[Paper Review] Semi-stable Higgs sheaves and Bogomolov type inequality

Jiayu Li, Chuanjing Zhang+1 more|arXiv (Cornell University)|Jan 5, 2016

Geometry and complex manifolds15 references3 citations

TL;DR

This paper establishes the existence of an approximate admissible Hermitian-Einstein structure on semi-stable reflexive Higgs sheaves over compact Kähler manifolds, proving a Bogomolov-type inequality for such sheaves via analytic methods. The result confirms a differential-geometric counterpart to the algebraic notion of semi-stability in the Higgs sheaf setting.

ABSTRACT

In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.

Motivation & Objective

• To establish the existence of an approximate admissible Hermitian-Einstein structure on semi-stable reflexive Higgs sheaves over compact Kähler manifolds.
• To extend the Donaldson-Uhlenbeck-Yau theorem to the setting of reflexive Higgs sheaves using analytic techniques.
• To prove a Bogomolov-type inequality for semi-stable reflexive Higgs sheaves on compact Kähler manifolds.
• To resolve Kobayashi's conjecture on approximate Hermitian-Einstein structures in the Higgs sheaf context.

Proposed method

• Constructing a family of Hermitian metrics $ H_{ ho} $ on the regular locus $ M\setminus\Sigma $ of the reflexive Higgs sheaf $ (\mathcal{E}, \phi) $, satisfying approximate Hermitian-Einstein conditions.
• Using the Hitchin-Simpson connection $ D_{H,\phi} $ and its curvature $ F_{H,\phi} $ to define the Hermitian-Einstein equation involving $ \sqrt{-1}\Lambda_{\omega}(F_H + [\phi, \phi^{*H}]) = \lambda \mathrm{Id}_{\mathcal{E}} $.
• Applying a limiting argument via a sequence of approximate solutions $ H_{\epsilon} $, showing convergence in $ C^{\infty} $-topology on $ M\setminus\Sigma $ to a limit metric $ H $.
• Proving that the limit metric $ H $ satisfies the admissible Hermitian-Einstein condition by verifying square-integrability of curvature and uniform boundedness of $ |\Lambda_{\omega}F_H|_H $.
• Using Simpson’s trick to construct a saturated Higgs subsheaf $ \mathcal{F} \subset \mathcal{E} $ with $ \mu_\omega(\mathcal{F}) \geq \mu_\omega(\mathcal{E}) $, leading to a contradiction if $ (\mathcal{E}, \phi) $ is stable.
• Establishing the Bogomolov-type inequality via the existence of the approximate structure and the resulting bounds on Chern classes.

Experimental results

Research questions

• RQ1 Does every semi-stable reflexive Higgs sheaf on a compact Kähler manifold admit an approximate admissible Hermitian-Einstein structure?
• RQ2 Can the Bogomolov inequality be proven for semi-stable reflexive Higgs sheaves on compact Kähler manifolds using analytic methods?
• RQ3 Is the existence of an approximate Hermitian-Einstein structure equivalent to semi-stability for reflexive Higgs sheaves?
• RQ4 Can Kobayashi’s conjecture on approximate Hermitian-Einstein structures be extended to the Higgs sheaf setting?
• RQ5 What are the necessary and sufficient conditions for a reflexive Higgs sheaf to admit an admissible Hermitian-Einstein metric?

Key findings

• A reflexive Higgs sheaf $ (\mathcal{E}, \phi) $ on a compact Kähler manifold is semi-stable if and only if it admits an approximate admissible Hermitian-Einstein structure.
• The limit of a sequence of approximate Hermitian-Einstein metrics $ H_\epsilon $ converges in $ C^{\infty} $-topology on $ M\setminus\Sigma $ to a Hermitian metric $ H $ satisfying the admissible Hermitian-Einstein equation.
• The metric $ H $ is admissible: $ |F_H|_{H,\omega} \in L^2 $ and $ |\Lambda_\omega F_H|_H $ is uniformly bounded on $ M\setminus\Sigma $.
• The Bogomolov-type inequality holds: $ \int_M \left(2c_2(\mathcal{E}) - \frac{r-1}{r}c_1(\mathcal{E}) \wedge c_1(\mathcal{E}) \right) \wedge \frac{\omega^{n-2}}{(n-2)!} \geq 0 $ for a semi-stable reflexive Higgs sheaf of rank $ r $.
• The proof relies on constructing a limiting $ L^1 $-section $ \tilde{u}_0 $ of $ \mathrm{End}(\mathcal{E}) $ with trace zero and using it to derive a contradiction if the Higgs sheaf is stable but not semi-stable.
• The method avoids algebraic-geometric techniques, providing a differential-geometric proof of the Bogomolov inequality in the Kähler setting, distinct from Langer’s algebraic approach.