A Characterization of Affine Hyperquadrics

Annali di Matematica 184, 555–564 (2005) Digital Object Identifier (DOI) 10.1007/s10231-004-0132-6

Giorgio Patrizio · Adriano Tomassini

A characterization of affine hyperquadrics
Received: April 15, 2003; in final form: May 12, 2004 Published online: April 15, 2005 – © Springer-Verlag 2005 Abstract. In this paper, using the existence of special exhaustions that satisfy the complex homogeneous Monge–Ampère equation and curvature properties are given characterizations of the affine hyperquadric and other special Stein manifolds. Mathematics Subject Classification (2000). 32W20, 32V40 Key words. Monge-Ampère foliations – adapted complex structures – Grauert tubes

1. Introduction The compact rank one symmetric spaces – i.e., spheres, projective spaces over the real, the complex and quaternionic numbers and the Cayley projective plane – admit canonical complexifications as affine algebraic manifolds. These Stein manifolds have been characterized by Morimoto and Nagano [14] as complex manifolds with a group acting as automorphisms such that the generic orbit is a hypersurface. There is yet another point of view in studying these classes of manifolds (see, e.g., [15]). For each compact rank one symmetric space S this complexification M(S) is diffeomorphic to the tangent bundle of S so that S sits in M(S) as the zero section. The complex structure is such that there exists a strictly plurisubharmonic exhaustion ρ : M → [1, +∞) such that u = cosh−1 ρ satisfies the complex homogeneous Monge–Ampère equation (1.1)

(dd c u)n = 0.

Furthermore, the metric on M(S) with Kähler form dd c ρ is restricted to the standard metric on the symmetric space S; in fact, each geodesic of S is complexified in M(S), and its complexification is exactly (the closure of) a leaf of the Monge– Ampère foliation associated to the solution u = cosh−1 ρ of (1.1). This type of complexification has been extensively studied in recent years, and in this sense M(S) is referred as a Grauert tube [6] and the special complex structure is called an adapted complex structure [12]. In particular [17] and [18], adapted complex G. Patrizio: Dipartimento di Matematica “U.Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy, e-mail: [email protected] A. Tomassini: Dipartimento di Matematica, Università di Parma, Via Massimo d’Azeglio 85/A, 43100 Parma, Italy, e-mail: [email protected] 

Research partially supported by MURST and by INdAM.

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structures exist on the full tangent bundle of compact, normal Riemannian homogeneous spaces (see also [1] for more examples). Here, replacing the group action geometry and the conditions on orbits with the Monge–Ampère geometry and curvature conditions, we obtain characterizations for the affine hyperquadrics with a flavor similar to the Morimoto–Nagano result. After some preliminaries about the curvature of Kähler manifolds, in Section 3 we give an account of the curvature properies of complexifications of generic rank one compact symmetric spaces using the approach suggested by the Monge– Ampère geometry. In Section 4 characterizations of the affine hyperquadric among unbounded Grauert tubes (Theorems 4.1 and 4.2) are given. Similar results hold for quotients of Cn and of the unit ball in Cn (Theorems 5.1 and 5.2). 2. Holomorphic sectional and bisectional curvatures of Kähler metrics We begin by recalling the definitions of holomorphic sectionals and bisectional curvatures of Kähler metrics and stating some useful relations. Let M be a complex manifold with complex structure J and let h be a Kähler metric on M. The curvature tensor R of the metric h satisfies the following algebraic identities: (2.1) (2.2) (2.3)

R(X, Y, Z, W ) = −R(Y, X, Z, W ) = −R(X, Y, W, Z ), R(JX, JY, Z, W ) = R(X, Y, JZ, JW ) = R(X, Y, Z, W ), R(X, Y, Z, W ) = R(Z, W, X, Y ),

and the first Bianchi identity (2.4)

