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Miyaoka-Yau equality and uniformization of log Fano pairs

Louis Dailly

TL;DR

The paper proves a uniformization criterion for log Fano pairs $(X,\Delta)$ with standard coefficients: such a pair is isomorphic to a quotient $(\mathbb{P}^n/G,\Delta_G)$ for a finite subgroup $G\subseteq\mathrm{PGL}(n+1,\mathbb{C})$ if and only if the canonical extension of the orbifold pair is semistable with respect to $-(K_X+\Delta)$ and the orbifold Miyaoka–Yau equality $(2(n+1)c_2(X,\Delta)-nc_1(X,\Delta)^2)\cdot c_1(X,\Delta)^{n-2}=0$ holds. The authors develop a framework of adapted canonical extensions for both orbifolds and klt pairs, along with a robust theory of orbisheaves, orbistructures, and pullbacks, to prove developability and to perform orbifold Chern-class computations. The two main implications (Quo) => (Unif) and (Unif) => (Quo) are established via a maximally quasi-étale cover, flat endomorphism bundles, and projective flatness, ultimately yielding that the universal cover is $\mathbb{P}^n$ and the base is a finite quotient of projective space. The paper also provides concrete examples, such as weighted projective spaces and log smooth quotients of $\mathbb{P}^2$, illustrating the construction of $\Delta_G$ and discriminants. Overall, the work extends uniformization results to positive-slope log Fano pairs, giving explicit structural and cohomological criteria for when a pair arises from a projective space quotient.

Abstract

Let $(X, Δ)$ be a log Fano pair with standard coefficients. We show that if it satisfies the equality case in the Miyaoka-Yau inequality, then its orbifold universal cover is a projective space.

Miyaoka-Yau equality and uniformization of log Fano pairs

TL;DR

The paper proves a uniformization criterion for log Fano pairs with standard coefficients: such a pair is isomorphic to a quotient for a finite subgroup if and only if the canonical extension of the orbifold pair is semistable with respect to and the orbifold Miyaoka–Yau equality holds. The authors develop a framework of adapted canonical extensions for both orbifolds and klt pairs, along with a robust theory of orbisheaves, orbistructures, and pullbacks, to prove developability and to perform orbifold Chern-class computations. The two main implications (Quo) => (Unif) and (Unif) => (Quo) are established via a maximally quasi-étale cover, flat endomorphism bundles, and projective flatness, ultimately yielding that the universal cover is and the base is a finite quotient of projective space. The paper also provides concrete examples, such as weighted projective spaces and log smooth quotients of , illustrating the construction of and discriminants. Overall, the work extends uniformization results to positive-slope log Fano pairs, giving explicit structural and cohomological criteria for when a pair arises from a projective space quotient.

Abstract

Let be a log Fano pair with standard coefficients. We show that if it satisfies the equality case in the Miyaoka-Yau inequality, then its orbifold universal cover is a projective space.
Paper Structure (27 sections, 13 theorems, 47 equations)

This paper contains 27 sections, 13 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Orbifolds
  3. Generalities on pairs
  4. Pairs and orbistructures
  5. Orbifold morphism and pullback orbistructure
  6. Fundamental group and universal cover in the orbifold setting
  7. Canonical extension of an orbifold (X,D)
  8. Generalities on orbisheaves
  9. Canonical extension of an orbifold
  10. Flatness and developability
  11. Orbi-locally freeness and sheaf of invariants
  12. Description of orbifolds with an orbi-locally free tangent bundle
  13. Flatness and developability
  14. Klt pairs
  15. Adapted canonical extension
  16. ...and 12 more sections

Key Result

Proposition 2.20

Let $\mathcal{S}$, $\mathcal{Q}$ be two orbisheaves on an orbifold $(X, \Delta)$, with transition morphisms denoted by $\varphi_{\alpha\beta}$ and $\psi_{\alpha\beta}$ respectively. Let $\Omega := \left( \omega_{\alpha\beta} \colon g_{\beta \alpha}^* \mathcal{Q}_\beta \longrightarrow g_{\alpha \beta then, it induces an extension $\mathcal{F}$ of $\mathcal{Q}$ by $\mathcal{S}$, it means a short exa

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 49 more