Admissible Hermitian-Yang-Mills connections over normal varieties

TL;DR

This paper extends the Donaldson-Uhlenbeck-Yau theorem to normal varieties, establishing a complete Hitchin-Kobayashi correspondence, and deriving inequalities and characterizations for reflexive sheaves and Hermitian-Yang-Mills connections.

Contribution

It provides the first complete proof of the Donaldson-Uhlenbeck-Yau theorem over normal varieties, including the Hitchin-Kobayashi correspondence and related inequalities.

Findings

Proves the complete Donaldson-Uhlenbeck-Yau theorem over normal varieties.

Establishes the Bogomolov-Gieseker inequality for reflexive sheaves.

Characterizes when a normal Kaehler variety with trivial first Chern class is a quotient of a torus.

Abstract

In this paper, we first prove a complete version of the Donaldson-Uhlenbeck-Yau theorem over normal varieties, including normal Kaehler varieties and projective normal varieties with multiple polarizations. In particular, this gives the polystability of reflexive sheaves under symmetric and exterior powers and tensor products. As a consequence of the singular Donaldson-Uhlenbeck-Yau theorem, the complete Hitchin-Kobayashi correspondence over normal varieties smooth in codimension two is built by showing that an admissible Hermitian-Yang- Mills connection defines a polystable reflexive sheaf. Furthermore, it is shown that the Hermitian-Yang-Mills connection gives a lower bound for the discriminants of any Kaehler resolutions, which gives a Bogomolov-Gieseker inequality over normal varieties and a characterization of the equality using projectively flat connections. We discuss typical cases including normal surfaces and varieties smooth in codimension two where we could simplify the Bogomolov-Gieseker inequality and endow it with topological meanings. We also prove the Bogomolov-Gieseker inequality for semistable reflexive sheaves and characterize the class of semistable sheaves that satisfy the Bogomolov-Gieseker equality. Finally, as another application, we give a new criteria for when a normal Kaehler variety with trivial first Chern class is a finite quotient of torus.