#### Volume 17, issue 2 (2013)

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

### Margarida Mendes Lopes, Rita Pardini and Gian Pietro Pirola

Geometry & Topology 17 (2013) 1205–1223
##### Abstract

Given a smooth complex projective variety $X$, a line bundle $L$ of $X$ and $v\in {H}^{1}\left({\mathsc{O}}_{X}\right)$, we say that $v$ is $k$–transversal to $L$ if the complex ${H}^{k-1}\left(L\right)\to {H}^{k}\left(L\right)\to {H}^{k+1}\left(L\right)$ is exact. We prove that if $v$ is $1$–transversal to $L$ and $s\in {H}^{0}\left(L\right)$ satisfies $s\cup v=0$, then the first order deformation $\left({s}_{v},{L}_{v}\right)$ of the pair $\left(s,L\right)$ in the direction $v$ extends 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 and Popa in higher dimension. In particular, we prove the inequality ${p}_{g}\left(X\right)\ge \chi \left({K}_{X}\right)+q\left(X\right)-1$ for a variety $X$ of maximal Albanese dimension without irregular fibrations of Albanese general type.

##### Keywords
paracanonical system, irregular varieties, varieties of maximal Albanese dimension, numerical invariants
##### Mathematical Subject Classification 2010
Primary: 14C20, 14J29, 32G10