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 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 Kähler threefolds.