structure we would like to understand and relate to X. The second object is Zp(X), the topological group of all p-cyc1es on X. This is a fundamental algebraic object attached to X. It would also be nice to understand the topological structure of Zp(X). In fact by Theorem 3.4 above we know that the homotopy type of Zp(X) is completely determined by the homotopy groups ?r.Zp(X). Each group ?r.Zp(o) is an interesting functor on the category of projective varieties. Now Zp(X) is simply the naIve topological group completion of Cp(X) and one might hope that the homotopy of Zp(X) is somehow a "completion" of that of Cp(X). In fact we can be quite specific. Let M be an abelian topological monoid, and set r = ?roM. Let

M= liMa aEr

denote the decomposition into connected components, and choose an element E Ma for each ct. Then we can define continuous maps

Xa

fa : M

X

r -+ M

X

r

by setting fa(x,{3) = (x + x a ,{3 + ct). The homotopy class of fa depends only on ct E r. Now r is a directed system (where {3 > ct {::::::} (3 = ct + 'Y for 'Y E r), and f a+{3 is homotopic to fa + f {3 for all ct, {3. Hence, for any covariant homotopy functor h we can define the direct limit lim h(M


X

{ct}).

H. BLAINE LAWSON, JR.

158

Suppose now that M = M x M / '" is the naIve topological group completion, where (x,y) ~(x',y') <==:} 3z E M with x + y' + z = x' + y + z, and where one takes on M the quotient topology in the compactly generated category (cf. [80]). Then one might naIvely hope that

h(M)

(8.1)

-

= lim

h(M x {a}),

o Er

and in particular that

(8.2)

-

7rk(M) = lim

7rk

(Mo)

a Er

for k > O. In fact for general M this is almost certainly not true. There are nice cases, such as

M =Co(X)

M

and

= Zo(X)

where it does hold (cf. Theorem 3.2). However the standard proofs are difficult and quite indirect. In general homotopy theorists ignore this question because there exists a substitute for M which has the good property (8.1). It is the homotopy-theoretic group completion

(8.3) where BM is the classifying space of the monoid obtained from the standard bar construction, and flY denotes the loop space of Y. (See [22] for a detailed discussion.) There is a canonical homotopy class of maps M --+ M+ which is an equivalence when M is a group, and there is a model [52] for M+ which admits a map

(8.4) so that

M commutes. The desired relationship (8.1) (and (8.2)) will hold if 1/J is a homotopy equivalence. The author conjectured several years ago that this should be true when M = Cp(X) for a projective variety X. The conjecture proved to be useful

but quite hard. Friedlander made important progress on it. The first complete proof was given by P. Lima-Filho in a beautiful paper [52] in which several other basic results are established. Following this a somewhat stronger result

SPACES OF ALGEBRAIC CYCLES

159

was proved by E. Friedlander and O. Gabber [28]. (This stronger result can also be obtained from methods in [52].) Theorem 8.1. (Lima-Filho [52], Friedlander, Gabber [28]) Let M = Cp(X)

be the Chow monoid in dimension p of a projective variety X. Then the map (8.4) between the homotopy-theoretic and naive group completions of M is a homotopy equivalence. Corollary 8.2. Let Cp(X) be as above. Set r

Cp(X) =

= 11'0 Cp(X)

and let

II Cp,Q(X) QEr

be the decomposition into connected components. Then 1I'0Zp(X) ~

f

(the algebraic group completion of the monoid r), and 1I'kZp(X)

!:::!!!!!} 1I'k Cp,Q(X) Q

for all k

> O. Furthermore, H.Zp(X) ~ H. (Cp(X)) ffiz[r] Z[I1.

The Friedlander-Gabber proof uses the fact that Zp can built out of quotients of varieties by algebraic equivalence relations. (They work in a nice category which contains varieties and is closed under push-out's). Lima-FiIho's arguments are couched in more topological terms, and give also the Dold-Thorn result that Zo(X) !:::! Z x SP(X) for any connected finite complex X. The result above shows that the functors 1T.Z.(X) and H.Z.(X) are related to the homotopy and homology of the Chow varieties of X. In the absence of this theorem one could replace Zp(X) by nBCp(X) (or equivalently just BCp(X)) and obtain interesting functors. In fact. this is important for extending the theory to varieties defined in characteristic p > 0 (See IV. 13.). However, in doing this one loses direct contact with the variety and such theorems as localization, discussed in Chapter IV, are difficult to establish. §9. Cycles and the Plateau problem. For varieties defined over C the components' of Cp(X) have a beautiful geometric interpretation. Fix 'Y E H2p (Xj Z) and let Cp,y(X) denote the set of all c E Cp(X) whose homology class is 'Y. This is a finite union of connected components of Cp(X) and is very possibly empty. However, whenever there exists a cycle C E Cp,y(X), H. Federer [26] proved the following. For any Kahler metric on X, c is a current of least mass (i.e., weighted volume) in its homology class 'Y. That is, (9.1)

Mass(c) ~ Mass(c')

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H. BLAINE LAWSON, JR.

for all rectifiable cycles c' on X which represent 'Y (cf. [27]). Furthermore, equality occurs in (9.1) if and only if d E Cp,-y(X). (Note: When X is not smooth, a "Kahler metric on X" means a Kahler metric defined on some neighborhood of X in pN.) Hence whenever it is non-empty, Cp,-y(X) is precisely the moduli space of all solutions to the Plateau problem for the homology class 'Y onX.

This says that when it is not empty, Cp,-y(X) embeds into the space 32P,-Y(X) of rectifiable 2p-cycles on X with homology class 'Y, as the set of minimum points of the continuous function Mass: 32P,-Y(X) - 7 IR+. One may wonder whether Cp,-y(X) is connected. This turns out not to hold in general. However, for certain basic spaces, such as projective spaces, Grassmannians, general flag manifolds, etc, this and much more are true. For such spaces the inclusion

becomes a homotopy equivalence as 'Y tends to infinity in the partially ordered monoid 1I'0Cp (X). These results are discussed in Chapter N. In recent years there have been examples of geometric variational problems where "as the degree goes to infinity" the set of absolute minima gives a homotopy approximation to the space. This was seen in the work of G. Segal and others [72], [12], [65] where the space of rational maps of p1 into a good variety (as above) approximates the space of all continuous maps. It also appears in the theory of SU2 gauge fields over 8 4 where as the degree of the bundle increases, the finite-dimensional space of self-dual connections approximates the space of all connections modulo gauge equivalence (cf. [3], [77], [11]). Algebraic cycles provide another example of this phenomenon.

Chapter II - Suspension and Join In this section we introduce two elementary constructions on projective subvarieties and discuss the Algebraic Suspension Theorem which is the key to much of the subsequent material. §1. Algebraic suspension. Fix a hyperplane

]pn C ]pn+1

and a point

]po

E

]pn+1 _]pn.

Definition 1.1. Let V C ]pn be a closed set. By the algebraic suspension of V (or complex cone on V) with vertex ]po we mean the set ~V

= U{l: l

is a projective line through]P° which meets V}.

161

SPACES OF ALGEBRAIC CYCLES

~--r---~+-----------~~4

The projection ]p>n+! _lP'0 -t ]p>n is a holomorphic line bundle. It is the normal bundle to lP'n and is equivalent to 0(1). Its fibres are the lines through lP'0 (with lP'0 removed). Thus $V is homeomorphic to the Thom space o/O(l)/v· The construction $V is particularly simple in terms of homogeneous coordinates. Suppose C'+2 is a choice of homogeneous coordinates for lP'n+! with projection 7r : cn+ 2 - {OJ -t lP'n+1. Given any subset S C lP'n+!, let C(S) ~f 7r- 1 (S) U {O},

(1.1)

and suppose the coordinates are chosen so that C(]p>n) C(lP'°) = {OJ x Co Then for any closed set V C lP'n, (1.2)

C(~V)

= C(V)

x C(lP'°)

= cn+1

X {OJ and

= C(V) x c.

From this we see that if lP'P C lP'n is a linear subspace, then

(1.3) is also a linear subspace. Furthermore, we see that if V is a projective subvariety of lP'n, then ~V is a projective subvariety of lP'n+!. In fact if V is defined by homogeneous polynomial equations P1(ZO, ..• ,Zn) = ... = PN(ZO, ... ,zn) = 0, then ~V is defined by exactly the same polynomials, now considered to be functions of an additional "hidden" variable Zn+1. Hence by linearity the algebraic suspension gives a homomorphism (1.4)

which is easily seen to be continuous. Consequently, we have

H. BLAINE LAWSON, JR.

162

Proposition 1.2. For any algebraic subset X gives a continuous monoid homomorphism

c

lP'n, the algebraic suspension

(1.5)

which extends to a continuous group homomorphism (1.6)

for all p, 0 ~ p ~ dim(X). §2. Algebraic join. Fix disjoint linear subspaces lP'n IIlP'm C lP'n+m+!. Definition 2.1. Let V C lP'n and We lP'm be closed subsets. By the algebraic join of V and W we mean the set

V#W

= U{l: l

is a projective line which meets both V and W}.

=

Suppose en+m+2 en+! x en+! is a choice of homogeneous coordinates such that C(lP'n) en+ 1 x {O} and C(Irm) = {O} x en+!. Then we have that

=

(2.1)

C(V#W)

= C(V) x C(W).

From this it is clear that the join takes linear subspaces to linear subspaces, i.e., (2.2)

for 0 $ p $ n and 0 $ q $ m. Furthermore, one has

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163

(2.3)

~m+1 V ~f ~(~( ••• (~V) ... ))

,

...

