Lecture 1: October 15 Introduction. Our topic this semester is the \generic vanishing theorem" and its applications. In the late 1980s, Green and Lazarsfeld studied the cohomology of topologically trivial line bundles on smooth complex projective varieties (and compact K¨ahlermanifolds). The generic vanishing theorem is one of their main results: it says that, under certain conditions on the variety, the cohomology of a generically chosen line bundle is trivial in all degrees less than the dimension of the variety. This theorem, and related results by Green and Lazarsfeld, are a very useful tool in the study of irregular varieties and abelian varieties. Here are some examples of its applications: (1) Singularities of theta divisors on principally polarized abelian varieties (Ein- Lazarsfeld). (2) Numerical characterization of abelian varieties up to birational equivalence (Chen-Hacon) (3) Birational geometry of varieties of Kodaira dimension zero (Ein-Lazarsfeld, Chen-Hacon) (4) Inequalities among Hodge numbers of irregular varieties (Lazarsfeld-Popa) (5) M-regularity on abelian varieties (Pareschi-Popa) There are now two completely different proofs for the generic vanishing theorem. The original one by Green and Lazarsfeld used deformation theory and classical Hodge theory; there is also a more recent one by Hacon, based on derived categories and Mukai's \Fourier transform" for abelian varieties. Here is a list of the sources: [GL87] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theo- rems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389{407. [GL91] , Higher obstructions to deforming cohomology groups of line bun- dles, J. Amer. Math. Soc. 4 (1991), no. 1, 87{103. [Sim93] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. Ecole´ Norm. Sup. (4) 26 (1993), 361{401. [Hac04] C. D. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173{187. [PS13] M. Popa and C. Schnell, Generic vanishing theory via mixed Hodge mod- ules, Forum Math. Sigma 1 (2013), e1, 60. The following articles contain various examples and applications of the theory. We will be discussing most of them over the course of the semester. [Bea92] A. Beauville, Annulation du H1 pour les fibr´esen droites plats, Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 1{15. [EL97] L. Ein and R. Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243{ 258. [CH01] J. A. Chen and C. D. Hacon, Characterization of abelian varieties, Invent. Math. 143 (2001), no. 2, 435{447. [Par12] G. Pareschi, Basic results on irregular varieties via Fourier-Mukai methods, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 379{403. [LP10] R. Lazarsfeld and M. Popa, Derivative complex, BGG correspondence, and numerical inequalities for compact K¨ahlermanifolds, Invent. Math. 182 (2010), no. 3, 605{633. 1 2 Vanishing theorems and their applications. Before I introduce the work of Green and Lazarsfeld, let me say a few words about the role of vanishing theorems in modern algebraic geometry. Vanishing theorems are very important because many questions can be phrased in terms of coherent sheaves, their global sections, and their cohomology. Here are three typical cases: (1) Lifting sections. Suppose we have a short exact sequence of coherent sheaves 0 ! F 0 ! F ! F 00 ! 0 on an algebraic variety X, and need to know whether or not every global section of F 00 can be lifted to a global section of F . The long exact sequence ···! H0(X; F ) ! H0(X; F 00) ! H1(X; F 0) !··· in cohomology shows that the vanishing of H1(X; F 0) is sufficient. (2) Existence of sections. Suppose we have a coherent sheaf F on an alge- braic variety X, and need to know whether or not F has nontrivial global sections. If Hi(X; F ) happens to vanish for every i > 0, then dim H0(X; F ) = χ(X; F ) is equal to the Euler characteristic of F , which can typically be computed with the help of the Riemann-Roch theorem. (3) Vanishing of obstructions. Suppose we need to do some global construction on an algebraic variety X, but only know that it works locally. In this situation, the obstruction to the global problem is often an element of some sheaf cohomology group; the vanishing of this cohomology group is therefore sufficient to get a solution. The most famous vanishing theorem is the Kodaira vanishing theorem for ample line bundles on complex projective varieties. (In