R(X, Y, Z, W ) + R(X, Z, W, Y ) + R(X, W, Y, Z ) = 0,

where X, Y, Z, W are real tangent vectors to M. If X and Y are two real h-orthogonal unit tangent vectors on M, then K(X, Y ) = R(X, Y, X, Y ), K H (X ) = R(X, JX, X, JX ), B(X, Y ) = R(X, JX, Y, JY ) define, respectively, the sectional curvature of the plane spanned by the (real) vectors X and Y , the holomorphic sectional curvature of the J-invariant plane spanned by the vectors X, JX, and the holomorphic bisectional curvature of the J-invariant planes spanned respectively by the vectors X, JX and Y, JY . We will need a result that is algebraic in nature relating sectional curvature, holomorphic sectional, and bisectional curvatures. Such a result is probably well known and is implicitly used in the proofs of [7] (for a similar formula, see [4]). For the reader’s sake we state it explicitly and indicate its proof here: Lemma 2.1. Let (M, J, h) be a Kähler manifold and X and Y any two h-orthogonal unit real tangent vectors on M. Then (2.5)




X +Y X −Y 4K(X, Y ) = −K H (X ) − K H (Y ) + 4B , X + Y  X − Y 

 + 2B(X, Y ) .

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√ Proof. If X = Y  = 1 and h(X, Y ) = 0, then X + Y  = X − Y  = 2 and h(X + Y, X − Y ) = 0. A straightforward computation using the definitions and (2.1)–(2.3) shows that
 X +Y X −Y 4B , = R(X + Y, J(X + Y ), X − Y, J(X − Y )) X + Y  X − Y  = K H (X ) + K H (Y ) + 2B(X, Y ) − 4K(X, JY ) . The first Bianchi identity (2.4) and (2.2) give K(X, JY ) = R(X, JY, X, JY ) = −K(X, Y ) + B(X, Y ) . By substituting the last expression for K(X, JY ) in the previous one, we get the formula for K(X, Y ).   3. Curvature of the complexification of compact rank one symmetric spaces Let M be a connected complex manifold of complex dimension n, and suppose that ρ : M → [1, R), 0 < R ≤ ∞ is a strictly plurisubharmonic function of class C ∞ such that, if S = { p ∈ M | ρ( p) = 1} and u = cosh−1 ρ ∈ C 0 (M) ∩ C ∞ (M \ S), then u is plurisubharmonic and satisfies on M \ S (3.1)

(dd c u)n = 0.

The pair (M, ρ) has a rich geometry that has been investigated in detail (see, for example, [3], [11], [12], [15]). We quickly recall the properties that will be useful for our purposes. Under the assumptions above it follows that in fact ρ is of class C ω and the minimal set S = { p ∈ M | ρ( p) = 1}, which we refer to as the center of M, is a totally real, C ω submanifold of M of real dimension n. The annihilator of the form dd c u is an integral distribution of constant complex rank 1 and, in fact, is the trivial line subbundle of T 1,0 (M \ S) spanned by the complex gradient Z of ρ (with respect to the metric dd c ρ). In local coordinates ∂ . ∂z α The integral curves of Z are Riemann surfaces and provide a foliation for M \ S, called the Monge–Ampère foliation associated to u. The metric defined by dd c ρ plays a very special role. The center S with the induced metric is a totally geodesic submanifold. Furthermore, S is a Lagrangian submanifold of M, meaning that if J is the complex structure of M, then, if X ∈ T p S for some p ∈ S, JX is orthogonal to T p S. The leaves of the Monge–Ampère foliation are also totally geodesic submanifolds. The closure of each leaf meets the center S along (the image of) a complete geodesic of S. In fact, there is a one-to-one correspondence between the oriented complete geodesic of S and the leaves of the Monge–Ampère foliation, and the tangent bundle TS is diffeomorphic to M (see [15] or [12] for an alternative presentation). (3.2)

