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Abstract
As an application of the theory of deformation of cohomology groups, we study divisors on a principally polarized abelian variety and we obtain a generalization of a theorem of Ein and Lazarsfeld characterizing abelian varieties whose theta divisor is very singular (ie l.c. but not l.t.).
Next, we study the numerical properties of ample vector bundles. We give examples of ample and generated vector bundles with small Chern numbers. These are counterexamples to well known conjectures of Ballico and of Beltrametti, Sommese and Schneider, and in particular they are examples of ample vector bundles with small Seshadri constants. We also produce bounds (under appropriate auxiliary conditions) for the Seshadri constants of these vector bundles.
