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Remarks on the minimal model theory for log surfaces in the analytic setting

Nao Moriyama

TL;DR

This work extends the relative log minimal model program, abundance, and finite generation from algebraic to analytic settings for log surfaces. By introducing and exploiting condition ($\bigstar$) and leveraging Fujino’s cone and contraction framework, the authors establish an MMP over a neighborhood of a compact subset $W$ of the base, yielding either a good minimal model with semi-ample $K_X+\Delta$ or a Mori fiber space. They prove the relative abundance first for $\mathbb{Q}$-divisors and then for $\mathbb{R}$-divisors via Shokurov’s polytope, overcoming the analytic challenges of non-global $\mathbb{Q}$-factoriality and lack of compactification. As a corollary, the log canonical ring is locally finitely generated over $Y$, supporting the analytic analog of canonical model theory for log surfaces. These results solidify the analytic minimal model theory and provide a robust foundation for further applications in complex geometry.

Abstract

We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of complex surfaces which are projective over complex analytic varieties.

Remarks on the minimal model theory for log surfaces in the analytic setting

TL;DR

This work extends the relative log minimal model program, abundance, and finite generation from algebraic to analytic settings for log surfaces. By introducing and exploiting condition () and leveraging Fujino’s cone and contraction framework, the authors establish an MMP over a neighborhood of a compact subset of the base, yielding either a good minimal model with semi-ample or a Mori fiber space. They prove the relative abundance first for -divisors and then for -divisors via Shokurov’s polytope, overcoming the analytic challenges of non-global -factoriality and lack of compactification. As a corollary, the log canonical ring is locally finitely generated over , supporting the analytic analog of canonical model theory for log surfaces. These results solidify the analytic minimal model theory and provide a robust foundation for further applications in complex geometry.

Abstract

We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of complex surfaces which are projective over complex analytic varieties.

Paper Structure

This paper contains 6 sections, 16 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal model program
  4. Abundance for Q-divisors
  5. Abundance for R-divisors
  6. Proof of the main theorem and corollaries

Key Result

Theorem 1.1

Assume that $(X, Y, W, \pi, \Delta)$ satisfies the condition ($\bigstar$) and that one of the following conditions holds: We shrink $Y$ around $W$ suitably. Then there is a sequence of at most $\rho(X/Y;W)-1$ contractions over $Y$ such that We note that $K_{X_i}$ is $\mathbb{Q}$-Cartier and every prime divisor on $X_i$ which is mapped into $W$ is $\mathbb{Q}$-Cartier for every $i$ in Case (A),

Theorems & Definitions (51)

  • Theorem 1.1: see, Fjn12
  • Remark 1.2
  • Definition 1.3: Mori fiber space
  • Theorem 1.4: Abundance theorem, see Theorem \ref{['thm: abundance for R-divisors']}
  • Corollary 1.5: see, Fjn12
  • Corollary 1.6: see, Fjn12
  • Definition 2.1
  • Remark 2.2: Stein factorization
  • Definition 2.3: Nefness, see Fjn22
  • Definition 2.4: Suitable neighborhoods
  • ...and 41 more