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 $s$ ?).