A stability of vector bundles with twisted sections and the Donaldson-Thomas instantons on compact K\"{a}hler threefolds

TL;DR

This paper establishes a correspondence between solutions to a generalized Hermitian-Einstein equation with a Higgs field, called Donaldson-Thomas instantons, and a stability condition for sheaves on compact K"ahler threefolds, extending Hitchin's theory.

Contribution

It proves a Hitchin--Kobayashi-type correspondence for Donaldson-Thomas instantons on compact K"ahler threefolds, generalizing known results from Riemann surfaces.

Findings

Established a stability condition for sheaves with sections on K"ahler threefolds.

Proved the existence of Donaldson-Thomas instantons under stability conditions.

Extended Hitchin's correspondence to higher-dimensional K"ahler manifolds.

Abstract

We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact K\"ahler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to K\"ahler threefolds. In the appendix of arXiv:0805.2192, following an analogy with the Hitchin equation, we introduced a stability condition for a pair consisting of a locally-free sheaf over a compact K\"ahler threefold and a section of the associated sheaf of the endomorphisms tensored by the canonical bundle of the threefold. In this article, we prove a Hitchin--Kobayashi-type correspondence for this and the Donaldson-Thomas instanton on compact K\"ahler threefolds.