¯

Z = ραβ ρβ¯

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Finally, the holomorphic sectional curvature along each leaf was computed in [15]:
 Z 2 (3.3) K Hol =− 3 Z ρ (notice that here we are using different normalizations and hence we need to multiply by a factor of 2 the result of the computation in [15]). In the special case of the complex affine hyperquadric, this computation follows, for instance, from curvature formulas for hypersurfaces in Vitter ([19]), which allow one also to compute the holomorphic sectional curvature in the directions contained in the holomorphic tangent bundle to any level set of ρ, showing that these curvatures are also functions of ρ only. A simple computation, based on the fact that (3.1) holds, shows that at every point p ∈ M \ S the holomorphic tangent space Ker∂ρ( p) is exactly the orthogonal complement of Z( p) with respect to the metric dd c ρ, i.e., Ker∂ρ( p) is orthogonal to the direction tangent to the leaf of the Monge–Ampère foliation that passes through p (see, e.g., [15]). In fact this behavior of the holomorphic sectional curvature is shared by a larger class of examples. Let S be a generic compact rank one symmetric space, i.e., S is either the unit sphere Sn or the real projective space RP n or the complex projective space CP n or the quaternionic projective space HP n . Let us also denote by M(S) the standard affine algebraic complexification of S (see [14] or [15] for examples). Then there exists a strictly plurisubharmonic function ρ : M(S) → [1, +∞) of class C ∞ with (dd c cosh−1 ρ)n = 0 on M(S) \ S so that S is the center of M(S). The metric dd c ρ is equal to the restriction of the euclidean metric of the ambient space and is invariant under the natural Lie group action defined on M(S). In fact the holomorphic sectional curvature also in this case is constant along the holomorphic tangent bundle to a level set of the exhaustion, a result that we were unable to find in the literature. We have the following: Theorem 3.1. Let M(S) be the complexification of a generic compact rank one symmetric space S (i.e., S is not the Cayley projective plane), and let ρ : M(S) → [1, +∞) be its associated exhaustion function. (i) (3.4)

If p ∈ M(S) \ S and W ∈ T p1,0 M \ {0}) with ∂ρ p (W ) = 0, then K Hol (W )( p) = −

2 . ρ( p)

If M(S) = Qn is the complex hyperquadric in Cn+1 , then (ii) The holomorphic bisectional curvature of the metric dd c ρ vanishes in the directions tangent to level sets of ρ; (iii) The Riemannian sectional curvature of planes spanned by a real vector orthogonal to the leaves of the Monge–Ampère foliation and by a real vector tangent to a leaf is constant on each level set of ρ. Proof. Let p ∈ M \ S and W as in the hypothesis. If Y is the real part of W, i.e., W = Y − i JY , it is not restrictive to assume that Y is a unit vector. We start

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by recalling that there is a unique (extended through S) leaf L of the Monge– Ampère foliation in M, through the point p. Then (see [15] for example) L can be parameterized by means of the exponential map for the metric dd c ρ. In fact, there exists a unit speed geodesic γ : R → S such that   L = Expγ(t) (−sJ γ˙ (t)) | s, t ∈ R and with p = Expγ(0) (−s0 J γ˙ (0)) for a suitable s0 > 0. Let p0 = γ(0). If Y(s) is the parallel transport of the real part Y = Y(s0 ) of the vector W along the geodesic s → Expγ(0) (−sJ γ˙ (0)) , then set Y0 = W(0) ∈ T p0 S. Fix also X 0 = γ˙ (0) ∈ T p0 S. If Z is the complex gradient of ρ, by its very definition we have (3.5)

γ

dd c ρ(Z, W ) = ραβ¯ z α wβ = ραβ¯ ραγ¯ ργ¯ wβ = δβ ργ¯ wβ = ρβ¯ wβ = 0 ,

since, by hypothesis, ∂ρ(W ) = 0. Therefore, as X 0 is the parallel transport along the same geodesic of the normalization of the real part Z( p), the vectors X 0 and Y0 are also orthogonal. Now, if S is a generic compact rank one symmetric space and X 0 and Y0 are two orthogonal unit vectors of T p M, then there exists a totally geodesic sphere S2 ⊂ S such that p ∈ S2 and T p S2 =Span{X 0 , Y0 } (see [2]). Therefore, if   Q 2 = Expσ(t) (−sJ σ˙ (t)) | σ(t) is a normal geodesic of S2 , s, t ∈ R , then Q 2 is a two-dimensional hyperquadric totally geodesically sitting in M and p ∈ L ⊂ Q 2 . By construction, if ρˆ = ρ| Q2 , then W = W(s0 ) ∈ T p1,0 (Q 2 \ {0}) with ∂ ρˆ p (W ) = 0. Since the quadric Q 2 is totally geodesic in M, we have that the curvature of the ambient space (M, dd c ρ) is the same as that of the submanifold (Q 2 , dd c ρ), ˆ i.e., K ddc ρˆ (W ) = K ddc ρ (W ) . Therefore, it will be sufficient to prove our statement for   M = Q2 = (z 1 , z 2 , z 3 ) ∈ C3 : z 21 + z 22 + z 23 = 1 . But this becomes a direct computation that can be performed, for instance, using formula (ii) of Proposition 1 of [19] for complex hypersurfaces in Cn+1 . In fact, the complex gradient vector field Z can be explicitly computed. In this case ρ(z) := z2| 2 ,   being the euclidean norm of C3 . Then (see [15] for instance): Q

