Biman Roy, USC

Title: Geometric implications of 𝔸^1-connectedness

Abstract: 𝔸^1-homotopy theory, introduced by Morel and Voevodsky, allows us to apply the algebraic topology techniques in algebraic geometry. From the standpoint of homotopy theory of a model category, there is an abstract notion of 𝔸^1-connected component sheaf of a variety. It is a natural goal to understand the 𝔸^1-connected component sheaf of a variety geometrically and this was initiated by Asok and Morel. They proved that a smooth proper variety X over a field k is 𝔸^1-connected if and only if for every finitely generated separable field extension F/k, any two F-points in X can be joined by a chain of 𝔸^1F’s in X. In this talk, we will see that if X is an 𝔸^1-connected smooth variety over an algebraically closed field k, then X is 𝔸^1-uniruled. Thus in particular, if k is of characteristic zero, then X has negative logarithmic Kodaira dimension. We will also see some useful consequences of this result. This is based on my Ph.D. thesis and this is a joint work with Utsav Choudhury.