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

Greb, Daniel¹ ; Sibley, Benjamin² ; Toma, Matei³ ; Wentworth, Richard⁴

¹ Fakultät für Mathematik, Universität Duisburg–Essen, Essen, Germany
² Departement de Mathematiques, Université Libre de Bruxelles, Brussels, Belgium
³ Institut Élie Cartan de Lorraine, Université de Lorraine, Nancy, France
⁴ Department of Mathematics, University of Maryland, College Park, MD, United States

Geometry & topology, Tome 25 (2021) no. 4, pp. 1719-1818.

Résumé

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.

Détail

DOI : 10.2140/gt.2021.25.1719

Classification : 14D20, 14J60, 32G13, 53C07
Keywords: Hermitian Yang–Mills connections, stability, moduli of coherent sheaves, Kobayashi–Hitchin correspondence, Donaldson–Uhlenbeck compactification

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