The Witten deformation of the Dolbeault complex

We introduce a Witten-Novikov type perturbation ${\bar{\mathrm{\partial }}}_{\bar{\omega }}$ of the Dolbeault complex of any complex K"ahler manifold, defined by a form $\omega$ of type $(1,0)$ with $\mathrm{\partial }\omega =0$. 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.