this course, we will only consider algebraic varieties that are defined over the complex numbers; I will therefore not explicitly mention that assumption from now on.) Kodaira Vanishing Theorem. Let L be an ample line bundle on a smooth pro- i jective variety X. Then H (X; !X ⊗ L) = 0 for every i > 0. The Nakano vanishing theorem extends this to the other sheaves of differential p forms ΩX , but the result is not as strong as in the case of the canonical bundle. Nakano Vanishing Theorem. Under the same assumptions on X and L, one q p has H (X; ΩX ⊗ L) = 0 for every p; q 2 N with p + q > dim X. Here are a few elementary applications, along the lines of what I said above. Example 1.1. If L is an ample line bundle on an abelian variety A, then H0(A; L) 6= 0. Indeed, the canonical bundle !A is trivial, and so all higher cohomology groups of L vanish by Kodaira's theorem. Together with the Riemann-Roch theorem, Lg dim H0(A; L) = χ(A; L) = 6= 0; g! where g = dim A. Example 1.2. Infinitesimal deformations of a smooth projective variety are parametrized 1 by H (X; TX ), where TX is the tangent sheaf; the obstructions to extending an 2 infinitesimal deformation to an actual deformation lie in H (X; TX ). Suppose that −1 X is a Fano manifold of dimension n, which means that !X is an ample line bundle. n−1 Because !X ⊗ TX ' ΩX , we obtain 2 2 n−1 −1 H (X; TX ) ' H (X; ΩX ⊗ !X ) = 0 from the Nakano vanishing theorem. Fano manifolds are therefore unobstructed. 3 It turns out that one can relax the assumptions in the Kodaira vanishing theorem and allow L to be only nef and big (= a birational version of being ample). Recall that L is nef if, for every curve C ⊆ X, the intersection number L · C ≥ 0. It is called big if the function m 7! dim H0(X; L⊗m) grows like mdim X ; when L is nef, this is equivalent to having Ldim X > 0 (by the Riemann-Roch theorem). Kawamata-Viehweg Vanishing Theorem. Let X be a smooth projective vari- i ety. If L is nef and big, then H (X; !X ⊗ L) = 0 for every i > 0. This result has had a great influence on birational geometry; in fact, there is even a more general version of the Kawamata-Viehweg vanishing theorem that involves R-divisors and multiplier ideals. The generic vanishing theorem. All of the vanishing theorems above depend on the fact that the first Chern class c1(L) is (in some sense) positive. The ques- tion that motivated the generic vanishing theorem is what happens when we con- sider instead line bundles with c1(L) = 0. We may call such line bundles \topo- logically trivial", because the underlying smooth line bundle is trivial. They are parametrized by the points of Pic0(X), which is an abelian variety of dimension 1 dim H (X; OX ) when X is smooth and projective (and a compact complex torus when X is a compact K¨ahlermanifold). To get a feeling for what can happen, let us consider a few simple examples. 0 Example 1.3. We clearly have H (X; L) = 0 unless L ' OX ; in fact, any nontrivial global section must be everywhere nonzero because c1(L) = 0. Example 1.4. Let n = dim X. Serre duality shows that n 0 −1 dim H (X; L) = dim H (X; !X ⊗ L ); and in many situations (for example, on curves of genus at least two), the coho- mology group on the right is nonzero for every L 2 Pic0(X). Example 1.5. Let X be a smooth projective variety, and suppose that we have a morphism f : X ! C with connected fibers to a curve of genus at least two. Then ∗ f∗OX ' OC , and if we look at the first cohomology of f L, we get (from the Leray spectral sequence) an exact sequence 1 1 ∗ 0 1 0 ! H C; L ! H X; f L ! H C; R f∗OX ⊗ L ! 0: Since H1(C; L) is nonzero for every L 2 Pic0(C), this means that Pic0(X) contains a whole subvariety isomorphic to Pic0(C) where the first cohomology is nontrivial. This example was studied very carefully by Beauville; it will appear again later in the course. The lesson to draw from these three examples is that we cannot expect to have a good vanishing theorem that works for every L 2 Pic0(X), because there may be special line bundles whose cohomology does not vanish for geometric reasons. Moreover, even if we exclude those special line bundles, the group Hdim X (X; L) will typically be nonzero. The following example shows another interesting phe- nomenon. Example 1.6. Suppose that X = C × Y , where C is a curve of genus at least two, 1 and H (Y; OY ) = 0.