Algebra Seminar: K-theory of logarithmic schemes via toric mirror symmetry

Federico Binda, University of Milan

Title: K-theory of logarithmic schemes via toric mirror symmetry

Abstract: Logarithmic geometry, developed by Fontaine, Illusie, Kato and many others, is a way to package information about degeneration and boundary phenomena in arithmetic and algebraic geometry. Since its introduction, it has been an open question to find a good notion of quasi coherent sheaf on a log scheme, leading to an appropriate categorical definition of K-theory and other localizing invariants (like topological Hochschild and cyclic homology). We offer a solution to this problem: building on the work of Talpo and Vistoli, we study a modified category of quasi-coherent sheaves on the infinite root stacks. In the case of toric varieties, a surprising result of Vaintrob (revisited by Scholze) relates this category to sheaves on topological spaces, in an instance of toric mirror symmetry. Using some recent results of Efimov on K-theory of dualizable categories, these invariants are in fact accessible to calculations. Joint work in progress with Ben Antieau, Tess Bouis, Brian Shin.