,

= v#pm.

m+l-times

Of course, one has symmetrically that pn# W ~ ~n+1 W, and this gives the basic relation

(2.4) that the join pairing is obtained by suspending and then intersecting. Note that this suspension always puts cycles in good position, i.e., so they intersect properly. Note from (2.1) that if V and W are projective subvarieties then so is V#W. In fact if V is defined by homogeneous polynomials Pl(Z) = ... = PN(Z) = 0 and W is defined by ql(() = ... = qM(() = 0, then V#W is defined by the vanishing of all Pi'S and qj'S simultaneously. The join extends to algebraic cycles by bilinearity. Suspension is continuous, and the proper intersection of cycles in pI: is continuous on the subset of pairs which meet properly. (See Fulton [34] or Barlet [6].) Hence, we have the following. Proposition 2.2. Let X C pn and Y C pm be algebraic subsets. Then the algebraic join defines a continuous biadditive pairing (2.5) which extends to a continuous biadditive pairing (2.6) for all 0 ~ P ~ dim(X) and 0 ~ r ~ dim(Y).

In particular, if we choose the notation (2.7) where n

= dim(X), then (2.5) and (2.6) give basic pairings:

(2.8) (2.9)

§3. The Algebraic Suspension Theorem. The importance of algebraic suspension comes from the following result.

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H. BLAINE LAWSON, JR.

Theorem 3.1. ([47]) For any algebraic subset X C ]pn and any p, 0 $ p S dim(X), the algebraic suspension homomorphism

is a homotopy equivalence.

Idea of Proof. Suppose X = ]pn. (The general case will follow by restricting to cycles in X.) For simplicity set Cp+1 = CP+1 (]pn+1) and Zp+1 = Zp+1 (]pn+l). Consider the subset

of cycles which meet the hyperplane ]pn in proper dimension. Let .JP+1 C be the subgroup generated by .1P+1' The proof breaks into two steps.

Assertion 1. The subset ~(Zp(]pn)) C

Assertion 2. The inclusion

ZP+1

.JP+l is a deformation retract.

.JP+1 C Zp+1

is a homotopy equivalence.

For the first step we recall that ]pn+1 - ]po -+ ]pn is a line bundle. Scalar multiplication by t > 0 in this bundle defines a one-parameter family of automorphisms CPt : ]pn+1 -+ ]pn+1 which fixes ]pn ll]p°. It induces a 1-parameter family of automorphisms

which leaves invariant the submonoid .1p+1 and fixes the submonoid ~(Cp(]pn)). The main point here is that on the subset .1P+1 the map ~t extends continuously to t = 00 where

is the retraction defined by

c . lP'n denoting the intersection of c E .1P+1 with the hyperplane lP'n. The continuity of this process, called "pulling to the normal cone" is established in

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165

the book of Fulton [34].

t=

t= 1

.Extending q,t, 0 < t $

00,

1000

t=-

to the group completions proves Assertion I.

To prove Assertion 2 it suffices to prove that the homamorphism

(3.1) induced by the inclusion ~+1 C Zp+1 is an isomorphism for all k ~ O. Note that the inclusion map on positive cycles JP+1 C Cp +1 is very far from being a homotopy equivalence. It does induce a bijection of connected components, but the corresponding components have very different dimensions in general. It is in this step that we must use the group completion strongly. For this we erect a superstructure. Fix a linear embedding pn+1 C pn+2 and two points xo, Xl E pn+2 - pn+1. The projections

(3.2) give each set pn+2 - {Xk} the structure of a holomorphic line bundle over pn+1, Consider now a positive divisor D on pn+2 of degree d with Xo ¢ D and ¢ D. One can think of D as a d-valued section of the bundle 11'"0 : pn+2 {xo} -t pn+1. The key observation is that any positive cycle c E Cp +1 (pn+1 ) can be "lifted" to a cycle with support in D. This lifting is defined to be the intersection Xl

of the divisor D with the suspension of c to the point Xo. This gives us a continuous map

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H. BLAINE LAWSON, JR.

Xo ,',

=

Note that (71'0).0 WD d (multiplication by the integer d in the monoid). However, the composition (71'1).0 WD is very interesting. It gives us a transformation of cycles in IP'n+l which makes most of them "transversal" to IP'n, i.e., which moves most of them into .11'+1' Consider now the family of divisors tD, 0:::; t :::; 1, given by scalar multiplication by t in the bundle 71'0 : IP'n+2 - {xo} -+ IP'n+l. We assume Xl f/:. tD for all such t. (This will be true for all divisors in a neighborhood of d .lP'n+l.) The above construction then gives us a family of transformations

FD,t ~f (7I't). for 0 :::; t :::; 1 such that Fo

0

WtD : Cp+l(lP'n+1) ---+ Cp+l(lP'n+l)

=d (multiplication by d).

Fix c E Cp+l (IP'n+l) and ask which divisors D of degree d have the property that FD,t(C) E .1P+l 1 .. for all t > O. Let Be C Cn+1,d(lP'n+2 ) ~ IP' (n+2+d) d be the subset of dlVlsors for which this fails, i.e., for which there is some t > 0 such that FD,t(C) ¢ .1p+l' Then the main algebro-geometric calculation is that

(3.3) We can now apply these transformations with d = 1 to prove that .1p+l '-t Cp+l induces a bijection on connected components. Hence, (3.1) is an isomorphism for k = O. Suppose now that f : S" -+ CP+l is a continuous map for k > O. We may assume f to be P L up to homotopy. Then for all d sufficiently large, we see

SPACES OF ALGEBRAIC CYCLES

that the map d· family

167

f is homotopic to a map Sk ~ .7P+l' Indeed just consider the Ft,D

o :5 t :5 1, where D

0

f : Sk --+ Cp+l

lies outside the union

which is a set of real codimension ~ 2(P+~+l) - (k + 2). Similarly, suppose we are given a map of pairs f : (Dk+ 1, sn) -+ (Cp+1, .7P +l)' Then for all sufficiently large integers d, the map d· f can be deformed through a map of pairs to one with image in .7P+l' From this we deduce that the map

'$ .. : ~ 7rk(Cp,a) --+ ~ a

7rk(Cp+l,a)

a

is an isomorphism for all k > O. Hence the induced map on homotopy group completions is a homotopy equivalence. One then applies Theorem 1.8.2 for the statement concerning naive group completions. 0 Note 3.2. With a little more care the arguments above can be applied to prove directly that ~ : Zp -+ Zp+l is a homotopy equivalence (without using Theorem 1.8.1). See [48] for example. The Algebraic Suspension Theorem can be thought of as a "stability result" . IT we choose notation (3.4)

where n

= dim(X)! then Theorem 3.1 can be restated by saying that

(3.5)

is a homotopy equivalence for all q

:5 dim(X).

§4. SOIne immediate applications. For cycles in projective space one can make a construction which strictly generalizes the Dold-Thorn construction of SP to the "p-dimensional points". Fix a linear subspace fo of dimension p in IP'n, and consider the sequence of embeddings

given by c 1-+

C

+ lo.

Define Cp(lP'n) == ~ Cp.d(lP'n) d

H. BLAINE LAWSON, JR.

168

to be the limiting space with topology generated by this family of compact sets. (A set C is closed iff its intersection with each Cp,d is closed.) Note that Co(lP'n) = SP(lP'n). As in (3.4) we write fq(lP'n) == Cn_q(lP'n) as the connected monoid of codimension-q cycles. Theorem 4.1. ([47]) There are homotopy equivalences

(4.1)

cq(lP'n) ~ K(Z,2) x K(Z,4) x .. · x K(Z,2q)

(4.2)

zq(lP'n)!:!:! K(Z,O) x K(Z,2) x K(Z,4) x .. · x K(Z,2q)

for all n

~

q~

o.

Proof. Apply Theorem 3.1 to see that zq(lP'n) !:!:! zq(lP'q) = Zo(lP'q) and then apply the Dold-Thom Theorem (cf. (1.3.12)). The space cq similarly reduces D down to Co(lP'q) = SP(lP'q). Theorem 4.2. ([47]) Let IP n - 1 C IP'n be a hyperplane, then there are homotopy equivalences

(4.3) for all n

~

q ~ O.

Theorem 4.3 ([47]). Let m > 0 be any positive integer, and let zq(lP'n) ® Zm = zq(lP'n)/mZq(lP'n) be the topological group of codimension-q cycles with coefficients in Zm = Z/mZ. Then there are homotopy equivalences

and (4.5) for all n

zq(lP'n)®Zm/zq-1(lP'n-1)®Zm ~

q~

o.!

K(Zm,2q)

o.

Theorem 4.1 can be applied to give results about the structure of the Chow sets Cp,d(lP'n). We say that a map f : A -+ B between spaces has a right homotopy inverse through dimension k, if there is a finite complex C and a map i : C -+ A so that the composition f 0 i is k-connected. Theorem 4.4 ([47]). The inclusion

Cp,d(pn) <-+ Cp(pn) has a right homotopy inverse through dimension 2d. In particular the induced maps

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169

are surjective, and the maps

are injective for k

~

2d and for any coefficient ring A.

This establishes the existence of a lot of "stable" cohomology in the classical Chow varieties. If A = 2';2, we pick up much ofthe Steenrod algebra as d, n -t 00.

