Abstract: I will explain a construction of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin– Mumford threefold. As an application, we will answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties.

Abstract: We want to study connected algebraic groups that act faithfully and by birational maps on a variety X. Enriques classified these groups when X is the plane, and Umemura classified them in the 80ies when X is the projective space. From these classification we obtain that any such connected algbebraic group is contained in a maximal connected algebraic group acting birationally on X. In this talk I want to motivate why this does no longer holds when X is rational of dimension at least 4. This is joint work with A. Fanelli and E. Floris.

Abstract: On smooth algebraic varieties there is a good theory of Chow groups, encoding algebraic cycles and their intersections, which is moreover closely related to algebraic K-theory and vector bundles through the Riemann-Roch formula. A question raised in the 1980s, and investigated in particular by Collino, Levine, Pedrini, Srinivas, and Weibel is whether the theory can be extended to singular varieties. We will explain how this can indeed be done, by using non-A¹-invariant motivic cohomology to define well-behaved cohomological Chow groups of arbitrarily singular varieties. Joint work with Elden Elmanto.