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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Vafa-Witten invariants for projective surfaces I: stable case

Authors: Yuuji Tanaka and Richard P. Thomas
Journal: J. Algebraic Geom. 29 (2020), 603-668
Published electronically: October 23, 2019
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Abstract | References | Additional Information


On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb {C}^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations.

When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.

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Additional Information

Yuuji Tanaka
Affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
MR Author ID: 836351

Richard P. Thomas
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
MR Author ID: 636321

Received by editor(s): January 19, 2018
Received by editor(s) in revised form: November 21, 2018
Published electronically: October 23, 2019
Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Scientific Research numbers JP15H02054 and JP16K05125, and a Simons Collaboration Grant on “Special holonomy in Geometry, Analysis and Physics”. He thanks Seoul National University, NCTS at National Taiwan University, Kyoto University, and BICMR at Peking University for their support and hospitality during visits in 2015–17 when part of this work was done.
Article copyright: © Copyright 2019 University Press, Inc.