Note. In this context it is natural to ask how much of the "stable" homology of Cp,d is represented by algebraic cycles. Recently Michelsohn has found such representatives for essentially all possible classes [94]. §5. The relation to topological cycles. Let X be a projective variety of dimension n. Fix a triangulation of X compatible with the smooth stratification and let X <-+ ~N be a linear embedding of this simplicial complex. Then for any k ~ 2n, the Lipschitz singular k-chains in X can be completed to a group of rectifiable k-currents

using the Federer mass norm. (See [26] and [27].) These currents retain certain manifold-like properties, and the spaces have nice compactness properties. The restriction of the de Rham differential d makes (R.(X), d) a complex whose homology is H*(X; Z) ([27]). In particular in each dimension k ~ 2n we have the topological group

of rectifiable k-cycles on X. The group depends only on the P L-structure of X. The topology is the restriction of the standard weak topology of the space of de Rham currents on ~N with support in X. Now there is a beautiful theorem of Fred Almgren which generalizes the Dold-Thom result. Theorem 5.1 (Almgren [1]). For each pair of non-negative integers k,l, there is an isomorphism

There is a natural continuous homomorphism (5.1) for each p, 0

~ p ~

n. Theorem 5.1 above can be restated in the following way.

Theorem 5.2. When X for all p.

= IFn ,

the inclusion (5.1) is a homotopy equivalence

H. BLAINE LAWSON, JR.

170

Thus in projective space the algebraic p-cycles carry the full homotopy-type of the space of all rectifiable 2p-cycles. This strongly generalizes the basic fact that every homology class is represented by an algebraic cycle. We will see that Theorem 5.2 remains true for a large family of varieties including Grassmannians and in fact all generalized flag manifolds. However, such a result is necessarily false (even at the level of connected components) for any variety X for which H2p (XjQ) ct. Hp,p(X). In this case the bigradingof1l'lZp(X) : i,p~ obecomes more interesting than it is in the topological case.

§6. The ring 1I'.Z(PO). Let X C pN be a projective subvariety. Then the algebraic join gives pairings

which, since O#C = 0 and C'#O = 0, descend to the smash product (6.1) Now the Suspension Theorem 3.1 gives a canonical homotopy equivalence Zq+ql (X) ~ Z9+ql (~n+l X), provided that q + q' ~ dirnX. Hence taking homotopy groups in (6.1) gives a pairing

(6.2) introduced by E. Friedlander and B. Mazur [29]. This pairing can be extended somewhat as follows. For any q ~ 0 we have canonical homotopy equivalences zq(:E k X) ~ zq(~n+1 X) e!! ••• for all n such that n + dim (X) ~ q. Let us define

zQ(X)

(6.3)

=~

zQ(~n X)

n

to be this well-defined homotopy type. For example

for any n

~

q. The pairing (5.2) now extends to a pairing

(6.4) defined for all k, k', q, q'

~

O.

TheorelIl 6.1. (Friedlander and Mazur [29]). When X =]pO, the pairing (6.4) gives :FM 'W 11'. Z· (PO) the structure 0/ a commutative bigraded ring. In fact this ring is isomorphic to a polynomial ring Z[s,h] on two generators where

171

SPACES OF ALGEBRAIC CYCLES

For any projective subvariety X c pN, the .pairing (6.4) gives 7r.Z·(X) the structure of a bigraded F M -module.

The operation h is related to the Lefschetz map in homology coming from intersection with a hyperplane. The operation s is more subtle and interesting. We will return to this in Chapter IV.

§7. Suspension and symmetry. In all of the discussion above it is natural to ask what happens in the presence of symmetries. Consider, for example, a projective variety X and a finite group G C Aut(X) of automorphisms. There is a naturally induced action of G on Zp(X) for each p, and in analogy with §3 above, one can consider the topological groups

ZG,p(X) ~f Zp(X)G /G . Zp(X) introduced in [58], where Zp(X)G = {c E Zp(X) : g.c = c'l;/g E G} denotes the fixed-point set and GZp(X) = {~9EG g.c : c E Zp(X)} is the subgroup of averaged cycles. These are functors on the category of G-varieties and Gequivariant morphisms, and therefore are the homotopy groups 7rkZG,P(X). If we are given a representation G -t GLN+1 (C) and a G-equivariant embedding, then algebraic suspension defines homomorphisms

Together with M.-L. Michelsohn it was proved that these maps are homotopy equivalences if one localizes away from the order of the group. In particular we have Theorem 7.1. ([58]) The suspension homomorphisms (7.1) and the induced homomorphism (7.2)

'$ : ZG,p(X)

-t

ZG,p+1 ('$X)

induce isomorphisms on the homotopy group 7rk (.) ® A for all k ring A in which the order of the group G is invertible.

~

0 and any

By "ring" we mean a commutative ring with unit. Examples of such rings are A = Z(q) (the integers localized at the prime q) and A = Z/qZ where q is any prime which does not divide the integer IGI. This condition that IGI be invertible is strictly necessary. It is shown by example in [58] that the suspension homomorphism fails to be a homotopy equivalence at primes dividing the order of the group. However, with LimaFilho and Michelsohn it has been proved that the m-fold suspension map is a G-homotopy equivalence, when m exceeds the codimension of the cycles. One can also suspend to a general representation. Here one obtains the delicate

172

H. BLAINE LAWSON, JR.

result that suspension to the regular representation of G is stably a G-homotopy equivalence [93].

Chapter III - Cycles on IPn and Classifying Spaces It is an interesting fact that the algebraic cycles in projective space can be used to construct models of certain universal spaces in topology - spaces that represent such everyday functors as K-theory and cohomology. Elementary constructions with cycles lead to Chern classes, Stiefel-Whitney classes and the cup product at the universal level. Families of cycles correspond to Steenrod operations.

§1. The total Chern class. It is a basic fact presented in most books on characteristic classes and K-theory that the space of linear cycles

(U.) i.e., the Grassmannian of linear subspaces of codimension-q in IPn is a classifying space for vector bundles. Specifically, for any finite complex Y of dimension ~ 2(n - q), there is an equivalence of functors (1.2)

where Vectq(y) denotes the set of equivalence classes of rank-q complex vector bundles on Y. The equivalence is given by associating to f : Y -+ gq (IPn ), the pull-back q ofthe tautological q-plane bundle q over gq (IPn).

re

e

Analogously we see from H.4.1 and (1.3.6) that the space zq (IPn) has the property that for any finite complex Y there is an equivalence of functors q

(1.3)

EBH2k(y;Z)!:!! [Y,zq (IPn)]. k=O

Observe now that we have a very natural map (1.4)

given by considering linear subspaces as cycles of degree 1. By (1.3) this corresponds to a cohomology class c on gq (IPn). In collaboration with M.-L. Michelsohn the following was proved. Theorem 1.1 ([57]). The cohomology class c = 1 + Cl + .,. + cq determined by the cycle inclusion (1.4) is the total Chern class of the tautological bundle over gq (JP'n)

eq

SPACES OF ALGEBRAIC CYCLES

173

Under algebraic suspension the maps (1.4) sit in a grid of inclusions:

n

n

gq (pn)

zq (pn)

~

n

~

(1.5)

C

gq

n

(pn+1) C

zq

n

~

~ ~

(pn+1)

n

~

where the vertical maps on the right are all homotopy equivalences. Hence we may pass to the limit BUq ~f ~

(1.6)

gq (pn) .

n

This space has the classifying property (1.2) for all finite complexes. Corollary 1.2 ([57]) Passing to the limit in (1.5) gives a map

(1. 7)

BUq --+

zq (POO)

~

K(Z, 0) x K(Z, 2) x ... x K(Z, 2q)

which represents the total Chern class of the universal q-plane bundle over BUq • Taking the limit as q

-t 00

in (1. 7) gives us map

(1.8) where BU and K (Z, 2*) have the property that K(Y)

= [Y,BU]

and

Heven(YjZ)

= [Y,K(Z,2*)]

for all finite complexes Y. That is BU and K(Z, 2*) are the classifying spaces for reduced K-theory and even cohomology respectively. From 1.2 we immediately have Corollary 1.3. The map (1.8) represents the universal total Chern class from K -theory to cohomology. Note that by Bott Periodicity the homotopy groups of BU and K(Z, 2*) are the same in positive dimensions. In fact Bott's fundamental results show that the homomorphism '1r2kBU

111

Z

~

'1r2k K

(Z,2*)

III

Z

174

H. BLAINE LAWSON, JR.

is multiplication by (k - I)! (See [57]). These results lead to some interesting questions. One can consider the spaces V(d)

(1.9)

= lim

Cp,d

(rn)

p-HXl

n-+oo

(with the compactly generated topology) for all d = 1,2"" ,00. There is a natural sequence of inclusion mappings (1.10)

BU

= V(l)

C V(2) C ... C V(oo)

= K(Z,2*).

