Mammola, Stefano (2018) Generic Vanishing in Geometria Analitica e Algebrica. [Magistrali biennali]
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The goal of this thesis is to illustrate the Generic Vanishing theorem (GVT), specifically the version that Green and Lazarsfeld proved in the late 80’s in [GL1], focusing on its proof and some of its applications, in particular on the so called Ueno’s conjecture K. To understand these problems we need some definitions and properties from the Hodge theory, in particular on a compact Kähler manifold. In fact these are the main objects of studying in this thesis. A Kähler manifold is a complex manifold equipped with a Hermitian metric whose imaginary part, which is a 2-form of type (1,1) relative to the complex structure, is closed. We are interested in these objects mostly because the Kähler identities, which are identities between certain operators on differential forms, provide the Hodge decomposition of the de Rham cohomology. We also prove the Kodaira Vanishing theorem, an important theorem on vanishing of cohomology of positive line bundle. While Kodaira’s theorem depends on the fact that the first Chern class c1(L) is positive, the GVT concerns line bundles with c1(L) = 0. This kind of line bundles are topologically trivial, because their underlying smooth line bundle is the trivial bundle X xC, and they are parametrized by the points of Pic0(X).
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