An explicit isomorphism of different representations of the Ext functor using residue currents

TL;DR

This paper constructs an explicit isomorphism between two different representations of Ext groups for coherent sheaves on complex manifolds, utilizing residue currents and twisted resolutions to connect cohomological and Dolbeault-based descriptions.

Contribution

It introduces a novel explicit isomorphism between two representations of Ext groups using residue currents and twisted resolutions, bridging cohomological and Dolbeault frameworks.

Findings

Explicit isomorphism between two Ext representations established

Residue currents used to describe the isomorphism concretely

Bridges cohomological and Dolbeault descriptions of Ext groups

Abstract

Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module over a complex manifold $X$, and let $G$ be a vector bundle on $X$. We describe an explicit isomorphism between two different representations of the global $\DeclareMathOperator{\Ext}{Ext}\Ext$ groups $\DeclareMathOperator{\Ext}{Ext}\Ext^k(\mathcal{F},G)$. The first representation is given by the cohomology of a twisted complex in the sense of Toledo and Tong, and the second one is obtained from the Dolbeault complex associated with $G$. A key tool that we introduce for explicitly describing this isomorphism is a residue current associated with a twisted resolution of $\mathcal{F}$.