Note that V(d) is the space of all positive projective cycles of degree d. There are a number of interesting questions concerning this filtration. A simple one is whether the maps 1I".V(d) ~ 1I".V(00) are injective, and ifso, at which levels d do the factors in the homomorphism 1I"2kV(1) ~ 1I"2kV(00), which is multiplication by (k - I)!, appear ? §2. Algebraic join and the cup product. From 11.2 we know that the algebraic join induces a continuous biadditive pairing (2.1) From Theorem 11.4.1. and its proof we obtain a canonical homotopy equivalence q

(2.2)

q

zq (rn) ~ II ~f II K(Z, 2k) k=O

for all n ~ q. It is natural to ask what the join map becomes when interpreted as a map of Eilenberg-MacLane spaces. Certainly the most basic pairing of such spaces comes from the cup-product in cohomology. This product

on spaces Y can be represented universally, via (1.3.6), by a map (2.3)

K(Z,a) x K(Z,b) ~ K(Z,a+ b),

which is determined up to homotopy by the fact that it classifies the cup product ttl ® t" of the fundamental cohomology classes, where tk E Hk(K(Z, k)j Z) ~ Z is the generator. The map (2.3) can be constructed explicitly by extending the smash product map Stl x S" ~ Stl AS" = Stl+" bilinearly to Zo (Stl) X Zo (S") ~ Zo (Stl+") and using (1.3.7). These basic cup product maps (2.3) assemble naturally to give a mapping (2.4)

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SPACES OF ALGEBRAIC CYCLES

which classifies the cup-product mapping in even-cohomology

In collaboration with M.-L. Michelsohn the following was proved. Theorem 2.1. ([57]) Under the canonical homotopy equivalence (2.2) the algebraic join pairing # is (homotopic to) the cup product mapping (2.4) Observe now that the inclusion of degree-l cycles into mutative diagram

gq (lp>n) (2.5)

X

gq'

(IPln')

zq (1I1In) gives a com-

~

1

The restriction of the join to linear subspaces corresponds to taking the direct sum (cf. (11.2.1)). Passing to the limit as q,q' -+ 00 and applying Theorem 2.1 gives a commutative diagram BUq X BUq , ~ BUq+ q,

(2.6)

exel q

q'

IIxII

Ie II

q+q'

~

where the map ffi classifies the Whitney sum of vector bundles. The commutativity of this diagram corresponds to the Fundmanental Whitney Duality Formula c(E ffi E') = c(E)c(E') for the total Chern class of complex vector bundles E, E' over a space Y. The importance of (2.5) was realized early by E. Friedlander. He pointed out that in conjunction with Theorem 1.1, it can be used to prove Theorem 2.1 over the rationals. §3. Real cycles and the total Stiefel-Whitney class. It was suggested by Deligne that if one worked with cycles modulo 2, some of the results above might carryover to real algebraic geometry. Indeed, with the correct formulation of "reality" this turns out to be true, and the results are surprisingly nice. Both the formulation of the theory and the proofs of the results are due to T.-K. Lam.

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H. BLAINE LAWSON, JR.

Following Atiyah [2] we define a Real projective variety to be a pair (X, C) where X is a projective variety and C : X -t X is an antiholomorphic map with C2 = I d. A basic example is that of projective space (I(»n, C) where C is defined by complex conjugation in homogeneous coordinates. The fixed-point set of C is real projective n-space

The choice of this real form corresponds to the choice of Real structure on pn. Observe that if V c X is an algebraic subvariety of a Real variety X, then its conjugate C(V) is also a subvariety. Thus C induces an involution C" on the set of subvarieties of X which extends by linearity to cycles. Definition 3.1. Let (X, C) be a Real projective variety. An algebraic p-cycle on X is said to be Real if it is fixed by C". Let Z:(X)

= {c E Zp(X) : C.(c) = c}

denote the topological group of all Real p-cycles on X. Note that any Real cycle can be uniquely written as E ~+ E mi (Wi + C. Wi) where the ~'s are C.-invariant subvarieties. It is enticing (and naturally suggested by Galois theory) to divide by the subgroup

(1 + C.) Zp(X) = {c + C.(c) : c E Zp(X)} of "averaged" cycles. Therefore following [48] we introduce the topological quotient group

(3.1) of reduced Real p-cycles on X. Algebraically RZp(X) is just the Z2-vector space generated by the Real (irreducible) subvarieties of X. However, this group is also furnished with a natural topology, and T.-K. Lam proves the following theorems. Theorem 3.2 ([48]). Let (X, C) ~ (pn, C) be a Real algebraic subvariety. Then C-equivariant algebraic suspension gives a homotopy equivalence ~: RZp(X) ~ RZp+l(~X)

lor all p, 0 :::; p :::; dim(X). As above we set RZq(X) = RZn_q(X) where n = dim(X). Theorem 3.3 ([48]). There are homotopy equivalences

SPACES OF ALGEBRAIC CYCLES

for all q

~

177

n.

A given Real structure C on I'" induces a real structure on gil (JIlin) whose fixed-point set is the real Grassmannian

of lR-linear subspaces of codimension-q in IRn+1. Theorem 3.4 ([48]). The natural inclusion (3.3)

gil (pn(IR» '-+ RZ' (pn)

!:!!

K (Z2. 0) X ••• X K (Z2. q)

represents the total Stiefel Whitney class of the tautological q-plane bundle over Therefore passing the limit as q -+ 00 in (3.3) gives a map

g' (pn(IR».

, BO q -+ II K (Z2' k)

(3.4)

01:=0

which represents the total Stiefel- Whitney class of the universal q-plane bundle over BO q = ~ gq (pn(IR)). n

Taking the limit as q -+

00

in (3.4) gives a map 00

BO -+ K (Z2. *) ~f

II K (Z2;k). k=O

Theorem 3.5. ([48]) The algebraic join RZq (P") x RZq' (pn') --+ RZp+q' (pn+n' +1 ) correspond via (3.2) to the map which classifies the cup-product in Z2-cohomology.

From 3.4 and 3.5 one retrives the classical Whitney Duality Formula weE ® E')

= w(E)w(E')

for the total Stiefel-Whitney class. §4. A conjecture of G. Segal. We have seen that the elementary inclusions (1.4) and (3.3) determine maps (4.1) which correspond to the universal total Chern and Stiefel-Whitney classes respectively. The question naturally arises whether these maps extend to transformations of generalized (connective) cohomology theories. In other words.

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H. BLAINE LAWSON, JR.

can we extend these to maps of spectra where the mappings (4.1) occur at the oth-Ievel? Finding such an extension amounts to finding infinite loop space structures on these spaces such that e and w are infinite loop maps. Now the spaces BU and BO have natural infinite loop space structures coming from Bott periodicity (e.g., BU = fl2 BU). Each K(G, n) is also an infinite loop space since it is an abelian topological group. However these structures are not even compatible at the I-loop level. H they were, then e and w would preserve the loop-product (flY x flY -t flY). However loop product in BU (and BO) is equivalent to Whitney sum, and in K(G, n) it is equivalent to addition. The Whitney Duality Formulas show that e and w are not additive homomorphisms. In fact if we fix Bott's loop space structures on BU, then a compatible loop structure on K (Z, 2*) will yield a quite different "addition" on even cohomology_ This different additive structure was pointed out and used by Grothendieck in 1958 [37]. It is given on H 2 ·(Yj Z) = HO(Yj Z) Ell H>O(Yj Z) by setting (ao, a'l (bo, b' ) = (eo, c') where

+

+

eo=ao+bo

and

(1 +

e/) =

(1 +

a'l u (1 +

b' ).

This is precisely the addition given by the cup product pairing on K (Z, 2*) discussed in §2. In 1975 G. Segal [71] asked the following question:

(4.2)

Do the cup product pairings on K(Z, 2*) and K(Z2' *) enhance to infinite loop space structures such that e and w become infinite loop maps?

Several such structures were proposed and shown not to work (cf.[73] , [75], [82]-[84], [86]. See [7] for a history). Question (4.2) is very complicated in nature. For any proposed structure one must check compatibility on an infinite pyramid of higher associativity relations. Fortunately topologists have found simpler sets of compatibility hypotheses which yield infinite loop space structures and infinite loop maps.' One such machine, due to Peter May, uses the linear isometries operator C. I will spare you the definition and say only that any C-space (a topological space with an action of C) is canonically an infinite loop space, and any C-map between C-spaces is an infinite loop map. Happily for us there is an elementary method for constructing C-spaces and C-maps. It involves the category I. whose objects are finite dimensional inner product spaces and whose morphisms are linear isometric embeddings. Let denote the category of compactly generated, Hausdorff topological spaces with base point. The sets of morphisms in T are given the compact-open topology.

r

Definition 4.1. An I.-functor (T,w) is a continuous functor T : I. -+ T together with a commutative, associative, and continuous natural transformation w : TxT -+ T 0 Ell such that

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179

H X E TV, and 1 E T{O} is the basepoint, then

a)

w(x,l)

= x E T(V EB {O}) ='TV.

H V = V' EB V", then the map TV' homeomorphism onto a closed subset.

b)

--+ TV given by x

1-+

w(x,l) is a

Theorem 4.2. ([63]) 1fT is an I.-functor, then T (COO)

= Vccao lim

T(V)

where the limit is taken over finite dimensional subspaces of COO, is an C-space. Any natural transformation ~ : T --+ T' of I.-functors induces a mapping ~ : T (COO) -+ T' (COO) of C-spaces. An illuminating example is given by the "Bott functor" TB which associates to each Hermitian V of dimension n the Grassmannian

=

of n-planes in V EB V, with distinguished point 1 V EB {OJ. Given an isometry W define TB(f) : TB(V) --+ TB(W) on a plane U by (TBI) (U) = ((fV)l.. EB {OJ) EB (f EB I)(U). The natural transformation WB is given by

f : V --+

WB(U, U') where T : V EB V EB V' EB V' an I.-functor, and clearly

= T(U EB U')

--+ V EB V' EB V EB V' is the obvious shufHe. This is TB (COO)

= BU.

It is shown in [63, p.16] that the induced infinite loop structure is the standard one of Bott. Now in parallel fashion one may define the Chow monoid functor Te by setting Tc(V) = en (JlD(V EB V» where n = dim(V) with distinguished point 1 = JlD(V EB {O}). In dimension 0 we set T {OJ = N with distinguished element 1. For an isometric embedding f : V -+ W, we define Te (f) : Tc(V) -+ Te (W) on a cycle c by

Te(f)c

= JlD (J(V).L EB {OJ) #(f EB I).(e).

