  Given a smooth complex projective variety X, a line bundle L of X an element
v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the
direction v, we give a sufficient condition on v in terms of Koszul cohomology
for this first order deformation to extend 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-Popa in higher dimension. In particular, we prove
the inequality p_g(X)>=\chi(K_X)+q(X)-1 for a variety X of maximal Albanese
dimension without irregular fibrations of Albanese general type.
