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On a vanishing theorem for surfaces

Osamu Fujino, Nao Moriyama

TL;DR

The paper introduces a new vanishing theorem for surfaces that follows from the Kawamata–Viehweg vanishing theorem and proves it via a pair of lemmas. This formulation suffices for the minimal model program and abundance theory of log surfaces, enabling results previously relying on deeper vanishing theories to be derived using only Kawamata–Viehweg vanishing in both algebraic and analytic settings. Consequently, the authors show how the core surface theory in Fujino1, Fujino2, Fujino5, and Moriyama can be obtained with a simpler vanishing toolkit. The work thus streamlines the conceptual framework for surface MMP/abundance and broadens accessibility to researchers focused on log surfaces.

Abstract

We propose a new formulation of a vanishing theorem for surfaces. Although this vanishing theorem follows easily from the well-known Kawamata--Viehweg vanishing theorem, it turns out to be remarkably useful. In particular, it is sufficient for the minimal model theory of log surfaces, and it allows one to carry out both the minimal model program and the abundance theorem for log surfaces without invoking any of the deeper vanishing theorems.

On a vanishing theorem for surfaces

TL;DR

The paper introduces a new vanishing theorem for surfaces that follows from the Kawamata–Viehweg vanishing theorem and proves it via a pair of lemmas. This formulation suffices for the minimal model program and abundance theory of log surfaces, enabling results previously relying on deeper vanishing theories to be derived using only Kawamata–Viehweg vanishing in both algebraic and analytic settings. Consequently, the authors show how the core surface theory in Fujino1, Fujino2, Fujino5, and Moriyama can be obtained with a simpler vanishing toolkit. The work thus streamlines the conceptual framework for surface MMP/abundance and broadens accessibility to researchers focused on log surfaces.

Abstract

We propose a new formulation of a vanishing theorem for surfaces. Although this vanishing theorem follows easily from the well-known Kawamata--Viehweg vanishing theorem, it turns out to be remarkably useful. In particular, it is sufficient for the minimal model theory of log surfaces, and it allows one to carry out both the minimal model program and the abundance theorem for log surfaces without invoking any of the deeper vanishing theorems.

Paper Structure

This paper contains 9 sections, 5 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas
  4. Proof of Theorem \ref{['a-thm1.1']}
  5. On the minimal model theory for log surfaces
  6. On the fundamental theorems established in fujino1
  7. On the minimal model theory for log surfaces in fujino2
  8. On the cone and contraction theorem established in fujino5
  9. On the minimal model theory for analytic log surfaces in moriyama

Key Result

Theorem 1.1

Let $X$ be a smooth complex analytic surface and let $\Delta$ be a boundary $\mathbb{R}$-divisor on $X$ such that $\mathop{\mathrm{Supp}}\nolimits \Delta$ is a simple normal crossing divisor. Let $f\colon X \to Y$ be a bimeromorphic morphism to a complex analytic space $Y$, and let $\pi\colon Y \to for every $p>0$ and every $q$.

Theorems & Definitions (12)

  • Theorem 1.1: Main Theorem
  • Definition 2.1: Log surfaces
  • Definition 2.2: Nefness
  • Definition 2.3: Bigness
  • Theorem 2.4: Relative Kawamata--Viehweg vanishing theorem
  • Lemma 3.1: Vanishing theorem of Reid--Fukuda type
  • proof : Proof of Lemma \ref{['a-lem3.1']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['a-lem3.2']}
  • proof : Proof of Theorem \ref{['a-thm1.1']}
  • ...and 2 more