The Witten deformation of the Dolbeault complex

TL;DR

This paper introduces a new Witten-Novikov type perturbation of the Dolbeault complex on Kähler manifolds, revealing a dependence on a specific form and analyzing heat invariants.

Contribution

It defines a novel perturbation of the Dolbeault complex and explicitly describes its index density, highlighting its dependence on the form .

Findings

Index density depends nontrivially on 

Lower order heat invariants are zero

Provides explicit description of the perturbation and its properties

Abstract

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.