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Total Cartier index of a bounded family

Jingjun Han, Chen Jiang

TL;DR

The paper proves that the total Cartier index is bounded in any bounded family of projective varieties of klt type, answering a folklore question. The authors reduce to bounded families of (Q-)Gorenstein klt models via a key lemma that produces a small birational model with $mK_Y$ Cartier and $Y$ being $\frac{1}{m}$-lc, then apply a framework from HLQ23 to obtain the uniform bound $N$. The approach relies on standard minimal model program techniques, the negativity and semi-ample lemmas, and the boundedness of crepant or related models to control Cartier indices across the family. This yields a foundational result with consequences for epsilon-lc perturbations and related boundedness questions in birational geometry.

Abstract

We prove that the total Cartier index of a bounded family of projective varieties of klt type is bounded.

Total Cartier index of a bounded family

TL;DR

The paper proves that the total Cartier index is bounded in any bounded family of projective varieties of klt type, answering a folklore question. The authors reduce to bounded families of (Q-)Gorenstein klt models via a key lemma that produces a small birational model with Cartier and being -lc, then apply a framework from HLQ23 to obtain the uniform bound . The approach relies on standard minimal model program techniques, the negativity and semi-ample lemmas, and the boundedness of crepant or related models to control Cartier indices across the family. This yields a foundational result with consequences for epsilon-lc perturbations and related boundedness questions in birational geometry.

Abstract

We prove that the total Cartier index of a bounded family of projective varieties of klt type is bounded.

Paper Structure

This paper contains 4 sections, 7 theorems.

Table of Contents

  1. Introduction
  2. Minimal model prgram
  3. Total Cartier index
  4. Examples and Open problems

Key Result

Theorem 1.2

Let $\mathcal{P}$ be a bounded family of projective varieties of klt type. Then there exists a positive integer $N$ depending only on $\mathcal{P}$ such that the total Cartier index of $X$ is bounded from above by $N$ for any $X\in \mathcal{P}$.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 11 more