A balanced proper modification of P 3

Comment. Math. Helvetici 66 (1991) 505-511

0010-2571/91/040505-0751.50 + 0.20/0 9 1991 Birkh~iuserVerlag, Basel

A balanced proper modification of/)3 LUCIA ALESSANDRINI AND GIOVANNI BASSANELLI*

I. Introduction Since a p r o p e r modification M o f a K~ihler m a n i f o l d is not necessarily K~hler, it is n a t u r a l to ask what kind of weaker geometrical properties are preserved in this situation. In particular we are interested in the existence o f a balanced metric on M, i.e. a metric such that the trace o f the torsion o f the Chern connection vanishes. We p o i n t out that balanced manifolds are, in some sense, d u a l to Kfihlerian ones: in fact, a metric is balanced if its K/ihler form is " c o - c l o s e d " (see Definition 3.1); moreover, balanced n-dimensional manifolds are characterized by means o f their (n - 1, n - 1)-currents as well as K~ihler manifolds can be characterized by (1, 1)-currents (see [Mi] and [HL]). The classical example o f H i r o n a k a ([Hi] or IS]) o f a M o i s h e z o n non algebraic manifold X is given by a p r o p e r modification o f P3; in this note we prove that X is balanced. F r o m a geometrical point o f view, the obstruction for X to be K/ihler is a curve h o m o l o g o u s to zero on the exceptional divisor E: in a dual manner, our p r o o f lies in showing that if E is not h o m o l o g o u s to zero then M is balanced. This result is a first c o n f i r m a t i o n o f the conjecture: " E v e r y p r o p e r modification o f a K/ihler manifold is b a l a n c e d . " A b o u t techniques, since positive currents which are ( p , p ) - c o m p o n e n t s o f b o u n d a r i e s are not necessarily closed (see Question on p. 170 in [HL]), we c a n n o t apply the well k n o w n results on (locally) flat currents, and therefore we must develope a parallel technique for positive d0--closed currents.

*This work is partially supported by MPI, 40%. The second author thanks the Department of Mathematics of the University of Trento for its kind hospitality during the preparation of this paper.

506

LUCIA ALESSANDRINI A N D GIOVANNI BASSANELLI

2. Preliminaries

Let M be a connected complex manifold of dimension n; we denote as usually by ~P the sheaf of germs of holomorphic p-forms and by gP the sheaf of germs of ~ p-forms. Moreover, let ~P'P(M)R denote the Fr6chet space of real (p, p)-forms, while 8~.p(M)R denotes its dual space of real currents of bidimension (p, p). We will say that T ~ g'p.p(M)R is the (p, p)-component o f a boundary if there exists a current S such that for every ~o e gP'P(M), T(~o) = S(d~o), i.e. there exists a (p, p + 1)-current S such that T = ffS + OS. Let V be a complex vector space of dimension n, p an integer and ~p.'= iP22 P. An element of A p'~ that can be expressed as v~ ^ 9 9 9 ^ vp with vj~ Vis called simple. 2.1 D E F I N I T I O N . ([AA] and [AB]) ~ E o~P'P(M)R is called transverse if Vx e M, Vv ~ AP(T'~M), v r 0 and simple, it holds O x ( a i l v ^ ~) > O. T ~ g'p.p(M)a is called positive if T(O) > 0 for every transverse (p, p)-form Q. 2.2 R E M A R K . ([H1] p. 325-326) A positive (p,p)-current T has measure coefficients, and moreover, if [tTt[ denotes the mass of T, there exists a HT[[-measurable function ir such that ][iPx [I = 1 for HTI[-a.e. x ~ M, and

= f (L)II TH for all test forms r Let us prove now our main technical tool. 2.3 T H E O R E M . Let M be a complex n-dimensional manifold, and let T be a positive (p, p)-current on M such that d J T = 0 and the Hausdorff 2p-measure o f supp T, H2p(T), vanishes. Then T = O. 2.4 R E M A R K S . In this statement, M is not assumed to be compact. If moreover T is closed, the result is well known (see for instance [F]). P r o o f o f Theorem 2.3. Let x0~ supp T; as in [H2] p. 72 we can choose a coordinate neighbourhood (U, Z l , . . . , zn) centered at x0 such that for every increasing multiindex/, [I I = p, it holds locally: (a) if nz is the projection defined by T~I(Z 1 . . . . . Z n ) = (Zil . . . . . Zip), supp TC~ Ker n, = {Xo} (b) there exist two balls A'I~CP, A ~ C n-p such that ( A ' , x b A T ) c ~ supp T = ~ . This implies that n~ [suppr : supp T c~(d) x A 7 ) ~ A ) is a proper map.

