The Witten deformation of the Dolbeault complex

Abstract

We introduce a Witten-Novikov type perturbation ${\bar{\partial }}_{\bar{\omega }}$ of the Dolbeault complex of any complex Kähler 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.

15 pages