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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

Given a smooth complex projective variety X, a line bundle L of X and v H1(OX), we say that v is k–transversal to L if the complex Hk1(L) Hk(L) Hk+1(L) is exact. We prove that if v is 1–transversal to L and s H0(L) satisfies s v = 0, then the first order deformation (sv,Lv) 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 pg(X) χ(KX) + q(X) 1 for a variety X of maximal Albanese dimension without irregular fibrations of Albanese general type.

paracanonical system, irregular varieties, varieties of maximal Albanese dimension, numerical invariants
Mathematical Subject Classification 2010
Primary: 14C20, 14J29, 32G10
Received: 25 August 2012
Accepted: 31 January 2013
Published: 29 May 2013
Proposed: Richard Thomas
Seconded: Jim Bryan, Ronald Stern
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
Rita Pardini
Dipartimento di Matematica
Università di Pisa
Largo B. Pontecorvo 5
I-56127 Pisa, Italy
Gian Pietro Pirola
Dipartimento di Matematica
Università di Pavia
Via Ferrata 1
I-27100 Pavia, Italy