Speaker:  Pierre Descombes (EPF Lausanne)

Abstract: Given a scheme X with an action of a one-dimensional torus, the hyperbolic localization functor, which restricts constructible complexes from X to the attracting variety X^+ and then projects with compact support to the fixed variety X^0, was introduced by Braden in order to study generalizations of Białynicki-Birula decompositions beyond the smooth case. Richarz has then proven that this functor commutes with vanishing cycles.
Using shifted symplectic geometry and a shifted Darboux theorem, moduli spaces of sheaves on Calabi-Yau threefolds are described locally by critical loci of functions on smooth spaces, which are related locally by adding quadratic forms to the functions. On such moduli spaces, a DT perverse sheaf, whose cohomology gives the cohomological DT invariants, was defined by Brav, Bussi, Dupont, Joyce, and Szendroï by gluing vanishing cycles on such local models, involving a subtle trivialization of the action of quadratic forms using orientation data.
We will explain here how to prove a formula for the hyperbolic localization of the DT perverse sheaf, combining the results of Białynicki-Birula and Richarz with a study of the behavior of hyperbolic localization with quadratic forms and orientations. One obtains in particular from this result a critical version of Białynicki-Birula decomposition in cohomological DT theory.
We will also explain how to obtain a stacky version of the above result, replacing X^+ and X^0 by the stacks of filtered and graded points, which has recently led to the proof of fundamental results in DT theory, namely the proof of the Kontsevich-Soibelman wall-crossing formula and the construction of the cohomological Hall algebra for CY3 categories.