Geometry Seminar
A formality result for logarithmic Hochschild (co)homology
Speaker: Marton Hablicsek (University of Leiden)Date: Monday, 03 February 2025 - 11:30Place: Aula Naranja, ICMAT
Abstract:
Hochschild homology is a foundational invariant for associate algebras, schemes, stacks, etc. For instance, for smooth, complex, projective varieties X, Hochschild homology and its variants, like cyclic homology, are closely related to Hodge cohomology and to de Rham cohomology. In this talk, via a geometric approach, we extend Hochschild homology to logarithmic schemes, in particular to compactifications, i.e, to pairs (X,D) where X is a smooth and proper variety and D is a simple normal crossing divisor. This geometric approach allows us to extend well-known facts about Hochschild homology and its variants to a logarithmic setting, in particular, (1) we generalize the celebrated HKR theorem to relate logarithmic Hochschild homology to logarithmic Hodge cohomology, (2) we define and provide a description of logarithmic cyclic homology showing how it is related to the logarithmic de Rham complex, (3) we show that logarithmic Hochschild homology is log derived invariant, and (4) we compute log Hochschild (co)homology of logarithmic orbifolds. This is a joint work with Francesca Leonardi and Leo Herr.
