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Remarks on log pluricanonical representations

Osamu Fujino, Jinsong Xu

TL;DR

The paper proves finiteness of log pluricanonical representations under the assumption that a projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ being $\mathbb{Q}$-Cartier admits a good minimal model. It reduces the problem to the established Fujino–Gongyo finiteness result for the good minimal model $(X',\Delta')$ and transfers the finiteness back to $(X,\Delta)$ via birational conjugation. The results extend to the quasi-projective PBir setting for a smooth $V$ and to the affine case where $\mathrm{PBir}(V)=\mathrm{Aut}(V)$, all under the same good minimal model framework. Collectively, these findings contribute to the understanding of automorphism representations in the context of abundance-type questions and broaden the applicability of log pluricanonical finiteness across lc, semi-log canonical, and affine/quasi-projective scenarios.

Abstract

We show the finiteness of log pluricanonical representations under the assumption of the existence of a good minimal model.

Remarks on log pluricanonical representations

TL;DR

The paper proves finiteness of log pluricanonical representations under the assumption that a projective log canonical pair with being -Cartier admits a good minimal model. It reduces the problem to the established Fujino–Gongyo finiteness result for the good minimal model and transfers the finiteness back to via birational conjugation. The results extend to the quasi-projective PBir setting for a smooth and to the affine case where , all under the same good minimal model framework. Collectively, these findings contribute to the understanding of automorphism representations in the context of abundance-type questions and broaden the applicability of log pluricanonical finiteness across lc, semi-log canonical, and affine/quasi-projective scenarios.

Abstract

We show the finiteness of log pluricanonical representations under the assumption of the existence of a good minimal model.

Paper Structure

This paper contains 3 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs

Key Result

Theorem 1.1

Let $(X, \Delta)$ be a projective log canonical pair such that $K_X+\Delta$ is $\mathbb Q$-Cartier. Assume that $(X, \Delta)$ has a good minimal model. Then there exists a positive integer $k$ such that the image of is a finite group for every positive integer $m$, where

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4: Log canonical pairs of log general type
  • Remark 1.5: Kawamata log terminal pairs
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8: Affine varieties
  • Definition 2.1: $B$-birational maps
  • Definition 2.2: Proper birational maps
  • ...and 4 more