The natural transformation We is given on cycles c, c' by

wc(e,c') = T.(e#c') with

T

as above. One sees that 00

Tc(C"') = V ~f

II V(d) d=O

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H. BLAINE LAWSON, JR.

where V(d) is the space of all degree d cycles (See (1.9)). One verifies that Te is an I.-functor and concludes the following (due to Boyer, Mann, Lima-Filho, Michelsohn and the author).

Theorem 4.3. ([7]). The stabilized cycle space V is an C-space where the structure maps are defined via the algebraic join pairing # : V(d) x V(d') -+ V(dd'). Furthermore, the infinite loop space structure induced on V(I) = BU agrees with the standard one 0/ Bott. Of course Te has values in abelian topological monoids. From this one can deduce that V is an Eoo-ring space in the sense of P. May [63]. Associated to V is an Eoo-ring spectrum. This quickly leads to a positive answer to (4.2). However in the spirit of the exposition here one can proceed as follows. Again in parallel with the above, we define an I.-functor Tz by setting

Tz(v)

= zn(JI»(V EEl V))

=

=

where n dim(V), and continuing as in the definition of Te. (Here Tc{O} Z with distinguished element 1). Note that Tz(V) is the naIve group completion of Te (V) and the limit 00

Tz (COO)

=Z =

II

Z(d)

d=-oo

is the additive group completion of V.

Theorem 4.4. ([7]). The natural map V -+ Z o/V into its additive group completion is a map 0/ C-spaces. In particular, the infinite loop structure induced by the complex join on Z(I) is such that the total Chern class map V(I) <-+ Z(I) is an infinite loop map. This also carries through for Real cycles and we have the following.

Theorem 4.5. ([7]). The multiplication on K(Z, 2*) and K(Z2, *) induced by the algebraic join enhances to an infinite loop structure with respect to which the maps (4.1) are infinite loop maps. Let MO(-) and MOO(-) denote the functors H 2·(_jZ) and H·(-jZ2) with the Grothendieck addition defined above.

Corollary 4.6. ([7]). The functors MO and MOo enhance to generalized cohomology theories M· and MO· such that the maps (4.1) extend to natural transformations and

w: ko·

-+ MO·.

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181

This implies for example the existence of transfer· maps in cohomology which commute with c and w. It has been shown by Totaro [79] that the maps c and w cannot extend to natural transformations of multiplicative theories. §5. Equivariant Theories. It is natural to ask what happens in the constructions above if one introduces the action of a finite group. The very pleasant answer is that one finds a new equivariant cohomology theory with some very nice properties. To be more specific let G be a finite group, and to each finite-dimensional complex representation space V of G associate the cycle group

Tc(V)

= zn«IP'(V ® V))

where n

= dim (V)

as in section 4 above. This space has a natural action of G which respects the algebraic join pairing. It thereby gives us a "G-equivariant I.-functor". Now the theory of May has been carried through in this case [59], and we find the following. Let U = Vo EB Yo EB ... be the direct sum of infinitely many copies of the regular representation Yo of G, and consider the limiting G-space

(5.1)

ZG ~f ~ zn (IP'(V EB V)). VCU

Theorem 5.1. ([55], [56]). The cycle space ZG is a G-equivariant Eoo-ring space (in the sense 0/ [59]) and gives rise to an Eoo·ring G-spectrum. In particular it determines an equivariant cohomology theory 1l'G (.) which is ringvalued (and indexed by R(G)). This theory admits a naturol trons/ormation to a canonically associated theory 0/ Borel type

(5.2)

1l'G(.)

-t

where

ll~(~)Borel

ll'G(.)Borel

= II H'tj(·j Z) i~O

and where H'fj denotes the usual Borel equivariant cohomology. The component ZG(l), which is closed under the join #, determines a related equivariant cohomology theory M'G (.) which is only group-valued. It admits a naturol trons/ormation to its canonically associated "Borel counterpart" (5.3)

where M~(.)Borel

= 1 + II Hbi(.jZ) i~l

H. BLAINE LAWSON, JR.

182

is the group

0/ units in llM. )Borel.

Remark 5.2. Given a G-spectrum S, we define the associated Borel Gspectrum to be SBorel = F (EG+, S), where F(X, Y) denotes the space of pointed G-maps from X to Y. This gives the "associated Borel theories" referred to in 5.1. We may now restrict our attention to the subspaces (5.4)

gn(IP'(V €a V))

c zn(IP'(V €a V)),

where n = dim(V),

of linear cycles, (i.e., the Grassmannians, which are contained in the degree-one component), together with the pairing given by the join which is simply the direct sum of subspaces. This is also a G-equivariant I.-functor. Hence the limiting space (5.5)

BUa

=!!!!l

gn(IP'(V €a V))

vcu

is the oth space of a canonically determined G-spectrum. This spectrum classifies reduced equivariant K-theory KG(.). The following theorem establishes, among other things, a solution of the equivariant Segal Problem (cf. (4.2)) for Borel cohomology. Theorem 5.3. ([55], [56]). The inclusion (5.4) determines a naturol trons/ormation 0/ equivariant cohomology theories

Composing with (5.9) gives a naturol trons/ormation to the associated Borel theory, which on the oth -level is the usual total Chern class map

Ka(X) ~ 1 +

II H'2j(X; Z) i~1

into Borel cohomology. This analogue of the results in §4 should have applications, for example, to the computation of Chern classes of induced representations.

a(.)

The ring functor ll is a new equivariant cohomology theory which arises quite naturally and may have some interesting uses. The coefficients of the theory have been computed in basic cases using techniques of degeneration via CZ-actions. (cf. [55], [56]). For example if G is abelian, then 1l~( pt) = JH[* (G)s where JH[* (G) is a ring functor on finite abelian groups such that (i) JHI.O (G) = Z and JH[1 (G) ~ G (ii) lHI* (G 1 Ell G 2 ) = lHI* (Gt) ® lHI* (G 2 ). (iii) If G is cyclic, then lHI* (G) = H2·(Gj Z).

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SPACES OF ALGEBRAIC CYCLES

and where S is the multiplicative system generated by the total Chern classes of the irreducible representations of G. Note. The discussion above provokes the interesting question of what functor is defined by taking the equivariant homotopy groups of Zo(X) for a Gspace X. Is there an equivariant Dold-Thom Theorem? The very nice answer, due to P.Lima-Filho, is that for G-spaces one gets Bredon cohomology with Z-coefficients, and for G-spectra one gets the ROG-graded homology with coefficients in the Burnside ring Mackey functor [92].

Chapter IV - The Functor L.H. The groups 11".Z. (X) constitute a set of interesting invariants attached to any projective variety X. Some work has been done recently in trying to systematically understand these invariants. At least part of this is presented below. Before embarking let me offer some general motivation. As we have seen, for any projective variety X, the p-dimensional subvarieties generate an interesting topological group Zp(X). This group is functorially related to X. Its geometry is a limit of the Chow sets of X. Specifically, we know from 1.8.2 that 1I"kZp(X)

= ~ 1I"kCp,Q(X)

and

HkZp(X)

= ~ HkCp,Q(X) Q

Q

for all k > O. In fact all "stable" topological invariants of the Chow sets of X are carried in this fashion by Zp(X). Now the homotopy type of Zp(X) is completely determined by the groups 1I"kZp(X), Such a statement is false for general spaces. However for an abelian topological group Z, the invariants 1I".Z are special. For example 1I".Z appears as primitive elements in the Hopf algebra H.(Zj Z). It can also be computed as the homology of the simplicial group Sing.(Z). So the groups 1I".Z are simpler than other invariants, like H.(Z), but nevertheless they determine Z up to homotopy equivalence. This makes 1I"kZp(X) natural to consider in studying Zp(X). It is useful to think of Zp(X) as a generalized torus associated to X, much like the intermediate Jacobians. In fact there is a homotopy equivalence Zp ~ Ap

X

(SI)b 1

X

(BS1)b 2

X

(B 2 S1)b 3 x· .. x Tor p

where Ap is the group of p-cycles on X modulo algebraic equivalence, bk = rank (1I"k Zp) , and Torp is a connected space with 1I"kTorp finite for all k. The various tori which are delooped in this picture can in fact be directly related to intermediate jacobians and their generalizations. From another perspective 1I".Z. is the direct analogue of 11".3. where 3.(X) denotes the rectifiable cycles on X. (cf. 11.5). Now by Almgren's theorem 11.5.1

H. BLAINE LAWSON, JR.

184

there is an equivalence: 7r.3.(-)

= H*+.(-jZ).

This indicates that the functor 7r.Z. might behave like a homology theory on the category of projective varieties. In fact the map 7r.Z. -+ 7r.32. constitutes a natural transformation to standard integral homology. Thinking of 11". Z. in this way gives a systematic approach to the study of these invariants. Note however that 11".Z. is far from being a simple topological theory. For example, 7roZp is the group of algebraic p-cyc1es modulo algebraic equivalence. This already shows these groups to be non-trivially related to the algebraic structure of X. We shall see below that the theory in fact emcompasses many new algebraic invariants. §1. Definitions and basic properties. With the motivation above E. Friedlander introduced in [22] the groups (1.1) for k ~ 2p ~ O. Here k denotes the homology dimension, and p the algebraic level (the number of algebraic parameters). From the Algebniic Suspension Theorem 11.3.1 we have canonical isomorphisms (1.2) for all k

~

2p

~

O.

From 1.7.1 we see that L.H. is a functor on the category of projective varieties, i.e., if f : X -+ Y is a morphism, then there are induced homomorphisms (1.3) for all k

~

2p

~

0, and if 9 : Y -+ Z is another morphism, then

(1.4)

(g

0

f).