A balanced proper modification of P~

507

Fix an I, I I [ = p, and call T,,= (z~, Isupp T). ( r ) ; 7", is a OJ-closed (p, p)-current on A), hence it can be identified with a OJ-closed distribution fr on A); since J) is pluriharmonic, it is smooth. But 7zI is a Lipschitzian map, and therefore H2p(suppf~) = 0, i.e. J) = 0. Let

c o , = a ~ d z ~ AdZ~ in B . ' = ~ (A} x AT). I

Since

T is positive, in order to prove

T=0

it is enough

to verify that

TO~B(e)P/p!)) = 0. But in B o) p

p! - ~, ap dz, ^ dS, = ~, (Ts

•I

A

de, ),

therefore

T Zo

<- ~ T(Z~I • ~i,(xz )*(t~p dz, ^ dY, )) = i

flap dz, A dY I = O.

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i

3. Hironaka's example Let us recall the definition o f balanced manifold, and the characterization o f them by means o f positive currents. 3.1 D E F I N I T I O N . ([Mi]) A balanced manifold M is a compact complex n-dimensional manifold which satisfies one o f the following equivalent conditions: (i) M admits a hermitian metric h such that, if o) is the K~ihler form of h, dco.- l = 0 . (ii) M admits a hermitian metric h such that, if ~o is the K/ihler form of h and 6 is the formal adjoint of d in the metric h, it holds 3e) = 0. (iii) M admits a hermitian metric h such that the torsion 1-form rh of the canonical hermitian connection of h is zero. (iv) there are no non trivial positive (n - 1, n - 1)-currents on M which are (n - 1, n - l)-components of boundaries. Let us recall also briefly the construction o f Hironaka's example, which can be found in his Thesis [Hi] or, for instance, in [S].

508

LUCIA ALESSANDRINI AND GIOVANNI BASSANELLI

3.2 Consider the projective space P3 with coordinates (x,y, z), and in it the curve C of equation

{

y2 ~ X 2 ~ X 3

z=0

In a little ball near zero, blow up one branch o f C first, then the other; outside of the origin, just blow up C - {O}. Then glue together to obtain the compact complex manifold X and the holomorphic m a p

f : X --~P3; there exists an analytic subvariety E of X such that f I x - e : X - E -~ P3 - C is biholomorphism. 3.3 T H E O R E M .

The 3-dimensional Hironaka's manifold X described in 3.2 is

balanced. Proof. Let T be a positive (2, 2)-current on X which is the (2, 2)-component of a boundary, and let co be a K/ihler form for P3. f * ~ is a closed real (1, l ) - f o r m which is positive semidefinite, so that from

o

= T(po ) = f(f*co) (L)tl

we get E ~_ supp T.

We need now the following lemma, that we shall prove later. 3.4 L E M M A . Let f : X ~ P~, C, E, T as before. Then there exists k e C such that T = k[E]. ([El denotes the current defined by [E](t~) = ~RegE q5 for all test forms The previous lemma tells us in particular that dT = 0, and therefore T

= 0 S 2'3 + 8 5 2 . 3

for a (2, 3)-current S 2'3 with 8FS2'3 = 0. c3S2'3 becomes a holomorphic 2-form on X, hence the equality

f SS2,3 ^ tgX2,3 ^ f*co = f d(S2,3 ^ OS2,3 A f*~o) =O

A balanced proper modification of P~

509

proves that OS 2"3 vanishes, and we get T

= d ( S 2'3 4- 8 2 ' 3 ) .

If k were different from zero, we should conclude that the exceptional set E is h o m o l o g o u s to zero, which is not the case (for a p r o o f o f this fact, use a Mayer-Vietoris argument, as is done for example in [Mfi]). In this way we conclude T = 0, i.e. X is balanced. Z]

Proof of Lemma 3.4. Since dim E = 2, for every x e Reg E we can choose a c o o r d i n a t e n e i g h b o u r h o o d (U, z~, z2, z3) o f x such that E c~ U = {z

e

U/z

3 = 0}.

