Mohamed Benzerga | Searcharxiv

arXiv2017

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

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 $Δ$ such that $(X,Δ)$ 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.

math.AG↗

arXiv2015

Real structures on rational surfaces and automorphisms acting trivially on Picard groups

In this article, we prove that any complex smooth rational surface $X$ which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if $X$ cannot be obtained by blowing up $\mathbb P^2_{\mathbb C}$ at $r\geq 10$ points). In particular, we prove that the group $\mathrm{Aut}^{\#}X$ of complex automorphisms of $X$ which act trivially on the Picard group of $X$ is a linear algebraic group defined over $\mathbb R$.

math.AG↗