Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 3

Special Issue in Honor of Duong H. Phong

Edited by Tristan Collins, Valentino Tosatti, and Ben Weinkove

Continuity of the Yang–Mills flow on the set of semistable bundles

Pages: 909 – 931

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n3.a3

Authors

Benjamin Sibley (Université Libre de Bruxelles, Belgium)

Richard Wentworth (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Abstract

A recent paper [16] studied properties of a compactification of the moduli space of irreducible Hermitian–Yang–Mills connections on a hermitian bundle over a projective algebraic manifold. In this follow-up note, we show that the Yang–Mills flow at infinity on the space of semistable integrable connections defines a continuous map to the set of ideal connections used to define this compactification. Part of the proof involves a comparison between the topologies of the Grothendieck Quot scheme and the space of smooth connections.

Keywords

Yang–Mills flow, semistable bundles, Donaldson–Uhlenbeck compactification

2010 Mathematics Subject Classification

Primary 32G13, 53C07. Secondary 14J60.

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R.W.’s research supported in part by NSF grant DMS-1906403. The authors also acknowledge support from NSF grants DMS-1107452, -1107263, -1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).

Received 17 March 2019

Accepted 15 November 2019

Published 14 June 2021