# Meeting in December 2014

• Damien Megy: The period map of certain families of singular hypersurfaces. This is a joint work with Philippe Eyssidieux. We consider a natural Deligne-Mumford stack parametrizing degree $d$ hypersurfaces of $\mathcal P^n$ with ADE singularities, and prove an infinitesimal Torelli property along the stacky strata. This construction gives rise to examples of smooth projective varieties with interesting fundamental groups and universal covers. If time permits, I will discuss the Toledo and Shafarevich conjecture for these examples.
• Jean-Philippe Furter: Some properties of the group ${\rm GL}_2 ( {\mathbb C}[x_1,\ldots,x_m])$ and some applications to the polynomial automorphism group ${\rm Aut}( {\mathbb A}^n)$ of the affine space ${\mathbb A}^n$. The groups ${\rm GL}_2 ( {\mathbb C}[x_1,\ldots,x_m])$ and ${\rm Aut}( {\mathbb A}^n)$ can naturally be considered as ind-groups (algebraic groups of infinite dimension). As such, they are endowed with the Zariski topology. We will describe several topological properties of these two groups. In particular, we will give examples of closed subgroups.
• Behrouz Taji: Semistable reflexive sheaves over klt spaces. We prove that any semistable reflexive sheaf over a klt projective variety with vanishing orbiflod Chern classes comes form a linear representation of $\pi_1(X_{\rm{reg}})$. In the smooth setting such a correspondence goes back to the classical results of Narasimhan-Seshadri , Donaldson-Uhlenbeck-Yau and subsequently Simpson. As an application we establish a characterization of finite quotient of Abelian varieties via vanishing of orbifold Chern classes. This is a joint work with Steven Lu.
• Paolo Cascini: Toroidal modifications. I will survey on a conjecture due to Shokurov on the ACC for the set of minimal log discrepancies and I will describe an approach towards this conjecture using toroidal modifications. Joint work with J. McKernan.