Space of surjective morphisms between projective varieties

Transcription

1 Space of surjective morphisms between projective varieties -Talk at AMC2005, Singapore- Jun-Muk Hwang Korea Institute for Advanced Study Cheongryangri-dong Seoul, , Korea Let X and Y be two complex projective varieties. Denote by Hom s (X, Y ) the set of surjective morphisms from X to Y. Each irreducible component of Hom s (X, Y ) has the structure of a quasiprojective algebraic variety. A natural question to ask is what kind of an algebraic variety this is. Given a surjective morphism f : X Y, the question can be reformulated as describing all possible deformations of f as morphisms between X and Y. Since any surjective morphism from X to Y can be lifted to the normalizations of X and Y, we may assume that X and Y are normal varieties. In this note, we will survey some recent results on this question. In particular, our joint-work with S. Kebekus and T. Peternell in the case when Y is non-uniruled and our joint-work with N. Mok in the case when Y is a uniruled manifold of Picard number 1 will be discussed. It is well-known that the tangent space to Hom s (X, Y ) at a point [f] is T [f] (Hom s (X, Y )) = Hom(f Ω 1 Y, O X ) where Ω 1 Y denotes the sheaf of differentials on the variety Y. When Y is non-singular, T [f] (Hom s (X, Y )) = H 0 (X, f T (Y )) where T (Y ) is the tangent bundle of Y. An element of this vector space can be viewed as a multi-valued vector field on Y. The essential point of our discussion here is to understand the nature of this multi-valuedness. 1 The case of curves To have an idea of what kind of an answer we can expect for our problem, let us examine the case when Y is of dimension 1. To start with, consider the case when Y = P 1. For a given normal variety X, Hom s (X, P 1 ) is just the set of base-point-free linear systems of dimension 1 on X. This is a very classical object of study. Even for a curve X, this is a huge subject. The geometry of Hom s (X, P 1 ) is very complicated and depends heavily on X. There is no simple description of the variety Hom s (X, P 1 ). This first example may be discouraging to us. However the situation is quite different when Y P 1. 1

2 When Y is an elliptic curve, T (Y ) = O Y. Thus for any f : X Y, H 0 (X, f T (Y )) = H 0 (X, O X ) = C = f H 0 (Y, T (Y )). This means all the multi-valued vector fields on Y appearing in H 0 (X, f T (Y )) are actually univalent vector fields on Y. Then T [f] (Hom s (X, Y )) = f H 0 (Y, T (Y )) = C. In other words, all deformations of f are of the form τ f where τ : Y Y is a translation on the elliptic curve Y. We conclude that each component of Hom s (X, Y ) is an elliptic curve. On the other hand, when Y is a curve of genus 2, T (Y ) is a negative line bundle. Thus H 0 (X, f T (Y )) = 0. This means that Hom s (X, Y ) is discrete. We can summarize the 1-dimensional case as follows. Y = P 1 Y P 1 Hom s (X, Y ) complicated, each component is depends heavily on X an elliptic curve or a point. 2 The case of higher dimension The most natural question to ask is whether there is a natural generalization of the the table at the end of the previous section when the dimension of Y is bigger than 1. The first step is to formulate the correct generalization of the dichotomy of the set of curves into P 1 and the rest. A simple-minded generalization would be the dichotomy of the set of n-dimensional varieties into P n and the rest. It turns out that this is too naive. Experiences show that the correct generalization is the dichotomy into uniruled varieties and non-uniruled varieties. Recall that a projective variety is uniruled if through each point on it, there exists a rational curve on the variety. In other words, the variety is swept out by rational curves. Obviously, in dimension 1, only P 1 is uniruled. Now the following gives a natural generalization of the 1-dimensional case. Theorem 1 [HKP] Let X, Y be any normal projective varieties. Then the following holds. Y uniruled Y non-uniruled Hom s (X, Y ) complicated, each component is depends heavily on X an abelian variety of dimension dim Y. In the first case when Y is uniruled, there is nothing to prove. For example, when Y = P n, the study of Hom s (X, P n ) is the study of base-point-free linear systems of dimension n on X. It is well-known that this is complicated and depends heavily on X. The non-trivial case is when Y is non-uniruled, which is the main content of [HKP]. For simplicity, let us assume that Y is non-singular. The main result of [HKP] is the following from which the above follows easily. Theorem 2 [HKP] Let Y be a non-uniruled non-singular projective variety and f : X Y be a surjective morphism from a projective variety X. Then there exists a finite etale cover 2

