Algebraic surfaces with zero-dimensional cohomology support locus

TL;DR

This paper investigates the structure of algebraic surfaces by analyzing their cohomology support loci, providing conditions that classify surfaces based on their Albanese map and Betti number.

Contribution

It establishes a necessary condition linking the cohomology support locus of finite abelian covers to the classification of smooth projective surfaces.

Findings

Surfaces with finite cohomology support loci have trivial first Betti number, are ruled of genus one, or are abelian surfaces.

Provides a criterion for the Albanese map to be a submersion based on cohomology support loci.

Connects the topology of surfaces to their algebraic and geometric properties.

Abstract

Using the theory of cohomology support locus, we give a necessary condition for the Albanese map of a smooth projective surface being a submersion. More precisely, assuming the cohomology support locus of any finite abelian cover of a smooth projective surface consists of finitely many points, we prove that the surface has trivial first Betti number, or is a ruled surface of genus one, or is an abelian surface.