In this talk I will discuss some approach to study A^1-connected smooth projective varieties X over a fixed field k (for instance rational ones) based on A^1-homotopy and the classical "surgery" approach.

Being A^1-connected was introduced by Voevodsky and the author long ago, and says roughly that any two F-points of X can be joined by a sequence of rational F-curves in X . One main difference with classical topology is that a smooth projective k-variety of positive dimension which is A^1-connected has a non trivial A^1-fundamental (sheaf of) group(s). I will give examples of such X and also of A^1-fundamental groups.

Then I will discuss a very concrete conjecture of Anand Sawant and myself determining when a smooth projective A^1-connected variety should be orientable (the maximal external power of the tangent space is a square) in terms of its top A^1-cellular homology, introduced in a joint work with Anand Sawant. I will give some evidence and examples of computations.

In particular I will explain the case of surfaces, which is entirely understood (over a perfect field).

### Penn Mathematics Colloquium

Wednesday, November 29, 2023 - 3:45pm

#### Fabien Morel

LMU Munich & IAS Princeton