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On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic

Fei Ye, Zhixian Zhu

Abstract

In this paper, we study the equivalence between Bogomolov's instability theorem and the Miyaoka-Sakai theorem on surfaces in positive characteristic. We show that Bogomolov's instability theorem can be derived from Miyaoka-Sakai theorem. Conversely, it implies a partial version of the Miyaoka-Sakai theorem that lacks the vanishing conclusion. This partial version is still sufficient to deduce the Mumford-Ramanujam vanishing theorem. Additionally, we identify a class of surfaces in positive characteristic for which the Miyaoka-Sakai theorem (or a weaker variant), or the Kawamata-Viehweg vanishing theorem holds. In particular, we present a new proof of the Kawamata-Viehweg vanishing theorem on smooth del Pezzo surfaces. As an application of the Miyaoka-Sakai theorem, we obtain Reider-type results concerning Fujita's conjecture.

On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic

Abstract

In this paper, we study the equivalence between Bogomolov's instability theorem and the Miyaoka-Sakai theorem on surfaces in positive characteristic. We show that Bogomolov's instability theorem can be derived from Miyaoka-Sakai theorem. Conversely, it implies a partial version of the Miyaoka-Sakai theorem that lacks the vanishing conclusion. This partial version is still sufficient to deduce the Mumford-Ramanujam vanishing theorem. Additionally, we identify a class of surfaces in positive characteristic for which the Miyaoka-Sakai theorem (or a weaker variant), or the Kawamata-Viehweg vanishing theorem holds. In particular, we present a new proof of the Kawamata-Viehweg vanishing theorem on smooth del Pezzo surfaces. As an application of the Miyaoka-Sakai theorem, we obtain Reider-type results concerning Fujita's conjecture.
Paper Structure (6 sections, 31 theorems, 57 equations)

This paper contains 6 sections, 31 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From Bogomolov to Miyaoka-Sakai and then Mumform-Ramanujam
  4. From Miyaoka-Sakai to Bogomolov
  5. Vanishing Theorems on Surfaces of Kodaira Dimension at Most Zero
  6. Weak Miyaoka-Sakai theorem

Key Result

Theorem 1.1

Let $S$ be a smooth surface over an algebraically closed field of characteristic zero and $D$ be a big divisor such that $H^1(S, \mathcal{O}_S(-D))\ne 0$. There exists a nonzero effective divisor $B$ such that

Theorems & Definitions (60)

  • Theorem 1.1: Miyaoka-Sakai Theorem (see Miyaoka1980, Sakai1990)
  • Definition 2.1
  • Theorem 2.2: Bogomolov1979, Shepherd-Barron1991, Langer2016, Koseki2023
  • Theorem 3.1: Effective Miyaoka-Sakai Theorem
  • proof
  • Corollary 3.2: Effective Ramanujam Vanishing Theorem
  • proof
  • Lemma 3.3: Ramanujam's Connectedness Lemma (see Ramanujam1974, Kawachi1998, or Sakai1990)
  • Corollary 3.4: Effective Mumford-Ramanujam Vanishing Theorem
  • proof
  • ...and 50 more