3 h : Z Y and a surjective morphism g : X Z such that f = h g and all deformations of f come from the identity component Aut o (Z) of the group of the automorphisms of Z. In other words, let Hom f (X, Y ) be a component of Hom s (X, Y ) containing [f]. Then the natural map is surjective. h Aut o (Z) g Hom f (X, Y ) Since Y is non-uniruled, Z is also non-uniruled. For a non-uniruled variety Z, it is easy to see that Aut o (Z) is an abelian variety of dimension dim Z. Thus Theorem 2 implies Theorem 1. Also note that if Aut o (Z) is an abelian variety of positive dimension, the fundamental group of Z is infinite. This observation implies the following Corollary of Theorem 2. Corollary Let Y be a non-uniruled non-singular projective variety with finite fundamental group. Then for any projective variety X, Hom s (X, Y ) is discrete. Now let us briefly discuss the proof of Theorem 2. Suppose H 0 (X, f T (Y )) = f H 0 (Y, T (Y )), namely, all the multi-valued vector fields on Y involved here are univalent. Then we can just take Z = Y. The natural map Aut o (Y ) f Hom f (X, Y ) is surjective because its differential is exactly the natural homomorphism Now what if In this case, we have the following. f H 0 (Y, T (Y )) H 0 (X, f T (Y )). H 0 (X, f T (Y )) f H 0 (Y, T (Y ))? Theorem 3 Let Y be a non-uniruled non-singular projective variety and f : X Y be a surjective morphism. Suppose there exists σ H 0 (X, f T (Y )) such that σ f H 0 (Y, T (Y )). Then there exists an etale cover h σ : Z σ Y of degree > 1 and a surjective morphism g σ : X Z σ such that f = h σ g σ. Certainly, Theorem 2 follows from Theorem 3 by induction on the degree of f. To prove Theorem 3, the key point is to construct the etale cover Z σ. Let F σ T (Y ) be the subsheaf spanned by the multiple values of σ regarded as a multi-valued vector field on Y. Then F σ is a semi-positive sheaf, namely, its restriction to a general curve is semi-positive, because it is generically generated by global (multi-valued) sections. On the other hand, by the famous theorem of Miyaoka, any quotient sheaf of the cotangent sheaf T (Y ) of a non-uniruled variety is semi-positive. It follows that both F σ and its dual are semi-positive. This implies that F σ is a flat bundle, defining an etale cover h σ : Z σ Y. This has degree > 1, because σ is genuinely multi-valued. Now it is easy to verify the existence of g σ with the desired property. 3

4 3 The case of uniruled manifolds of Picard number 1 In the discussion of Theorem 1, we remarked that the case when Y is uniruled is complicated and depends heavily on X, as one can see when Y = P n. However, it is not immediate that this is so for uniruled varieties other than P n. When Y is of the form P k Y for some lowerdimensional variety Y or more generally when Y is a P k -bundle over a lower dimensional variety, one can see that the structure of Hom s (X, Y ) can be complicated and depends heavily on X, just as in the case when Y = P n. But suppose Y is not of this form. Then we expect that the structure of Hom s (X, Y ) is rather simple. One natural condition to exclude the possibility of P k -bundle structure on Y is that the Picard number of Y is 1. In this setting we have the following conjecture. Conjecture Let Y be a uniruled non-singular variety with Picard number 1. If Y is different from a projective space, then for any surjective morphism f : X Y, H 0 (X, f T (Y )) = f H 0 (Y, T (Y )), namely, each component of Hom s (X, Y ) is biregular to Aut o (Y ). Note that when Y is uniruled and of Picard number 1, Aut o (Y ) is an affine algebraic group. Thus Conjecture predicts that each component of Hom s (X, Y ) is an affine variety. This is in contrast with the non-uniruled case in Theorem 1 where each component of Hom s (X, Y ) is an abelian variety. The strongest evidence for Conjecture is the following result. Theorem 4 [HM2] Let Y be a uniruled non-singular variety of Picard number 1 different from a projective space. Assume that Y contains no immersed projective space with trivial normal bundle. Then for any surjective morphism f : X Y, H 0 (X, f T (Y )) = f H 0 (Y, T (Y )). Theorem 4 proves Conjecture with the additional assumption that Y contains no immersed projective space with trivial normal bundle. There are many examples where Y contains immersed rational curves with trivial normal bundle. In this case, we have the following result. Theorem 5 [HM1] Let Y be a uniruled non-singular variety of Picard number 1 different from a projective space. Assume that Y contains no immersed rational curves with trivial normal bundle. Then for any surjective morphism f : X Y, H 0 (X, f T (Y )) = f H 0 (Y, T (Y )). On the other hand, no example is known where Y contains immersed projective space of dimension 2 with trivial normal bundle. Thus Theorem 4 and Theorem 5 prove Conjecture for all known examples of Y. 4

