A weak compactness theorem of the Donaldson-Thomas instantons on compact K\"ahler threefolds

TL;DR

This paper establishes a weak compactness theorem for solutions to the Donaldson-Thomas equations on compact Kähler threefolds, showing subsequential convergence outside a finite measure set under bounded Higgs field norms.

Contribution

It extends analytic compactness results to higher-dimensional Kähler manifolds for Donaldson-Thomas instantons, incorporating Higgs fields and non-linear terms.

Findings

Solutions have converging subsequences outside a finite 2D Hausdorff measure set.

Proves an n/2-compactness theorem for solutions on Kähler threefolds.

Bounded Higgs field norms imply partial convergence of solutions.

Abstract

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact K\"ahler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact K\"ahler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact K\"ahler threefolds.