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arXiv · 0805.2195

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

Yuuji Tanaka
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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 Kähler 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 Kähler 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 Kähler threefolds.

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Yuuji Tanaka. 2013-06-18. A weak compactness theorem of the Donaldson-Thomas instantons on compact Kähler threefolds. https://doi.org/10.1016/j.jmaa.2013.05.059

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