From 1.7.2 we have Gysin maps. If f are induced homomorphisms

= g. :X

0

f •.

-+ Y is a flat morphism, then there

(1.6) where r (1.7)

= dim X -

dim Y. If 9 : Y -+ Z is also flat, then (g

0

f)* =

r

0

g*.

Note. For those who are simplicially minded we note that the definition in (1.1) could be replaced by Hk_2pZp(X) where Zp(X) = NSing. (Zp(X)) is the normalized chain complex of the simplicial group Sing. (Zp(X)).

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185

§2. The natural transforlDation to H.(.jZ). The continuous homomorphism

(2.1) defined in II.5 induces a map on homotopy groups which is independent of the choice of smooth triangulation. Applying Almgren's theorem 11.5.1 gives the following. TheorelD 2.1. There is a natuml tmnsfonnation of functors

(2.2) for all 0

~

2p

~

k.

Note that it is integml (not rational or real) homology that appears here. §3. Coefficients in Zm. In the preceeding two sections, one could replace 3. (X)/m3. (X) for a fixed integer m > O. This yields a functor

3.(X) with the quotient group

(3.1) with a natural transformation

(3.2) Most results discussed below will carry through in this case. §4.

R~lative

groups. Let Y

c

X be an algebraic subset of a projective variety

X. For each p, Zp(Y) is a closed subgroup of Zp(X) and we can consider the quotient group Zp(X)/Zp(Y) with the quotient topology. We set

(4.1)

for k ;::: 2p ;::: O. Then we have the following. TheorelD 4.1 ([47], [50], [51]). There is a long exact sequence

which is functorial for morphisms of pairs. This sequence tenninates with

where Ap denotes the group of p-cycles modulo algebmic equivalence.

H. BLAINE LAWSON, JR.

186

§5. Localization. Fundamental results in the theory theorems of P. Lima-Filho.

ar~

the localization

Theorem 5.1 (Lima-Filho [50], [51]). Let X, X' be projective varieties with algebraic subsets Y C X and Y' C X', and suppose f:X-y~X'-Y'

is an isomorphism of quasi-projective varieties. Then there is a naturally induced isomorphism of groups

which is a homeomorphism. In particular there is a naturally induced isomorphism !!!! L.H* ( f. : L.H.(X, Y ) --t X' ,Y')

This theorem enables us to extend the theory to quasi-projective varieties. Definition 5.2. Let U C pN be a quasi-projective variety with closure U. Then we define the topological group of p-cyc1es on U to be the quotient

and we set for all k

~

2p

~

O.

By 5.1, Zp(U) and LpHk(U) are independent of the projective embedding of U. They are, in fact, functors on the category of quasi-projective varieties and proper morphisms. Furthermore, the following holds. Theorem 5.3. (Lima-Filho [50], [51]). Let V CUbe a Zariski open subset of a quasi-projective variety U. Then there is a long exact "localization" sequence:

From this one can inductively build a Zariski open covering and do computations. The proof of Theorem 5.1 uses strongly that one can work with the nai:ve group completion. The idea is as follows. Suppose r.p : X - Y -+ X' - Y' is an isomorphism. By replacing X with the closure of the graph of r.p in X x X' we can assume r.p extends to a morphism on X. One then has a well-defined map r.p. : Zp(X)/ Zp(Y) -+ Zp(X')/ Zp(yl), which a direct technical argument shows to be a homeomorphism.

SPACES OF ALGEBRAIC CYCLES

187

The proof of Theorem 5.3 amounts to proving that a short exact sequence of groups is a principal fibration. Theorem 5.3 is quite useful. One can inductively build a space from a suitable open covering and apply the localization sequence step by step. In this way for example one can "untwist" the Suspension Theorem to get the following pretty result. Theorem 5.4. (Friedlander-Gabber [28]). Let U be a quasi-projective variety. Then algebraic suspension induces isomorphisms (5.1)

for all p ~ 2p ~ O. More generally if 11" : E -t U is an algebraic vector bundle of rank rover U, then the flat pull-back of cycles induces isomorphisms (5.2)

for all k

~

2p

~

O.

There is' a related "projective bundle theorem" which we will discuss soon. §6. Computations. With the results discussed thus far one can compute the groups L.H.(X) in a number of interesting cases. We begin with the following. A projective variety X is said to admit a cell-decomposition if there exists a nested family Xo C Xl C ... C XN = X of algebraic subsets with the property that X/c - XIc-1 is isomorphic to en. . for all k (where 0 = no ~ n1 ~ n2 ~ ••• ). Spaces of this type include : Grassmannians and in fact all generalized flag manifolds, hermitian symmetric spaces, and varieties on which a reductive group acts with isolated fixed points. Theorem 6.1. (Lima-Filho [50], [53]). Let X be a projective variety which admits a cell decomposition. Then the inclusion

is a homotopy equivalence and the natural transformation

is an isomorphism for all p

~

2p ~ O.

This represents a vast generalization of the fact that on such spaces every homology class is represented by an algebraic cycle unique up to algebraic equivalence. (This fact corresponds to the isomorphism 1ToZ. (X) ~ 1T032* (X)).

188

H. BLAINE LAWSON, JR.

Of course such a result does not hold for general projective manifolds. It is precisely for this reason that the groups L.H. are interesting. A good example where it fails is a product of elliptic curves, or more generally any abelian variety. This follows directly from Hodge theory, since the homology class of an algebraic cycle is always of type (P,p). Other examples can be constructed from the following result (cf. (I.4.2)). Theorem 6.2 (Friedlander [22], [28]). Let X be a non-singular projective variety of dimension n. Then there are isomorphisms ~

Z, Ln-1H2n-1(X) ~ H 2n- 1(XjZ), Ln-1H2n-2(X) ~ H n-1,n-1(Xj Z) L n- 1H 2n (X)

and Ln-1Hk(X)

= NS(X)

= 0 for k > 2n.

This computes the groups completely for smooth algebraic surfaces. In [28] Friedlander and Gabber extend the Algebraic Suspension Theorem to a refined intersection theorem with divisors (cf. §8). This enabled them to prove the following "projective bundle theorem".

Theorem 6.3 (Friedlander-Gabber [28]). Let E be an algebraic vector bundle of rank r over a quasi- projective variety U. Then for each p ~ r - 1 there is a homotopy equivalence r-1 Zp(JlD(E))) ~ Zp-k(U)

II

k=O

where JlD(E) denotes the projectivization of the bundle E.

A direct consequence of localization and Theorem 6.2 is the following : Theorem 6.4. Let X be a smooth projective 3-fold. Then each of the groups L1Hk(X) for k ~ 6 is a birational invariant of X. It is not unreasonable to conjecture that LpHk(X) = 0 for all p > 2 dimc(X). This would be interesting if true. IT false, then in the first dimension for which it fails one finds non-trivial birational invariants.

§7. A local-to-global spectral sequence. By using the Localization Theorem of Lima-Filho (Theorem 5.3), Friedlander and Gabber are able to construct an analogue of Quillen's local-to-global spectral sequence in algebraic K-Theory [69]. Fix a quasi-projective variety X and, as before, let X(P) denote the set of p-dimensional subvarieties of X. For each x E X (P), set L;ii;.(x) ~f limucz LrHk(U)

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SPACES OF ALGEBRAIC CYCLES

where the limit is taken over all Zariski open subsets U of x. From the localization exact sequence one constructs an exact couple which yields the following. Propostion 7.1 ([28]). Let X be a quasi-projective variety and r ~ 0 an integer. Then there is a spectral sequence of homological type of the form :

E!,q

= E9

L--::H;:q(x) ~ LrHp+q(X).

zEX(p)

Following ideas of Quillen [69] and Bloch-Ogus [8]' one can compute the E2-term of this spectral sequence. Let Crllk denote the Zariski sheaf on X associated to the presheaf

Theorem 7.2 ([28]). Let X be a quasi-projective variety of dimension n, and fix 0::5 2r ::5 k. Then there is an exact sequence of sheaves on X :

0-+ Crllk -+

EBzEX(n) iz EBzEX(n-l) iz

(L,:H;(x») -+

(L~l (x») -+ ... -+

EB zEX (n-k+2r) iz

(£;ii;..(x») -+

0

where iz (:L,:':i'f;(x») denotes the constant sheaf:L,:':i'f;(x) on x extended by zero to all of X, and the spectral sequence of 7.1 has the form E~,q

= Hn-p (X, Crlln+Q)

~ LrHp+q(X).

§8. Intersection Theory. In [28] E. Friedlander and O. Gabber succeed in extending the Algebraic Suspension Theorem to a beautiful intersection pairing defined at the level of the groups Z.. (Recall that intersection theory is conventionally defined in the quotient A.. == Z. / ,..., of cycles modulo rational equivalence (cf. [34])). This pairing enables us to define a graded commutative ring structure on L .. H.(X) for X smooth. To begin suppose ED .:; X is a line bundle associated to a divisor Don X. Let i : X '-t ED be the inclusion as the zero-section. Then composing with the homotopy inverse in (5.2) gives a map (8.1)

This represents "intersection with D". In fact if one lets Cp(X, D) denote the effective p-cycles which meet D in dimension :5 p - 1, then the restriction of (8.1) to the naive group completion of Cp(X, D) is homotopic to the intersection product (8.2)

ct-+c·D

H. BLAINE LAWSON, JR.

190

which is continuously defined as in [34]. It would be a sharper and more useful result to know that the image of the composition (8.1) consisted of cycles in the support IDI of D. In [28] this and much more are accomplished. We recall that an effective Cartier divisor on X is one which is defined by the vanishing of a regular section of a line bundle on X. Theorem 8.1 (Friedlander-Gabber [28]). Let D be an effective Cartier divisor on a quasi-projective variety X. Then for each p ~ 1 there is a canonical homotopy class of maps

which on the subgroup generated by Cp(X, D) is induced by the intersection map (8.2). The composition

( where iD : IDI <-+ X denotes the inclusion) depends only on the isomorphism class of the line bundle L = O(D). If D, D' are two such divisors then

and

Note 8.2. In [28] the authors work in the category of chain complexes localized with respect to quasi isomorphisms (maps of chain complexes inducing isomorphisms in homology). This makes the statements slightly neater and stronger. Note 8.3. Let p : IP'(E) -+ U be the projectivization of a bundle of rank r, and let LE denote the standard line bundle on IP'(E). Then the equivalence in the Projective Bundle Theorem 6.3 is given by r-l

L

k=O

r-l Cl

(L E )"

0

p. :

IT Zp_,,(U) ~ Zp(IP'(E)).

