Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces

作者：Tanaka Yuuji^*

来源：Manuscripta Mathematica, 2015, 146(3-4): 351-363.

DOI：10.1007/s00229-014-0706-6

This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair , where is a locally-free sheaf over a surface X and is a section of End; and the existence of a solution to certain gauge-theoretic equations, the Vafa-Witten equations, for a Hermitian metric on . It turns out to be a special case of results obtained by Alvarez-Cnsul and Garcia-Prada on the quiver vortex equation. In this article, we give an alternative proof which uses a Mehta-Ramanathan style argument originally developed by Donaldson for the Hermitian-Einstein problem, as it relates the subject with the Hitchin equations on Riemann surfaces, and surely indicates a similar proof of the existence of a solution under the assumption of stability for the Donaldson-Thomas instanton equations described in Tanaka (2013) on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.

• 出版日期2015-3