On complex surfaces with definite intersection form

TL;DR

This paper classifies compact complex surfaces based on the definiteness of their intersection form, identifying specific types for positive and negative definite cases, and describing their lattice properties.

Contribution

It provides a complete classification of complex surfaces with definite intersection forms, detailing their types and lattice characteristics.

Findings

Surfaces with positive definite intersection form are either projective planes or false projective planes.

Negative definite cases include non-minimal secondary Kodaira, elliptic, and class VII surfaces.

All such lattices are odd and diagonalizable over the integers.

Abstract

A compact complex surface with positive definite intersection lattice is either the projective plane or a false projective plane. If the intersection lattice is negative definite, the surface is either a non-minimal secondary Kodaira surface, a non-minimal elliptic surface with $b_1=1$, or a class VII surface with $b_2>0$. In all cases the lattice is odd and diagonalizable over the integers.