On the spectrum of the twisted Dolbeault Laplacian on line bundles over K\"ahler manifolds

TL;DR

This paper establishes a precise lower bound for the first eigenvalue of the twisted Dolbeault Laplacian on line bundles over compact K"ahler manifolds using Dirac operator methods.

Contribution

It introduces a novel approach employing Dirac operator techniques to derive sharp eigenvalue bounds in the context of K"ahler geometry.

Findings

Derived a sharp lower bound for the first eigenvalue

Applied Dirac operator methods to complex geometry

Enhanced understanding of spectral properties of Laplacians on line bundles

Abstract

We use Dirac operator techniques to establish a sharp lower bound for the first eigenvalue of the twisted Dolbeault Laplacian on holomorphic line bundles over compact K\"ahler manifolds.