Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville. (English) Zbl 0659.14007

Green, Mark; Lazarsfeld, Robert

Invent. Math. 90, 389-407 (1987).

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.

Reviewer: D.Popescu

MSC:

14C22	Picard groups
14C20	Divisors, linear systems, invertible sheaves
53C55	Global differential geometry of Hermitian and Kählerian manifolds
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14D15	Formal methods and deformations in algebraic geometry

Keywords:

topologically trivial line bundle; Kähler manifold; Picard group

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