Talks & Abstracts

Title: Motivic Hecke Algebras
Abstract: The cohomology of locally symmetric spaces, that is, quotients of symmetric spaces by arithmetic groups, carries an action of a highly interesting algebra: the one generated by Hecke operators. When a locally symmetric space S arises from an algebraic variety, as is the case for modular curves, one may ask whether this algebra can be defined by purely algebraic-geometric means—for instance, if the variety is smooth and projective, in terms of algebraic cycles on S×S. In this talk, we will introduce this problem and show how the theory of motives makes it possible to answer this question positively, at least when the variety under consideration can be interpreted as a moduli space of abelian varieties endowed with additional structures.

Title: Algebraic families of higher dimensional A1-contractible affine varieties non-isomorphic to affine space
Abstract: In this talk, we will see positive-dimensional moduli of smooth affine A1-contractible varieties of every dimension n ≥ 4 over fields of characteristic zero which are non-isomorphic to affine spaces and potential counterexamples to the Zariski Cancellation Problem. We further show that these families of varieties are also counterexamples to the generalized Cancellation Problem. 

Title: Lagrangian Fibrations onto varieties with isolated quotient singularities
Abstract: We prove that if $M$ is a holomorphic symplectic manifold of dimension four and if $f\colon M \rightarrow X$ is a projective Lagrangian fibration onto a normal surface, then $X$ must be smooth. In particular, in case $M$ is compact, this gives a new and short proof of a theorem of Ou and Huybrechts-Xu stating that $X$ is isomorphic to $\mathbb{P}^2$. This talk is based on joint work with Zheng Xu.

Title: Amitsur groups of G-varieties
Abstract: The Amitsur subgroup of a variety with a finite group action measures the failure of the action to lift to the total spaces of its line bundles. It is an equivariant birational invariant and, as such, can be used to distinguish conjugacy classes of finite subgroups of Cremona groups. The Amitsur groups of varieties have been studied for curves and rational surfaces over algebraically closed fields. A natural next step is to extend this classification to higher dimensions. In this talk, I will present recent results on the classification of Amitsur groups

for toric varieties and smooth Fano threefolds. Part of this work is joint with Alexander Duncan.