Z=

3 1 ∂ (ρz α − z¯α ) | 2. ρ ∂z α Q α=1

If Wˆ is the vector field defined by   1 ∂ ∂ ∂ ˆ :=  W (z¯2 z 3 − z 2 z¯3 ) + (z¯3 z 1 − z 3 z¯1 ) + (z¯1 z 2 − z 1 z¯2 ) , ∂z 1 ∂z 2 ∂z 3 ρ2 − 1 ˆ is a norm 1 holomorphic tangent vector a straightforward computation shows that W field on the quadric Q2 , which is dd c ρ-orthogonal to Z (see, for instance, [15]).

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But then Wˆ ( p) is parallel to W and hence, using the above-mentioned curvature formulas of [19] for complex hypersurfaces in Cn+1 , we get ˆ ( p)) = − K(W ) = K(W

2 . ρ( p)

Finally, if M(S) = Qn is the complex hyperquadric in Cn+1 , then claim (ii) follows from formula (iii) of Proposition 1 of [19]. Claim (iii) can be derived from Vitter’s computation of real sectional curvature (formula (i) of Proposition 1 of [19]) or arguing as follows. If, for p ∈ M \ S, X ∈ T p M is a real vector tangent to the leaf through p and Y ∈ T p M is orthogonal to the leaf through p, then X ⊥ Y and X ⊥ JY , so that X + Y ⊥ X − Y and X + Y ⊥ J(X − Y ). Then, using (ii),
 X +Y X −Y 4B , = 2B(X, Y ) = 0, X + Y  X − Y  from which the conclusion is immediate using (3.3), (3.4), and formula (2.5).

 

For the exceptional compact rank one symmetric space, the projective plane over the Cayley numbers, the above proof does not work as it is not true that one can find totally geodesic 2-spheres through any point. In fact, we expect that the result will turn out not to be true as the isotropy group at every point does not have the right dimension. 4. Characterizations of the hyperquadric In this section we shall describe how the curvature properties outlined above may be used to characterize the complexifications of the affine hyperquadrics. A characterization of the affine hyperquadrics in terms of the “constancy” properties of the curvature illustrated in Theorem 3.1 is provided by the following: Theorem 4.1. Let M be a connected, simply connected complex manifold of dimension n. Then M is biholomorphic to the affine hyperquadric   Qn = (z 1 , . . . , z n ) ∈ Cn | z 21 + · · · + z 2n = 1 . if and only if there exists a strictly plurisubharmonic exhaustion function ρ : M → [1, ∞) of class C ∞ such that u = cosh−1 ρ satisfies (dd c u)n = 0 on {ρ > 1} and either: (i)

The bisectional curvature of the metric dd c ρ is constant in the directions tangent to the minimal set S of ρ;

or (ii) For all level set Sr = {ρ = r}, r > 1, the holomorphic sectional curvature of the metric dd c ρ is constant in directions in the holomorphic tangent bundle to Sr ; and (iii) The Riemannian sectional curvature of planes spanned by a real vector orthogonal to the leaves of the Monge–Ampère foliation and by a real vector tangent to a leaf is constant on each level set of ρ.

