The motivic fundamental groupoid at tangential basepoints

Talk by Sofian Tur-Dorvault

Date: 14.04.26  Time: 13.15 - 14.45  Room: Y27H25

If U is a smooth scheme over a subfield of the field of complex numbers, it is known from the work of Pierre Deligne and Alexander Goncharov that the prounipotent completion of the fundamental group of U^an based at any point has a motivic incarnation. More precisely, its coordinate ring arises as the degree-zero homology of the Betti realization of a Hopf algebra object in Voevodsky's triangulated category of motives. More generally, Deligne conjectured a similar property for the fundamental group based at a "point at infinity", i.e. the datum of a point x on a smooth compactification of U with normal crossings boundary, together with the datum of a tangent vector at x, normal to the boundary. While Deligne and Goncharov proved this conjecture in the case of the projective line minus three points, the general case remained open. In this talk, I will explain how logarithmic geometry, together with the notion of virtual morphisms between log schemes, allows one to construct the motivic fundamental group in full generality and to compute its realizations.