A weak compactness theorem of the Donaldson-Thomas instantons on compact Kähler threefolds

In Tanaka [18], we introduced a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and called it a Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler threefolds, and study some of the analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima (1987)  [14], Nakajima (1988)  [15] and Tian (2000)  [20] with some additional arguments concerning an extra nonlinear 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 Kähler threefold has a converging subsequence outside a closed subset whose real two-dimensional Hausdorff measure is finite, provided that the L2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact Kähler threefolds.