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

Abstract

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 \({(\mathcal{E}, \varphi)}\), where \({\mathcal{E}}\) is a locally-free sheaf over a surface X and \({\varphi}\) is a section of End\({(\mathcal{E}) \otimes K_{X}}\); and the existence of a solution to certain gauge-theoretic equations, the Vafa–Witten equations, for a Hermitian metric on \({\mathcal{E}}\). It turns out to be a special case of results obtained by Álvarez-Cónsul and García-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.

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Correspondence to Yuuji Tanaka.

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Tanaka, Y. Stable sheaves with twisted sections and the Vafa–Witten equations on smooth projective surfaces. manuscripta math. 146, 351–363 (2015). https://doi.org/10.1007/s00229-014-0706-6

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Mathematics Subject Classification

  • Primary 53C07
  • Secondary 14J60