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Stability properties of adapted tangent sheaves on Kähler--Einstein log Fano pairs

Louis Dailly

TL;DR

This work extends Tian’s polystability results to the setting of log Fano pairs with standard coefficients and singular Kähler–Einstein metrics by working on a log resolution endowed with an orbifold structure. The authors develop a framework of orbibundles for the orbifold tangent bundle and its canonical extension, employing orbifold regularizations of the KE metric and adapted covers to transfer stability results from the resolution back to the original pair. They prove semistability first for the orbifold tangent bundle and its extension, then derive polystability via holomorphic splitting arguments and descent along strictly Δ-adapted morphisms. The results pave the way to uniformization conclusions under Miyaoka–Yau type equalities and generalize Tian–Druel–Guenancia–Paun type stability theorems to broader singular settings.

Abstract

Let $(X, Δ)$ be a log Fano pair with standard coefficients endowed with a singular Kähler--Einstein metric. We show that the adapted tangent sheaf $\mathcal{T}_{X, Δ, f}$ and the adapted canonical extension $\mathcal{E}_{X, Δ, f}$ are polystable with respect to $f^*c_1(X, Δ)$ for any strictly $Δ$-adapted morphism $f: Y \to X$.

Stability properties of adapted tangent sheaves on Kähler--Einstein log Fano pairs

TL;DR

This work extends Tian’s polystability results to the setting of log Fano pairs with standard coefficients and singular Kähler–Einstein metrics by working on a log resolution endowed with an orbifold structure. The authors develop a framework of orbibundles for the orbifold tangent bundle and its canonical extension, employing orbifold regularizations of the KE metric and adapted covers to transfer stability results from the resolution back to the original pair. They prove semistability first for the orbifold tangent bundle and its extension, then derive polystability via holomorphic splitting arguments and descent along strictly Δ-adapted morphisms. The results pave the way to uniformization conclusions under Miyaoka–Yau type equalities and generalize Tian–Druel–Guenancia–Paun type stability theorems to broader singular settings.

Abstract

Let be a log Fano pair with standard coefficients endowed with a singular Kähler--Einstein metric. We show that the adapted tangent sheaf and the adapted canonical extension are polystable with respect to for any strictly -adapted morphism .
Paper Structure (20 sections, 7 theorems, 115 equations)

This paper contains 20 sections, 7 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Singular Kähler--Einstein metric and resolutions
  4. Regularizations of a Kähler--Einstein metric on a resolution
  5. Stability of orbifold bundles on a log resolution
  6. Semistability of the orbifold tangent bundle
  7. Analysis of the different terms
  8. Semistability of an extension of the orbifold tangent bundle
  9. Extensions of the orbifold tangent bundle
  10. Inclusion of suborbisheaf
  11. Curvature of the deformation
  12. Semistability of the extension
  13. Proof of Theorem A
  14. Reduction to a log resolution
  15. Polystability of the adapted tangent sheaf
  16. ...and 5 more sections

Key Result

Theorem 1.1

Tia92 Let $X$ be a Fano manifold endowed with a Kähler--Einstein metric $\omega$. Then, the canonical extension $E_X$ is polystable with respect to $c_1(X)$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • ...and 7 more