On a vanishing theorem for surfaces

TL;DR

This paper introduces a new, simplified vanishing theorem for surfaces that streamlines the minimal model program and abundance theorem for log surfaces, bypassing more complex vanishing results.

Contribution

It presents a novel formulation of a vanishing theorem that is easier to apply and enhances the minimal model theory for log surfaces.

Findings

Simplifies the proof of minimal model and abundance theorems for log surfaces

Eliminates the need for deeper vanishing theorems in these proofs

Provides a more accessible tool for surface classification and minimal model programs

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.