Algebraic Geometry Seminar
For complex smooth threefolds, there are some enumerative theories of curves defined using sheaves, such as Donaldson-Thomas (DT) theory using ideal sheaves and Pandharipande-Thomas (PT) theory using stable pairs. These theories are conjecturally related among themselves and conjecturally related to other enumerative theories of curves, such as Gromov-Witten theory. The conjectural relation between DT and PT theories is known only for Calabi-Yau threefolds, where one can use the powerful machinery of motivic Hall algebras due to Joyce and his collaborators.
Bryan-Steinberg (BS) defined enumerative invariants for Calabi-Yau threefolds Y with certain contractions maps Y\to X. I plan to explain how to extend their definition beyond the Calabi-Yau case and what is the conjectural relation to the other enumerative theories. This conjectural relation is known in the Calabi-Yau case by work of Bryan-Steinberg using the motivic Hall algebra. In contrast to the DT/ PT correspondence, we manage to establish the BS/ PT correspondence in some non-Calabi-Yau situations.