Finiteness of real structures on KLT Calabi-Yau regular smooth pairs of dimension 2
Résumé
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 $\mathrm{Aut\;} X$ 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 https://arxiv.org/abs/1409.3490 does not predict this finiteness.