5 To understand the basic idea of the proof of Theorem 4, consider the special case when Y is the quadric hypersurface Q n in P n+1. The key point here is that Q n has a natural holomorphic conformal structure. There are many ways to describe this structure, but what is relevant to us is the description via lines on the quadric. Given a point y Y = Q n P n+1, let C y PT y (Y ) be the set of tangents to lines of P n+1 through y lying on the quadric Y. Then C y is a smooth quadric hypersurface in PT y (Y ). C y gives a conformal class of holomorphic Riemannian metric on T y (Y ). Then the union C = C y PT (Y ) y Y defines a holomorphic conformal structure on Y. Why is this conformal structure relevant to our problem? Given σ H 0 (X, f T (Y )) viewed as a multi-valued vector field on Y, a local holomorphic vector field on Y defining a value of σ turns out to be a conformal vector field, namely, the local flow of biholomorphisms generated by σ preserves the conformal structure. This follows from certain deformation-theoretic properties of lines on Q n. Now we recall the classical Liouville theorem in conformal geometry, which states that any conformal vector field on a connected analytic open subset of Q n can be extended to a global holomorphic vector field on Q n. Since a local representative of σ on Y is conformal, Liouville theorem shows that σ is actually globally well-defined holomorphic vector field on Y, i.e., an element of f H 0 (Y, T (Y )). This proves Theorem 4 for the special case when Y is a quadric hypersurface. How can we generalize this proof to general Y? The key idea is to construct a natural geometric structure on Y, which generalizes the conformal structure on Q n. For this, we need to generalize the notion of lines. The correct notion for general Y is minimal rational curves, i.e., rational curves covering Y which have minimal degree with respect to the ample generator of the Picard group of Y. For a general point y Y, let C y PT y (Y ) be the set of tangent vectors to minimal rational curves through y. Then the closure C PT (Y ) of the union general y Y C y PT (Y ) defines a holomorphic geometric structure on Y. This is non-trivial, i.e., C y PT y (Y ), if Y is different from the projective space. Given a surjective morphism f : X Y and σ H 0 (X, f T (Y )), a local representative of σ viewed as a multi-valued vector field on Y preserves the geometric structure C. This is because minimal rational curves have the same deformation-theoretic properties as lines on Q n. Now the crucial point is whether an analogue of Liouville theorem holds for C. The essential property of the conformal structure which is necessary for Liouville theorem is the non-degeneracy of the holomorphic Riemannian metric. It turns out that if the projective variety C y PT y (Y ) is not linear, an analogue of Liouville theorem holds for the structure C. When C y is linear, they can be integrated to give an immersed projective space with trivial normal bundle in Y. This proves Theorem 4. The proof of Theorem 5 uses a different approach. When Y has rational curves with trivial normal bundles, these rational curves locally give a foliation on Y. This local foliation gives rise 5

6 to a multi-valued foliation globally. From the viewpoint of geometric structures, this is a web structure. Given a generically finite morphism f : X Y, the web-structure on Y gives rise to a web-structure on X. One can show that such web-structures cannot be deformed. Any deformations of f must preserve these web-structures and from this we can prove Theorem 5. It is tempting to generalize Theorem 5 to the case when Y has immersed projective space with trivial normal bundles, which would give a complete proof of Conjecture. However there are many difficulties in this direction, which we have been unable to overcome so far. References [HKP] J.-M. Hwang, S. Kebekus and T. Peternell, Holomorphic maps onto varieties of nonnegative Kodaira dimension. to appear in J. Alg. Geom. [HM1] J.-M. Hwang and N. Mok, Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles. J. Alg. Geom. 12 (2003) [HM2] J.-M. Hwang and N. Mok, Birationality of the tangent map for minimal rational curves. Asian J. Math. 8, Special issue dedicated to Y.-T. Siu on his 60th birthday (2004)