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Complex algebraic compactifications of the moduli space of Hermitian Yang–Mills connections on a projective manifold

Daniel Greb, Benjamin Sibley, Matei Toma and Richard Wentworth

Geometry & Topology 25 (2021) 1719–1818

We study the relationship between three compactifications of the moduli space of gauge equivalence classes of Hermitian Yang–Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson–Uhlenbeck–Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker–Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. Following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge-theoretic compactification by adding certain gauge equivalence classes of ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf-theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves, allows us to endow the gauge-theoretic compactification with the structure of a complex analytic space.

Hermitian Yang–Mills connections, stability, moduli of coherent sheaves, Kobayashi–Hitchin correspondence, Donaldson–Uhlenbeck compactification
Mathematical Subject Classification 2010
Primary: 14D20, 14J60, 32G13, 53C07
Received: 5 May 2019
Revised: 28 February 2020
Accepted: 17 June 2020
Published: 12 July 2021
Proposed: Gang Tian
Seconded: Simon Donaldson, Dan Abramovich
Daniel Greb
Fakultät für Mathematik
Universität Duisburg–Essen
Benjamin Sibley
Departement de Mathematiques
Université Libre de Bruxelles
Matei Toma
Institut Élie Cartan de Lorraine
Université de Lorraine
Richard Wentworth
Department of Mathematics
University of Maryland
College Park, MD
United States