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

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 Hk−1(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.