On a vanishing theorem for surfaces

2025/11/24 by Fujino, Osamu, Moriyama, Nao
Mathematics · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory

paper · doi:10.48550/arxiv.2511.19783

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.

Citations

• Remarks on the minimal model theory for log surfaces in the analytic setting
• Variation of mixed Hodge structure and its applications
• Cone and contraction theorem for projective morphisms between complex analytic spaces
• Vanishing theorems for projective morphisms between complex analytic spaces
• Minimal model theory for log surfaces
• Fundamental theorems for the log minimal model program

Related

• Minimal model theory for log surfaces
• Minimal models and abundance for positive characteristic log surfaces
• Kawamata-Viehweg Vanishing on Rational Surfaces in Positive Characteristic
• Remarks on the minimal model theory for log surfaces in the analytic setting
• Counterexamples to the Kawamata-Viehweg Vanishing on Ruled Surfaces in Positive Characteristic