Weekly Bulletin

In this talk, I will introduce a virtual pullback formula between Oh-Thomas virtual cycles for moduli spaces of sheaves on Calabi-Yau 4-folds. I will explain a natural compatibility condition between obstruction theories which provides the formula. In the perspective of derived algebraic geometry, this compatibility condition can be understood as Lagrangian correspondences of (-2)-shifted symplectic derived schemes. The two main applications are the Lefschetz principle and the Pairs/Sheaves correspondence. In particular, these prove various conjectures in DT4 theory under some assumptions. Examples include (1) the Cao-Kool conjecture on tautological Hilbert scheme invariants, (2) the Cao-Kool-Monavari conjecture on DT/PT correspondence, and (3) the Cao-Maulik-Toda conjecture on genus zero Gopakumar-Vafa type invariants.

I will talk about how relative quantum cohomology, defined by Tseng-You and Fan-Wu-You, varies under birational transformations. Relation with FJRW theory and extremal transitions of absolute Gromov--Witten theory will also be discussed.