Manuscripta Mathematica

, Volume 146, Issue 3–4, pp 351–363 | Cite as

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

  • Yuuji TanakaEmail author


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.

Mathematics Subject Classification

Primary 53C07 Secondary 14J60 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.National Center for Theoretical Sciences (South)National Cheng Kung UniversityTainanTaiwan

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