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

Mendes Lopes, Margarida; Pardini, Rita; Pirola, Gian Pietro

doi:10.2140/gt.2013.17.1205

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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

DOI: 10.2140/gt.2013.17.1205

Abstract

Given a smooth complex projective variety $X$ , a line bundle $L$ of $X$ and $v \in H^{1} (O_{X})$ , we say that $v$ is $k$ –transversal to $L$ if the complex $H^{k - 1} (L) \to H^{k} (L) \to H^{k + 1} (L)$ is exact. We prove that if $v$ is $1$ –transversal to $L$ and $s \in H^{0} (L)$ satisfies $s \cup v = 0$ , then the first order deformation $(s_{v}, L_{v})$ of the pair $(s, L)$ 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 $p_{g} (X) \geq χ (K_{X}) + q (X) - 1$ 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

References

Publication

Received: 25 August 2012

Accepted: 31 January 2013

Published: 29 May 2013

Proposed: Richard Thomas

Seconded: Jim Bryan, Ronald Stern

Authors

Margarida Mendes Lopes
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos Departamento de Matemática Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais 1049-001 Lisboa, Portugal
http://www.math.ist.utl.pt/~mmlopes/
Rita Pardini
Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5 I-56127 Pisa, Italy
http://www.dm.unipi.it/~pardini/
Gian Pietro Pirola
Dipartimento di Matematica Università di Pavia Via Ferrata 1 I-27100 Pavia, Italy
http://www-dimat.unipv.it/~pirola/

Volume 17, issue 2 (2013)