As in ([HL] L e m m a 32), we get in U 3

T =

Z

^

~,fl = 1

where the measure t ~ can be written as

t,B(z ) = Gp(zl, z2) | b0(z3). Since T is a (2, 2 ) - c o m p o n e n t o f a b o u n d a r y , 0 = 8•T =

a~

~

( - - ~ 8,~t~ + ~6 8~t~7 + ~ ~t6/~ -- O~ ~t6~.) dG ^ dz~ ^ ds

^ d2~.

a < &/:r < ~

For 7=6=3

we get

rma=0 r~

fora, fl~{1,2} is h o l o m o r p h i c for ~ e {1, 2}

r33

is p l u r i h a r m o n i c

I

As (r~tr) is a non negative matrix, it follows r~s = 0 for a ~ {1, 2} and therefore T = o'l(r33(Zl, z2) (~) 60(z3)) dz3 A d773 in U.

510

LUCIA ALESSANDRINI

AND GIOVANNI

BASSANELLI

In this way we have proved that there exists a non negative pluriharmonic map h : Reg E ~ R such that

Tlx

Singe

=

h[Ellx- sm~E.

The next step is to prove that h is constant. For every x0 e C, x0 r O, h~--i(xo) is harmonic and therefore constant, so that we can project h to a non negative harmonic map h l : C - {O}--*R. The following parametrisation o f C x(t) = t 2 - 1

z = 0

y(t) = t(t 2 - 1)' induces biholomorphisms C-

{O} _---PI - {a, b} _-__C -

{O}

which allow us to use the following remark (see [B] p. 402-403): "Let B(O, b) be the ball o f center O and radius b > 0 in C, and let g : B ( O , b ) - {O}--*R be harmonic and non negative. Then there exist a constant A and a holomorphic function F in B(O, b) such that

g(z) = Re F(z) + A log

Izl for

z ~ B(O, b) - {O}."

Let h i ( z ) = Re F ( z ) + A log [z[; F and A are independent on the radius o f the chosen ball and moreover, replacing z by 1/w, we get F(z) = constant. Since hi is non negative, it must be A = 0, hence hi is constant and h is constant too. Now, the two currents T and h[E] are 8~--closed and coincide on X - Sing E; if we prove that T - h [ E ] is positive, the statement will follow from Theorem 2.3. To do this, let f2 be a transverse (2, 2)-form, let x E Sing E, let K be a compact neighbourhood o f x, let {V,} be a fundamental family o f neighbourhoods of K c~ Sing E. Since

h fE~(K- v,,) f2 = T(ZK - v, f2) ~ T(;~K Si,ge Q) we get

h f e ~ x 12 < T(Z x _ si,g e t2) < T(ZxI2).

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A balanced proper modification of P3

511

REFERENCES [AA] L. ALESSANDRINI and M. ANDREATTA, Closed transverse (p,p)-forms on compact complex manifolds, Compositio Math. 61 (1987) 181-200. Erratum ibidem 63 (1987) 143. lAB] L. ALESSANDRINI and G. BASSANELLI, Compact p-Kdhler manifolds, Geom. Ded. 38 (1991) 199- 210. [B] R. B. BURCKEL, An Introduction to classical Complex Analysis, vol. 1 Pure and Applied Mathematics, Academic Press, New York-San Francisco, 1979. [F] H. FEDERER, Geometric Measure Theory, Springer Verlag, New York, 1969. [H1] R HARVEY, Holomorphic chains and their boundaries, Proc. Symp. Pure Math., 30 Part 1, AMS Providence R.I., (1977) 309 382. [H2] R. HARVEY, Removable Singularities for positive currents, Amer. J. Math. 96 (1974) 67 78. [HL] R. HARVEY and J. R. LAWSON, An intrinsic characterization of Kiihler manifold~, Inv. Math. 74 (1983) 169-198. [Hi] H. HmONAKA, On the theory ofbirational blowing up, Thesis, Harward 1960, unpublished. [Mi] M.L. MICHELSON, On the existence of special metrics in complex geometry, Acta Math. 143 (1983) 261 295. [Mfi] S. MOLLER-STACH, Bimeromorphe Geometric yon dreidimensionalen Moishezonmannigfaltigkeiten, Diplomarbeit, Bayreuth 1987. [S] I . R . SHAFAREVICH, Basic Algebraic Geometry, Springer Verlag, Berlin Heidelberg New York, 1977. Dipartimento di Matematica Universitd di Trento 1-38050 PO VO (Trento) (Italy) and Dipartimento di Matematica e Fisica Universitd di Camerino 1-62032 CAMERINO (Macerata) (Italy) Received February 5, 1990