Complex algebraic compactifications of the moduli space of HermitianYangMills connections on a projective manifold
Abstract
In this paper we study the relationship between three compactifications of the moduli space of HermitianYangMills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the DonaldsonUhlenbeckYau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by GiesekerMaruyama semistable torsionfree sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, 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 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1810.00025
 Bibcode:
 2018arXiv181000025G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 53C07;
 14D20;
 32G13
 EPrint:
 minor changes to the exposition based on referee's comments