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A note on Kodaira vanishing on surfaces

Cristian Martinez

Abstract

We give a proof of the Kodaira vanishing theorem on smooth complex surfaces using geometric stability conditions. Likewise, we give a new proof of a result of Xie characterizing the counterexamples of the Kodaira vanishing theorem in positive characteristic.

A note on Kodaira vanishing on surfaces

Abstract

We give a proof of the Kodaira vanishing theorem on smooth complex surfaces using geometric stability conditions. Likewise, we give a new proof of a result of Xie characterizing the counterexamples of the Kodaira vanishing theorem in positive characteristic.

Paper Structure

This paper contains 4 sections, 9 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Stability functions and Schur's lemma
  3. A quick review of stability conditions on surfaces in characteristic zero
  4. Kodaira-type vanishing for surfaces over arbitrary algebraically closed fields

Key Result

Theorem 1.1

Let $X$ be a smooth projective surface over an algebraic closed field $k$. There is a constant $C_{[X]}\geq 0$, depending only on the birational class of $X$ and that vanishes unless $char(k)>0$ and $X$ is either of general type or a quasielliptic surface of Kodaira dimension 1 (see Theorem KosekiCo

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.3: Schur's Lemma
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 4.1
  • Corollary 4.2
  • ...and 8 more