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AG Seminar -- 15.12.2022

Shubhodip Mondal -- Unipotent homotopy theory of schemes

Abstract. Building on Toen's work on higher (affine) stacks, I will discuss a notion of homotopy theory for algebraic varieties, which we call ``unipotent homotopy theory". Over a field of char. p>0, I will explain how these unipotent homotopy group schemes recover (1) unipotent completion of the Nori fundamental group scheme, (2) p-adic 'etale homotopy groups, and (3) certain formal group laws arising from algebraic varieties constructed by Artin--Mazur. Time permitting, I will discuss unipotent homotopy types of Calabi--Yau varieties and show that the unipotent homotopy group schemes \pi^U_i of Calabi--Yau varieties (of dimension n) are derived invariant for all i; the case i = n corresponds to a recent result of Antieau--Bragg. Joint work with Emanuel Reinecke (in progress).