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Generalized Clifford-Severi inequality and the volume of irregular varieties. (English) Zbl 1409.14013
Summary: We give a sharp lower bound for the self-intersection of a nef line bundle \(L\) on an irregular variety \(X\) in terms of its continuous global sections and the Albanese dimension of \(X\), which we call the generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when \(X\) is of maximal Albanese dimension the bound is \(\mathrm{vol}(X)\geq2n!\chi(\omega_X)\) and it is sharp.

MSC:
14C20 Divisors, linear systems, invertible sheaves
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14J29 Surfaces of general type
14J30 \(3\)-folds
14J35 \(4\)-folds
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