Index of Kato surfaces

TL;DR

This paper introduces a method to compute the rational coefficients of the first Chern class for Kato surfaces, providing geometric criteria for integrality and relating exponents of the contracting germ to self-intersection numbers.

Contribution

It offers a new computational approach for the Chern class coefficients and links geometric properties of Kato surfaces to their algebraic invariants.

Findings

A formula for the rational coefficients of c_1(S).

A geometric obstruction criterion for c_1(S) to be integral.

An expression for the exponents of the contracting germ in terms of self-intersection numbers.

Abstract

The compact curves of an intermediate Kato surface $S$ form a basis of $H^2(S,\mathbb Q)$. We present a way to compute the associated rational coefficients of the first Chern class $c_1(S)$. We get in particular a simple geometric obstruction for $c_1(S)$ to be an integral class, or equivalently index$(S)=1$. We also find an expression for the exponents of the contracting germ of $S$ in terms of self-intersection numbers of the compact curves.