"=0

Note 8.4. The intersection pairing above leads to a general pairing Zp(X) " Div(X)+ -+ Zp-l (X) where, in the case that X is smooth, Div(X)+ £! ZI (X). The induced pairing on homotopy groups yields the operators cl(L) above (corresponding to elements L E 71'0 Div(X)+ ~ NS(X)), and also yields the operator 8 of Theorem 11.6.1 (corresponding to the generator of 71'2 Div(X)+ ~ Z).

SPACES OF ALGEBRAIC CYCLES

191

Theorem 8.1 can be generalized from divisors to subvarieties of general codimension. Theorem 8.5. (Friedlander-Gabber [28]). Let X be a quasi-projective variety and iv : V <-t X a regular (closed) embedding of codimension-q. Then for all p ~ q there is a naturally defined homotopy class of mapping

This map has the property that on the subgroup generated by the effective cycles which meet V in proper dimension, it is homotopic to the intersection-theoretic mapping c I-t c . V. These maps behave as expected with respect to composition and flat pull-back of cycles, namely

. .)! (tv 0 lV'

·1 = tv' tv .1

0

and iL 0 g* = y*

v

if g: Y -+ X is flat and IT f

rI

<-t

y: V '!2y Xx V

0

i~

-+ V is the pull-back of 9 via iv.

: X -+ Y is a morphism of varieties where Y is smooth, then the inclusion X x Y of the graph of f into the product is a regular embedding. Theorem

8.5 thereby leads to the following basic result. Theorem 8.6 (The Intersection Pairing, Friedlander-Gabber [28]).

Let

f : X -+ Y be a morphism of quasi-projective varieties where Y is smooth and of dimension n. Then if p + p' ~ n, f determines a natural pairing

In particular when X is smooth and of dimension n, one obtains a pairing (8.3)

which extends, up to homotopy, the usual intersection pairing on cycles which meet in proper dimension. This pairing is homotopy commutative and associative. Corollary 8.7. For any smooth quasi-projective variety X, the pairing L.H.(X) ® L.H.(X) --+ L.H.(X) induced by (8.3) gives L.H.(X) the structure of a bigraded commutative ring. Restricted to

EB LpH2p(X) (= cycles modulo algebraic equivalence), this is p

the standard ring structure given by intersection product.

H. BLAINE LAWSON, JR.

192

§9. Operations and filtrations. Using the complex join and the Suspension Theorem, E. Friedlander and B. Mazur [29] have introduced a ring of operators on L.H.. These operators lead to filtrations [29], [30], [23] which are compatible with, and conjecturally equal to certain standard filtrations. Throughout this section X will denote a projective variety with a fixed embedding X C IP'N. The Algebraic Suspension Theorem gives us canonical isomorphisms

Hence, we can extend our groups LpHk(X) to negative indices by setting (9.1)

for all k ~ 2p. We saw in Chapter 11.5 that L.H.. (X) is a bigraded module over the ring (9.2)

where (9.3)

are the additive generators. These homomorphisms are functorial, i.e., they are natural transformations of L.H. on the category of projective subvarieties and polorization-preserving morphisms. They constitute therefore a ring of "homology operations" which we call Friedlander-Mazur operations. The first operator h is an operator of Lefschetz type. In fact under the natural transformation c) of §2 we have a commutative diagram h

LpHk(X) ---+ Lp- 1 Hk-2(X) c)

.l.

,j,

c)

Hk(XjZ) ~ Hk-2(XjZ) where A is the Lefschetz map given by cap-product with the hyperplane class C1 (Ox (1)) (See [52], [29]). This operator h evidently depends on the projective embedding since A does. We recall that if X is smooth of dimension n, then Ak : Hn+k (X j Q) -t Hn-k (X j Q) is an isomorphism. The second operator s is a special feature of this theory. It preserves homology degree and lowers algebraic level. In fact we have commutative diagrams (cf. [23])

Hk(XjZ)

for all p and k. Note in particular that

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193

Theorem 9.1. The operation 8 of Friedlander and Mazur is a natural transformation of covariant functors. In particular it is independent of projective embedding and is also compatible with fiat pull-back of cycles and localization.

Work on the operation s developed as follows. It was introduced in [29] where it was proved that s1' : L1'H21'(X) ~ A1'(X) -t H 21' (Xj Z) is the "cycle map" which associates to an algebraic cycle (modulo algebraic equivalence) its homology class. In [52] Lima-Filho generalized this to prove that the map s1' : L1'Hk(X) -t Hk(XjZ), for any k ~ 2p, agrees with Almgren's map ([1]) and, in particular, is independent of the projective embeddings of X. Friedlander and Gabber [28] then proved that every s: L1'Hk(X) -t L1'-1Hk(X) is independent of projective embedding. In [23], Friedlander established a number of interesting properties and interpretations of this operation, some of which involve the intersection theory discussed above. Now the powers of s give rise to very interesting filtrations. Consider the case where k = 2p is even. We have the sequence of homomorphisms

(9.4)

where A1' denotes the group of algebraic p-cycles modulo algebraic equivalence. This gives us two filtrations

(9.5) (9.6)

defined by setting.

(9.7)

and

on A1' and in H21' (Xj Z) respectively. There are of course filtrations induced in the intermediate groups LjH21'(X) as well. However the filtrations above are on classically defined groups and can be compared with well-known filtrations. Note that 9",1' is the Griffiths group of p-cycles homologically equivalent to zero modulo those algebraically equivalent to zero. There is a filtration of this group due to Bloch and Ogus defined by setting

9:'? = {c E A1':

c is hamologous to zero in an algebraic subset of dimension ~ p + 1 + j EX}.

There is an analogus geometric filtration on H21'(Xj Z) defined by setting .rg~j

= span{a E H 21' (Xj Z):

a is the image of a class supported on an algebraic subset of dimension ~ 2p - j EX}.

Over Q this is the dual of Grothendieck's arithmetic filtration: on H2 1' (Xj Q). The following basic results have been proved.

H. BLAINE LAWSON, JR.

194

Theorem 9.2 (Friedlander [23]). For any projective variety X, we have

gp,; ~ g:'7 for all p,j. Theorem 9.3 (Friedlander-Mazur [29], [30]). For any projective variety X, we have

:F2p,; ~ :F2~~; for all p, j. Furthennore, the analogous result holds on odd-degree cohomology groups. Theorems 9.2 and 9.3 are in fact proved in a much stronger form. It is shown that the filtrations gp,. and :F2p,. coincide with certain geometrically defined filtrations. More specifically, in [30] one associates to a morphism f : Y --+ Cp(X) from any projective variety Y, an induced Chow correspondence homomorphism ~I : H .. (Yj Z) --+ H2p+.(Xj Z) which generalizes very classical constructions. Let :Ffp,; C H2p+;(Xj Z) be the subgroup generated by the images of all such maps. Then in [30] it is proved that

:F2p';

= ;Ffp,;.

This shows that Friedlander's functor L .. H .. , which is close to ordinary homology theory, has the property that its homologically and geometrically defined filtrations coincide. There is an analogous story for

g.. ,... In

[23] it is proved that the subgroup where f : Y --+ Cp-k(X) and c is an algebraic p-cycle homologous to zero on the projective variety Y.

gp,; is generated by cycles of the form

~/(c),

It is a classical result that the geometric filtration is subordinate to the Hodge filtration (as strengthened by Grothendieck in [39]). In particular, if X is smooth we can define

:Ft!,; = Pel (

EB

Hr,s(X))

Ir-sl:S2(p-;)

where Pc : H2p (Xj Z) -+ H2p (Xj C) is the coefficient homomorphism, and the decomposition H2p(Xj C)

=

EB

Hr,s(X) is dual to the standard Dolbeault

r+s=2p

decomposition of H 2p(XjC). Define :Fg,5 C :Fg,,; to be the pull-back by pc of a maximal Hodge substructure. (See [49] for a nice discussion.) A special consequence of the above is that

Question 9.4. Does one have equality in 9.2 and 9.3 when X is a !lmooth projective variety?

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It turns out this is not so unreasonable to ask. Theorem 9.5 (Friedlander [23]). If Grothendieck's standard conjecture B holds (cf. [38]) then for a smooth projective variety X, F2Plj

®Q

= .rg,~j ® Q

for all P and j. More modestly one might ask for an example where the filtrations Fp,j and Fp,j, consider a product of elliptic curves X = Tl X ••• x Tn. For each sequence (PI, ... ,Pn) of zeros and ones with Epj = P we have a map

9p ,j are at least non-trivial. For the case

ZPl (Td A ... A Zp" (Tn)

---+ Zp(X)

inducing a map

where E k j = k. This map commutes with the natural transformation «I> giving a diagram

(Sl;=l H"i (Tj; Z) ---+ H" (X; Z) from which one can deduce that in this case the filtrations :F.,. and and coincide with the trancendental Hodge filtration.

