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

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
Full Text: DOI EuDML
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