Monday, March 4, 2024 3:30pm to 4:30pm

About this Event

3620 South Vermont Avenue, Los Angeles, CA 90089

**Fabien Morel, LMU, Munich and IAS**

Abstract: In this talk I will discuss some computations related to the study *A*^1-connected smooth projective schemes *X* over a fixed field *k* (mostly perfect). For a smooth projective *k*-scheme, *A*^1-connected means *A*^1-chain connected, and rational smooth projective *k*-schemes are *A*^1-connected on nice field (char 0 for instance) as we observed some time ago with A. Asok. After briefly introducing the cellular *A*^1-homology sheaves (with Anand Sawant) I will explain why these are good for, for instance by understanding the top dimensional one. I will explain some of the basic facts and difficulties involved concerning strictly *A*^1-invariant sheaves, and give some examples and computations of those. In particular I will explain the case of smooth projective rational surfaces, where Poincare' duality holds (over a perfect field).