# Algebra/Topology Seminar

Speaker: Lukas Brantner

Title: On unobstructedness and liftings of Calabi–Yau varieties in characteristic p

Abstract: The Bogomolov–Tian–Todorov theorem asserts that every Calabi–Yau variety Z over an algebraically closed field k of characteristic zero is unobstructed. In joint work with Taelman, we use derived algebraic geometry to establish analogues of this result when k has characteristic p. More precisely, we show that if Z has degenerating Hodge–de Rham spectral sequence and torsion-free crystalline cohomology, then its mixed characteristic formal deformations are unobstructed. If Z is ordinary, we moreover prove that its deformation space is a formal torus, which implies unobstructedness and the existence of a canonical lift. Our work generalises ealier results by Achinger–Zdanowicz, Deligne–Nygaard, Ekedahl–Shepherd-Barron, Schröer, Serre–Tate, and Ward.