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2021 Complex algebraic compactifications of the moduli space of Hermitian Yang–Mills connections on a projective manifold
Daniel Greb, Benjamin Sibley, Matei Toma, Richard Wentworth
Geom. Topol. 25(4): 1719-1818 (2021). DOI: 10.2140/gt.2021.25.1719


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.


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Daniel Greb. Benjamin Sibley. Matei Toma. Richard Wentworth. "Complex algebraic compactifications of the moduli space of Hermitian Yang–Mills connections on a projective manifold." Geom. Topol. 25 (4) 1719 - 1818, 2021.


Received: 5 May 2019; Revised: 28 February 2020; Accepted: 17 June 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.1719

Primary: 14D20 , 14J60 , 32G13 , 53C07

Keywords: Donaldson–Uhlenbeck compactification , Hermitian Yang–Mills connections , Kobayashi–Hitchin correspondence , moduli of coherent sheaves , stability

Rights: Copyright © 2021 Mathematical Sciences Publishers


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Vol.25 • No. 4 • 2021
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