Grauert's direct image theorem via superconnections and desingularizations

TL;DR

This paper presents a new differential-geometric proof of Grauert's theorem on the coherence of higher direct images, utilizing superconnections and desingularizations to handle both smooth and singular cases.

Contribution

It introduces a novel proof method for Grauert's theorem using antiholomorphic superconnections and desingularization techniques, extending the approach to singular spaces.

Findings

Provides a differential-geometric proof of Grauert's theorem

Extends the proof to singular spaces via desingularization

Connects superconnection theory with complex analytic geometry

Abstract

We give a new differential-geometric proof of Grauert's theorem on the coherence of the higher direct image of a coherent sheaf under a proper holomorphic morphism between complex analytic spaces. In the smooth case, our approach is based on the antiholomorphic superconnection introduced by Block and further developed by Bismut-Shen-Wei. The required finiteness results follow from elliptic theory. In the singular case, we reduce the problem to the smooth setting using Hironaka's desingularization.