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Characterising Ball Quotients through their (higher) Chern Numbers

Niklas Müller

TL;DR

The paper provides a complete Chern-number–based characterization of ball quotient varieties among minimal smooth projective varieties of general type. Building on GKPT and Miyaoka–Yau type ideas, it employs stringy Euler numbers and crepant resolutions to derive higher-Chern inequalities; equality cases identify when the canonical model is a ball quotient and when the original variety is isomorphic to it away from a codimension-$k$ locus. The main result shows that, under big and nef canonical bundle, a sequence of lower-Chern equalities up to index $k-1$ forces a sharp lower bound on $c_k(X)\cdot K_X^{n-k}$ with equality characterising ball quotient canonical models and birational maps that are isomorphisms on open sets. This extends known two-dimensional criteria to higher dimensions and provides a practical numerical criterion for recognizing ball quotients from Chern data.

Abstract

In this short note we provide a characterisation of ball quotients among all minimal smooth projective varieties of general type purely in terms of their characteristic numbers. This generalises earlier work of Miyaoka, Yau and Greb--Kebekus--Peternell--Taji.

Characterising Ball Quotients through their (higher) Chern Numbers

TL;DR

The paper provides a complete Chern-number–based characterization of ball quotient varieties among minimal smooth projective varieties of general type. Building on GKPT and Miyaoka–Yau type ideas, it employs stringy Euler numbers and crepant resolutions to derive higher-Chern inequalities; equality cases identify when the canonical model is a ball quotient and when the original variety is isomorphic to it away from a codimension- locus. The main result shows that, under big and nef canonical bundle, a sequence of lower-Chern equalities up to index forces a sharp lower bound on with equality characterising ball quotient canonical models and birational maps that are isomorphisms on open sets. This extends known two-dimensional criteria to higher dimensions and provides a practical numerical criterion for recognizing ball quotients from Chern data.

Abstract

In this short note we provide a characterisation of ball quotients among all minimal smooth projective varieties of general type purely in terms of their characteristic numbers. This generalises earlier work of Miyaoka, Yau and Greb--Kebekus--Peternell--Taji.
Paper Structure (3 sections, 5 theorems, 28 equations)

This paper contains 3 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Stringy Euler number
  3. Proof of main results

Key Result

Theorem 1

(Miyaoka Miyaoka_ChernNumbersSurfacesGeneralType) Let $X$ be a smooth complex projective surface such that $K_X$ is big and nef. Then $X$ is isomorphic to a quotient of the complex unit ball $\mathbb{B}^2$ if and only if

Theorems & Definitions (9)

  • Theorem
  • Theorem A
  • Theorem B
  • Example 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm-inequalities-higher-Chern-classes']}
  • proof : Proof of Theorem \ref{['thm-main-result']}