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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 |
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Geometry & Topology 25 (2021) 1719–1818
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Abstract |
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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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Keywords
Hermitian Yang–Mills connections, stability, moduli of
coherent sheaves, Kobayashi–Hitchin correspondence,
Donaldson–Uhlenbeck compactification
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Mathematical Subject Classification 2010
Primary: 14D20, 14J60, 32G13, 53C07
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References |
Publication
Received: 5 May 2019
Revised: 28 February 2020
Accepted: 17 June 2020
Published: 12 July 2021
Proposed: Gang Tian
Seconded: Simon Donaldson, Dan Abramovich
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Authors |
| Daniel Greb | |
| Fakultät für Mathematik Universität Duisburg–Essen Essen Germany |
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| https://www.esaga.uni-due.de/daniel.greb/ | |
| Benjamin Sibley | |
| Departement de Mathematiques Université Libre de Bruxelles Brussels Belgium |
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| https://geometry.ulb.ac.be/ben-sibley/ | |
| Matei Toma | |
| Institut Élie Cartan de
Lorraine Université de Lorraine Nancy France |
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| https://iecl.univ-lorraine.fr/membre-iecl/toma-matei/ | |
| Richard Wentworth | |
| Department of Mathematics University of Maryland College Park, MD United States |
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| https://www.math.umd.edu/~raw/ | |