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Proof. It is clear that if M is biholomorphic to Qn , then the other conditions hold. Suppose that on M there exists a strictly plurisubharmonic exhaustion function ρ : M → [1, ∞) of class C ∞ such that u = cosh−1 ρ satisfies (dd c u)n = 0 on {ρ > 1} such that (i) holds. The manifold S is the minimal set of the exhaustion ρ. Hence, in particular S is compact and simply connected as, under the assumption, ρ is a Morse function (see, e.g., [15]). The Riemannian metric g induced on S by dd c ρ has constant Riemannian sectional curvature. To prove it, we have to show that for some constant k, given any p ∈ S and X, Y ∈ T p S, orthogonal unit vectors Kˆ (X, Y ) = k, where Kˆ denotes the curvature of the submanifold S. As S is totally geodesic in M, it is enough to prove K(X, Y ) = k, where K denotes the Riemannian curvature of the metric dd c ρ. This is an immediate consequence of the hypothesis and of Lemma 2.1 and (3.3). As it is compact and simply connected, S necessarily has constant positive curvature and hence is isometric to a sphere. In fact, it is not restrictive to assume that S has Riemannian sectional curvatures equal to one and hence is isometric to the unit sphere. This amounts to replace ρ = cosh u with ρk = cosh ku for a suitable constant k as in this case, if g and gk are the metrics induced on S by dd c ρ and dd c ρk respectively, then gk = k 2 g. But then as an immediate consequence of Theorem 5.2 of [15] (or of Theorem 4.2 in [12]) it follows that M is biholomorphic to Qn . To complete the proof, it is enough to show that assuptions (ii) and (iii) imply (i). We have to show that there exists a constant k such that, given any p ∈ S and X, Y ∈ T p S, orthogonal unit vectors B(X, Y ) = k, where B is the bisectional curvature of the metric dd c ρ. To take advantage of Lemma 2.1 and of the hypothesis, we shall compute the bisectional curvature along S by taking limits. Let p ∈ S and X, Y ∈ T p S be orthogonal unit vectors. There exists a unique unit speed geodesic γ : R → S such that γ˙ (0) = X. Let X(s) and Y(s) be the parallel transport along the geodesic s → Expp (−sJX) respectively of the vectors X = X(0) and Y = Y(0). Then for every s ≥ 0, X(s) ⊥ Y(s) and Z(s) = X(s) − i JX(s) ⊥ W(s) = Y(s) − i JY(s). Now, along the geodesic σ(s) = Expp (−sJX), the vector field −i JX(s) is equal to the normalized real gradient vector field Y of ρ with respect to the metric dd c ρ, and hence it is known that Z(s) = X(s) − i JX(s) is tangent to the extended leaf   L = Expγ(t) (−sJ γ˙ (t)) | s, t ∈ R (for details of the computation based on the fact that ρ satisfies (3.1), see, for instance, the statement and proof of Theorem 3.1 of [15]). On the other hand, if Z(s) = X(s) − i JX(s) and W(s) = Y(s) − i JY(s), then Z(s) ⊥ W(s), and hence the same computation as in 3.5 shows that ∂ρ(W(s)) = 0. But then as the holomorphic sectional curvature along the leaves is a function of ρ alone (Theorem 3.2 in [15]), using the hypothesis and the formula (2.5) of Lemma 2.1, it follows that the value of B(X(s), Y(s)) depends only on the level set of ρ. Therefore, taking s → 0 as the limit, it follows that B(X, Y ) = B(X(0), Y(0)) is independent of p and of the choice of orthogonal X, Y ∈ T p S, as claimed.  

