# On the Kaehler rank of compact complex surfaces

## TL;DR

This paper extends the computation of the Kaehler rank to Kato surfaces, using holomorphic contraction dynamics, thereby completing the classification for all known compact complex surfaces.

## Contribution

It introduces a method to compute the Kaehler rank for Kato surfaces, filling a gap in the classification of compact complex surfaces.

## Key findings

- Kaehler rank computed for Kato surfaces
- Utilizes holomorphic contraction dynamics
- Completes classification for all known surfaces

## Abstract

Harvey and Lawson introduced the Kaehler rank and computed it in connection to the cone of positive exact currents of bidimension (1,1) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions.