  We introduce a Witten-Novikov type perturbation $\bar\partial_{\bar\omega}$
of the Dolbeault complex of any complex K\"ahler manifold, defined by a form
$\omega$ of type $(1,0)$ with $\partial\omega=0$. We give an explicit
description of the associated index density which shows that it exhibits a
nontrivial dependence on $\omega$. The heat invariants of lower order are shown
to be zero.