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As a consequence of well-known pinching theorems for positively curved Riemannian manifolds (see, e.g., [10]) and from a differentiable sphere theorem due to Suyama([16]), we have this characterization under relaxed conditions on the holomorphic bisectional curvature. Theorem 4.2. Let M be a connected, simply connected complex manifold of dimension n, and suppose that there exists a strictly plurisubharmonic exhaustion function ρ : M → [1, ∞) of class C ∞ such that u = cosh−1 ρ satisfies (dd c u)n = 0 on {ρ > 1}. If − 15 ≤ B(X, Y ) ≤ 0 for any p ∈ S = S1 and X, Y ∈ T p S, then diam(S) ≥ π and M is diffeomorphic to the affine hyperquadric Qn . If, furthermore, diam(S) = π, then M is biholomorphically isometric to the affine hyperquadric Qn . Proof. Formula 2.5, together with the assumptions on the bisectional curvature of the metric dd c ρ and 3.3, implies that if K is the Riemannian sectional curvature of S, we have 0.7 ≤ K ≤ 1, and hence Suyama’s differentiable sphere theorem ([16]) implies that S is diffeomorphic to the sphere Sn . As M is diffeomorphic to the tangent bundle of S, it follows that M is diffeomorphic to the affine hyperquadric Qn . As S is 0.7pinched, from Theorem 2.8.7 in [10] we can conclude that diam(S) ≥ π and that if diam(S) = π, then S is isometric to a compact symmetric space of rank one. Since S is diffeomorphic to a sphere, the only possibility is that S is isometric to a sphere and therefore M is biholomorphically isometric to the complex hyperquadric Qn .   Remark. Under weaker curvature conditions for M as in Theorem 4.1 one may still gather topological information. If it is assumed that − 21 ≤ B(X, Y ) ≤ 0 for any p ∈ S = S1 and X, Y ∈ T p S, then it follows that 14 ≤ K ≤ 1. Then necessarily (Theorem 2.6.A.1 in [10]) diam(S) ≥ π. If diam(S) = π, then S is isometric to a compact symmetric space of rank one and, as a consequence, it may be concluded that M is biholomorphically isometric to the complexification of a compact symmetric space of rank one. If diam(S) > π, using Theorems 2.8.7 and 2.8.10 in [10], we can conclude that S is homeomorphic to the unit sphere Sn . Unfortunately, in general, from this it does not follow that the corresponding tangent bundles are homeomorphic ([13]). On the other hand, no example of an exotic sphere with strictly positive curvature is known, although recent results show that the curvature may be positive almost everywhere ([20]).

5. Final remarks In a similar vein, it is possible to prove characterizations of complexifications of nonpositive constant curvature Riemannian manifolds. Here we outline the results that can be proved along the same lines of the characterization of the affine

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hyperquadrics. Using the description provided for instance in [9], we briefly illustrate the assumptions in the case of constant negative curvature. Let S0 be a compact Riemannian manifold of real dimension n of constant negative curvature. Then there exists a complex manifold N, covered by the unit ball, and a real analytic, strictly plurisubharmonic exhaustion ρ : M → [−1, 0) such that

if u = √12 sin−1 (ρ) + π2 , then (dd c u)n = 0, the metric defined by dd c log(−ρ) has constant negative holomorphic sectional curvature, and S0 is isometric to the minimal submanifold S = {ρ = −1} with the induced metric. Since for Kähler manifold (M, h) of constant holomorphic sectional curvature k, the bisectional curvature may be computed by (see, e.g., [5]) (5.1)

B(X, Y ) =

k h(X, X )h(Y, Y ) + h(X, Y )2 + h(X, JY )2 . 2

If X and Y are orthogonal, length 1 vectors tangent to the minimal submanifold S = {ρ = −1}, then X ⊥ JY , and hence B(X, Y ) = 2k . With obvious changes, as for the characterization of the hyperquadric, it is possible to derive the following uniformization result: Theorem 5.1. Let M be a connected complex manifold of dimension n with a strictly plurisubharmonic exhaustion ρ : M → [−1, 0) of class C ∞ such that, if

1 π −1 u = √2 sin (ρ) + 2 , then (dd c u)n = 0 on {ρ > −1}. Suppose (i) The bisectional curvature of the metric dd c ρ is constant in the directions tangent to the minimal set S = S−1 of ρ; (ii) With the induced metric, the minimal set S = S−1 of ρ has negative Riemannian sectional curvature at least at one point. Then M with the metric dd c log (−ρ) is isometrically biholomorphic to a quotient of Bn with the Bergmann metric. A similar result holds also for the flat case: Theorem 5.2. Let M be a connected complex manifold of dimension n with a strictly plurisubharmonic exhaustion ρ : M → [0, +∞) of class C ∞ such that, √ if u = ρ, then (dd c u)n = 0 on {ρ > 0}. Suppose (i) The bisectional curvature of the metric dd c ρ is constant in the directions tangent to the minimal set S = S0 of ρ; (ii) With the induced metric, the minimal set S = S0 has zero Riemannian sectional curvature at least at one point. Then M with the metric dd c ρ is biholomorphically isometric to a quotient of Cn with the flat metric. Acknowledgements. The authors would like to thank the referee for pointing out an error in a previous version of the paper and for the helpful comments on the presentation of the paper.

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