Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

TL;DR

This paper proves vanishing theorems for $L^2$-cohomology groups on complete Hessian manifolds and convex cones with the Cheng-Yau metric, extending previous results in complex geometry.

Contribution

It establishes new vanishing theorems for $L^2$-cohomology groups on Hessian manifolds and convex cones, generalizing Kodaira-Nakano type results.

Findings

Vanishing theorems for $L^2$-cohomology on Hessian manifolds.

Vanishing results for $L^2H^{p,q}( ext{Omega})$ when $p>q$ on convex cones.

Extension of complex geometric vanishing theorems to Hessian and convex cone settings.

Abstract

We show vanishing theorems of $L^2$-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}(\Omega)$ on a regular convex cone $\Omega$ with the Cheng-Yau metric for $p>q$.