# ag.algebraic geometry – invariance by deformation of \$H^1(dlog)\$

Let $$X$$ be an algebraic variety. The canonical morphism of sheaves
$$dlog:{cal O}_X^*to Omega_X$$ defines a map $$c:Pic(X)to H^1(Omega_X)$$.
Is this map invariant by deformation ? (i.e if $$({cal L_s})_{sin S}$$ is a family of line bundles on $$X$$ parametrized by a smooth curve $$S$$, is $$c({cal L}_s)$$ independent of $$s$$ ?).