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

Mendes Lopes, Margarida¹ ; Pardini, Rita² ; Pirola, Gian Pietro³

¹ 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
² Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, I-56127 Pisa, Italy
³ Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

Geometry & topology, Tome 17 (2013) no. 2, pp. 1205-1223.

Résumé

Given a smooth complex projective variety $X$, a line bundle $L$ of $X$ and $v\in H^1\left({\mathcal{O}}_X\right)$, 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.

Détail

DOI : 10.2140/gt.2013.17.1205

Classification : 14C20, 14J29, 32G10
Keywords: paracanonical system, irregular varieties, varieties of maximal Albanese dimension, numerical invariants

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