Continuous families of divisors, paracanonical systems and a new inequality for varieties of maximal Albanese dimension

TL;DR

This paper develops a criterion for extending first-order deformations of line bundles on complex varieties and applies it to improve inequalities related to the paracanonical system of varieties with maximal Albanese dimension.

Contribution

It introduces a new sufficient condition based on Koszul cohomology for deformation extension and improves known inequalities for varieties of maximal Albanese dimension.

Findings

Extended deformation criteria using Koszul cohomology.

Proved a new inequality p_g(X) >= χ(K_X) + q(X) - 1.

Applied results to varieties without irregular fibrations of Albanese general type.

Abstract

Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this first order deformation to extend to an analytic deformation. We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld-Popa in higher dimension. In particular, we prove the inequality p_g(X)>=\chi(K_X)+q(X)-1 for a variety X of maximal Albanese dimension without irregular fibrations of Albanese general type.