.r;:! agree

The non-triviality of the filtration 9. ,• is related to recent work of M. Nori, [67]. For smooth varieties X, Nori introduces a filtration on Ap(X) which he proves to be non-trivial on certain projective hypersurfaces. It is shown in [23] that 9:'j C 9p ,j, in fact 9:'j is generated in the same fashion as 9p ,j, by Chow correspondence homomorphisms associated to maps f : Y ---+ Cp _ j (X) where Y is now assumed to be smooth.

9:'.

For the motivically minded, one should mention that there is an intriguing spectral sequence of homology type defined in [23], which incorporates both filtrations F and 9. Its abutment is the associated graded of H*(X; Z) with respect to the :F-filtration, and there are isomorphisms E~;~1 ~ Ap /9p ,,,. §10. Mixed Hodge Structures. It is a fundamental and useful fact that the groups LpH" (X) ®Q carry mixed Hodge structures. This fact, due to Dick Hain, was exploited for example in the work of Friedlander and Mazur mentioned in the previous section. We recall that a Mixed Hodge Structure over Q is a finite dimensional vector space W over Q provided with an increasing weight filtration

196

H. BLAINE LAWSON, JR.

and a decreasing Hodge filtration of the space We

= W ®Q C

···Fi- 1 ;2 Fi;2 Fi- 1 ;2'"

so that (W. ® C, F. , F.) form a triple of opposing filtrations (cf. [15], (1.2.7) and (1.2.13).). A morphism of mixed Hodge structures is a linear map of rational vector spaces which is filtration preserving. The mixed Hodge structures form an abelian category MHS which is closed under tensor products. One can expand this category to include infinite dimensional vector spaces which are direct limits in MHS, with morphisms which are direct limits of morphisms from MHS. This is again an abelian category called limits of mixed Hodge structures and denoted LMHS (See [40], [41]). Deligne showed that the functor X

I-t

Hk(X; Q) takes values in MHS.

Theorem 10.1 (Dick Hain). The functors LpHk(X)®Q take values in LMHS. In other words each group LpHk (X) ® Q is naturally equipped with a direct limit of mixed Hodge structures, which is respected by the maps induced by mOFphisms of varieties. The idea of the proof, which is given in [29], is that the groups Hi (Zp(X); Q) = ~ Hi (Cp,cr(X); Q) cr

are naturally limits of mixed Hodge structures, and the homotopy groups 7r.Zp (X)® Q can be identified with the primitive elements in the Hopf algebra H. (Zp (X); Q). Since LMHS is an abelian category, the subspace of primitive elements, which the kernel of the morphism

given by a

I-t

A.(a) - a ® 1-1 ® a, is also in LMHS.

Theorem 10.2 (Friedlander-Mazur [29]). The operators 8 and h on L.H. (X)® Q, which are discussed in §9 above, respect the (limits of) mixed Hodge structures. In particular, the natural transformation 4» : LpHk(X) ® Q ~ Hk(X; Q) is a transformation of functors with values in LMHS. §1l. Chern classes for higher algebraic K-theory. Friedlander and Gabber have defined Chern classes for the higher algebraic K-groups of a variety which have values in L.H.. One of the key steps is to replace a projective variety X by an "equivalent" affine variety, i.e., an affine variety with the same K-theory and LH-theory. This affine variety, whose construction is due to Jouanolou, is the total space of an affine em-bundle 7r : Jx ~ X. If X C lPN, then J x is merely the restriction of 7r : JpN ~ lPN defined as follows. Let

JpN

= {A E MN+l,N+l

: A2

=A

and

rankA

= I}

where MN+l,N+l is the space of (N + 1) x (N + 1) complex matrices. This is defined in C(N+l)3 by the vanishing of the 2 x 2 minors and the equation

197

SPACES OF ALGEBRAIC CYCLES

=

TrA 1 ; hence it is an affine variety. We set n(A) that the fibres are affine subspaces.

=

Im(A) ErN, and note

Now since Jx is an affine variety, it is of the form Spec R where R is the ring offunctions on Jx. Quillen shows that the map K.(X) -+ K.(Spec R) = n.(BGL(R)+) is an isomorphism. The homotopy property (cf. Theorem 5.4) which comes from the Suspension Theorem and Localization, give an isomorphism L.H.(X) -+ L.H.(Spec R) (with a shift in degrees). The Projective Bundle Theorem (cf. 8.3) and ideas of Grothendieck, lead to the following. Theorem 11.1 (Friedlander-Gabber [28]). Let X be a smooth, n-dimensional quasi-projective variety. Then for all j > 0 and all i with 0 ~ i ~ n, there exist natumlly constructed Chern classes

§12. Relation to Bloch's higher Chow groups. In [5] Spencer Bloch introduced higher Chow groups for a quasi- projective variety as follows. For each k ~ 0 consider the "algebraic simplex" Ie

~[k]

= {Z E CH1

: :LZj

= I}

j=O

with combinatorial structure given as in the real case (Le., "faces" are defined by intersections with coordinate planes). For a quasi-projective variety X, let zq(X, k) denote the free abelian group generated by irreducible subvarieties of codimension-q on X x ~[k] which meet X x F in proper dimension for each face F of ~[k]. Using intersection and pull-back of cycles, one defines face and degeneracy relations in the standard way, making zq(X, *) a simplicial abelian group. Let Izq(X,*)1 denote its geometric realization, and let (:zq(X,*),8) denote the chain complex naturally associated to zq(X, *) using the additive structure of each zq(X, *). Then by definition we have

Theorem 12.1 (Friedlander-Gabber [28]). Let X be a quasi-projective variety of dimension n. Then there exist natuml homomorphisms CHn-P(X, k) ---+ L p H 2p+k(X)

for all 0

~

p :5 n and all k.

This map is induced by a map zn- p -+ Zp in the derived category of chain complexes associated to simplicial abelian groups. When X is smooth and projective they show that Zl(X, *) ® ('Ljm) and Zn-l(X) ® ('Ljm) are quasiisomorphic for any integer m > O.

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H. BLAINE LAWSON, JR.

§13. The theory for varieties defined over fields of positive characteristic. The discussion in this article has been intentionally restricted to complex varieties. Nevertheless for many results stated above there are analogues which hold for varieties defined over arbitrary algebraically closed fields. This highly non-trivial achievement is due to Eric Friedlander. The reader should see the announcement [21] and the main paper [22] for details. Very roughly the main ideas are these. Suppose X is defined over a field of characteristic p ~ 0 and L is a prime :I p. Then the Chow monoids Cr(X) are well defined, and one can construct group completions OBCr(X)l via etale homotopy theory. Taking 11'. gives L-adic.homology groups which we shall denote by LrHk(X}t. When 2r = k, this is the group of algebraic equivalence classes of r-cycles ; and when r = 0 it is isomorphic to the kth etale l-adic homology group of X. IT X is defined over C, this group is just the tensor product of LrHk(X) with the L-adic integers. It is proved in [22] that the Algebraic Suspension Theorem is valid for L.H.(X)l, and the Friedlander-Mazur operations are defined. The map sp : LpH2p(X)l -+ LoH2p(X)l = limn H2p (Xet , 'L/.en), for p > 0, is just the cycle map. One has filtrations and mixed Hodge structures as in §§9-10 above.

One of the nice features of these groups is that they are Galois modules. Suppose X is defined over a field F and is provided with an embedding X C PW. Let K be the algebraic closure of F. Then Gal(K/F) acts naturally on L.H.(XK)l, and the operations and cycle maps discused above are all Gal(K/ F)-equivariant. So also are the maps f. : L.H.(XK)l -+ L.H.(YK)l induced by a morphism f : X -+ Y of varieties over F. §14. New directions. There have been some recent enhancements of the above LH-constructions which are both algebraically and geometrically more sophisticated but, of course, less manageable than the basic theory. The first is due to Friedlander and Gabber [28] who construct functorial spaces where 11'0 gives algebraic cycles modulo rational equivalence instead of the coarser algebraic equivalence. Their theory is therefore a "rational equivalence analogue" of LH-theory. The basic idea is to consider the simplicial monoid .rp(X) = M or(a[.], Cp(X)) where ark] is the algebraic simplex mentioned in §12 above. In the case p = 0, this becomes the Suslin complex .ro(X) = SUB.(X) of algebraic singular chains of the infinite symmetric product of X. It has recently been shown by Suslin and Voevodsky [76] that for all n,

H.(SU8.(X)j 'L/n)

~

H.(Xj 'L/n),

giving an algebraic computation of the singular homology of the variety. Furthermore the result extends to higher dimensional cycles to prove that

H.(Fp(X)j 'LIn)

~

LpH.(Xj 'LIn),

for all p ~ O. (See [24], [76].) In [24] the groups H.(Fp(X)) are computed for dim(X) - 1.

p =

SPACES OF ALGEBRAIC CYCLES

199

The rational theory of Friedlander-Gabber actually has a bivariant formulation in analogy with the constructions of the next section (see [90]). There has also recently been work of P. Gajer aimed at constructing intersection versions of L.H.(X). He has found workable definitions and has succeeded in formulating and proving an intersection homology version of the Dold-Thom Theorem [36]. P.Gajer and C.Flannery have also established the LH-groups [87], [91].

Chapter V - The Functor L· H* (Morphic Cohomology) Recently E. Friedlander and the author [31], [32] introduced the notion of an effective algebraic co cycle on a variety X as a morphism