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

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 $\mathrm{Aut\;} X$ 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.
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https://hal.archives-ouvertes.fr/hal-01494662
Contributor : Mohamed Benzerga <>
Submitted on : Thursday, March 23, 2017 - 6:01:20 PM
Last modification on : Monday, March 9, 2020 - 6:15:55 PM

### Identifiers

• HAL Id : hal-01494662, version 1
• ARXIV : 1702.08808

### Citation

Mohamed Benzerga. Finiteness of real structures on KLT Calabi-Yau regular smooth pairs of dimension 2. 2017. ⟨hal-01494662⟩

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