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Minimal projective varieties satisfying Miyaoka's equality

Masataka Iwai, Shin-ichi Matsumura, Niklas Müller

TL;DR

The paper extends Miyaoka's Chern-class inequalities to minimal projective klt varieties, proving that the Miyaoka equality enforces strong structural consequences: K_X is semi-ample and κ(K_X)=ν(K_X) ∈ {0,1,2}, and X becomes smooth after a finite quasi-étale cover with a precisely described Iitaka fibration. The authors develop and apply a robust framework based on Harder–Narasimhan filtrations of the cotangent sheaf, Q-Chern classes, Higgs theory, and numerically projectively flat structures, to handle both ν(K_X)≤1 and ν(K_X)≥2 cases. They also obtain a parallel result for varieties with nef -K_X under a vanishing condition on ẑc_2(T_X), showing a fibered picture over an abelian base with fiber dimension at most one, and yielding explicit product-type descriptions after suitable covers. Collectively, these results unify and extend prior classifications in the smooth and terminal settings, providing structural theorems that bridge abundance, fibration geometry, and topological considerations via maximally quasi-étale covers and orbifold fibrations.

Abstract

In this paper, we establish a structure theorem for minimal projective klt varieties $X$ that satisfiy Miyaoka's equality $3c_2(X) = c_1(X)^2$. Specifically, we prove that the canonical divisor $K_X$ is semi-ample and that the Kodaira dimension $κ(K_X)$ is either $0$, $1$, or $2$. Furthermore, based on this abundance result, we show that a maximally quasi-étale cover of $X$ is smooth, and we describe explicitly the structure of the Iitaka fibration. Additionally, we prove a similar result for projective klt varieties with a nef anti-canonical divisor.

Minimal projective varieties satisfying Miyaoka's equality

TL;DR

The paper extends Miyaoka's Chern-class inequalities to minimal projective klt varieties, proving that the Miyaoka equality enforces strong structural consequences: K_X is semi-ample and κ(K_X)=ν(K_X) ∈ {0,1,2}, and X becomes smooth after a finite quasi-étale cover with a precisely described Iitaka fibration. The authors develop and apply a robust framework based on Harder–Narasimhan filtrations of the cotangent sheaf, Q-Chern classes, Higgs theory, and numerically projectively flat structures, to handle both ν(K_X)≤1 and ν(K_X)≥2 cases. They also obtain a parallel result for varieties with nef -K_X under a vanishing condition on ẑc_2(T_X), showing a fibered picture over an abelian base with fiber dimension at most one, and yielding explicit product-type descriptions after suitable covers. Collectively, these results unify and extend prior classifications in the smooth and terminal settings, providing structural theorems that bridge abundance, fibration geometry, and topological considerations via maximally quasi-étale covers and orbifold fibrations.

Abstract

In this paper, we establish a structure theorem for minimal projective klt varieties that satisfiy Miyaoka's equality . Specifically, we prove that the canonical divisor is semi-ample and that the Kodaira dimension is either , , or . Furthermore, based on this abundance result, we show that a maximally quasi-étale cover of is smooth, and we describe explicitly the structure of the Iitaka fibration. Additionally, we prove a similar result for projective klt varieties with a nef anti-canonical divisor.
Paper Structure (17 sections, 30 theorems, 154 equations)

This paper contains 17 sections, 30 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Overview and outline of proofs
  4. Preliminary results
  5. Notation and Conventions
  6. Maximally quasi-étale covers
  7. Q-Chern classes
  8. Higgs sheaves
  9. Numerically projectively flat sheaves
  10. Filtrations of reflexive sheaves
  11. Harder-Narasimhan filtrations of generically nef reflexive sheaves
  12. Case of
  13. Case of
  14. Proof of Theorem \ref{['thm-main1']}
  15. Case of
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $X$ be a projective klt variety of dimension $n$ with nef canonical divisor $K_{X}$. Then, the following statements hold$:$$(A)$ Miyaoka's inequality holds for ample divisors $H_1, \ldots, H_{n-2}$ on $X$$:$ where $\widehat{c}_2(\Omega_{X}^{[1]})$ and $\widehat{c}_1(\Omega_{X}^{[1]})$ denote the $\mathbb{Q}$-Chern classes of the cotangent sheaf $\Omega_{X}^{[1]}$. $(B)$ Assume that Miyaoka's e

Theorems & Definitions (69)

  • Conjecture 1.1: Abundance conjecture for minimal klt varieties
  • Theorem 1.2: Main Result
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1: GKP22
  • Definition 2.2
  • Proposition 2.3: Hodge index theorem
  • Remark 2.4
  • proof
  • Lemma 2.5
  • ...and 59 more