A note on Kodaira vanishing on surfaces

TL;DR

This paper presents new proofs of the Kodaira vanishing theorem on smooth complex surfaces and characterizes counterexamples in positive characteristic using geometric stability conditions.

Contribution

It introduces novel geometric stability methods to prove Kodaira vanishing and provides a new characterization of counterexamples in positive characteristic.

Findings

Proof of Kodaira vanishing on complex surfaces using stability

New characterization of counterexamples in positive characteristic

Advances understanding of vanishing theorems in algebraic geometry

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.