# Finiteness of real structures on KLT Calabi-Yau regular smooth pairs of dimension 2

## TL;DR

This paper proves that regular smooth projective complex surfaces with certain Calabi-Yau pairs have finitely many real forms, using CAT(0) geometry and automorphism group actions, with an example illustrating the limits of previous results.

## Contribution

It establishes finiteness of real forms for specific Calabi-Yau pairs on regular surfaces using geometric group actions and provides a counterexample to previous finiteness predictions.

## Key findings

- Finiteness of real forms for regular KLT Calabi-Yau pairs.
- Construction of a CAT(0) metric space for automorphism actions.
- Counterexample with large automorphism group but finite real forms.

## Abstract

In this article, we prove that a smooth projective complex surface $X$ which is regular (i.e. such that $h^1(X,\mathcal O_X)=0$) and which has a $\mathbb{R}$-divisor $\Delta$ such that $(X,\Delta)$ is a KLT Calabi-Yau pair has finitely many real forms up to isomorphism. For this purpose, we construct a complete CAT(0) metric space on which $\text{Aut }X$ acts properly discontinuously and cocompactly by isometries, using Totaro's Cone Theorem. Then we give an example of a smooth rational surface with finitely many real forms but having a so large automorphism group that our previous result (see https://arxiv.org/abs/1409.3490) does not predict this finiteness.