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Admissible Hermitian-Yang-Mills connections over normal varieties

Xuemiao Chen

TL;DR

The paper extends the Donaldson-Uhlenbeck-Yau theory to normal varieties by introducing a complete singular DUY theorem for reflexive sheaves under multipolarizations and establishing a full Hitchin– Kobayashi correspondence on varieties smooth in codimension two. It defines and utilizes a discriminant via all resolutions to prove a Bogomolov–Gieseker inequality in the singular setting, with equality characterizing projectively flat bundles and energy-vanishing scenarios. The work leverages balanced metrics of Hodge–Riemann type, perturbation techniques, and Uhlenbeck–type compactness to control limits, enabling polystability results for reflexive sheaves and new criteria for when a normal Kaehler variety with trivial first Chern class is a finite torus quotient. Overall, the results provide a robust analytic and algebro-geometric framework for stability, Chern-class inequalities, and structural characterization of singular varieties in higher dimensions.

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.

Admissible Hermitian-Yang-Mills connections over normal varieties

TL;DR

The paper extends the Donaldson-Uhlenbeck-Yau theory to normal varieties by introducing a complete singular DUY theorem for reflexive sheaves under multipolarizations and establishing a full Hitchin– Kobayashi correspondence on varieties smooth in codimension two. It defines and utilizes a discriminant via all resolutions to prove a Bogomolov–Gieseker inequality in the singular setting, with equality characterizing projectively flat bundles and energy-vanishing scenarios. The work leverages balanced metrics of Hodge–Riemann type, perturbation techniques, and Uhlenbeck–type compactness to control limits, enabling polystability results for reflexive sheaves and new criteria for when a normal Kaehler variety with trivial first Chern class is a finite torus quotient. Overall, the results provide a robust analytic and algebro-geometric framework for stability, Chern-class inequalities, and structural characterization of singular varieties in higher dimensions.

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.
Paper Structure (34 sections, 39 theorems, 108 equations)

This paper contains 34 sections, 39 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Singular Donaldson-Uhlenbeck-Yau theorem
  3. Hitchin-Kobayashi correspondence over normal varieties smooth in codimension two
  4. Bogomolov-Gieseker inequality
  5. Simplifications of Bogomolov-Gieseker inequalities
  6. Over normal surfaces
  7. Smooth in codimension two
  8. Bogomolov-Gieseker inequality for semistable sheaves
  9. Quotients of torus
  10. Sketch of the proof
  11. Regarding singular Donaldson-Uhlenbeck-Yau theorem
  12. Bogomolov-Gieseker inequality
  13. Preliminary results
  14. Varieties with mutiple Kähler metrics
  15. Donaldson-Uhlenbeck-Yau theorem for balanced metrics of Hodge-Riemann type
  16. ...and 19 more sections

Key Result

Theorem 1.1

Suppose $\mathcal{E}$ is a stable reflexive sheaf over a normal variety $(X, \omega_1 \wedge \cdots \omega_{n-1})$. Then there exists an admissible Hermitian-Yang-Mills metric on $\mathcal{E}$. Moreover, such a metric is unique up to scaling.

Theorems & Definitions (77)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Corollary 1.9: When $\mathcal{E}$ is locally free over a normal surface
  • ...and 67 more