Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville (Q1112122)

Let X be a compact, connected Kähler manifold of dimension n, \(Pic^ 0(X)\) the identity component of the Picard group of X, \(S^ i(X)\subseteq Pic^ 0(X)\) the analytic subvariety given by \(S^ i(X)=\{L\in Pic^ 0(X)| H^ i(X,L)\neq 0\},\) \(i\geq 0\) and \(a: X\to Alb(X)\) the Albanese map of X. Then \(co\dim (S^ i(X),Pic^ 0(X))\geq \dim(a(X))-i\). In particular, if \(L\in Pic^ 0(X)\) is a generic line bundle, then \(H^ i(X,L)=0\) for \(i<\dim (a(X))\) (this is a positive answer to some conjectures of Beauville and Catanese). If X is an irregular surface without irrational pencils then the trivial bundle \({\mathcal O}_ X\) is an isolated point of \(S^ 1(X)\) and consequently any (effectively parametrized) irreducible family of curves on X containing at least one canonical divisor has dimension \(\leq p_ g(X)\) (this gives an upper bound on the dimensions of algebraic deformations of a canonical divisor on X as sought by Enriques). These results are proved by studying the deformation theory of the groups \(H^ i(X,L)\) as L varies.