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##### Department of Mathematics, University of California San Diego

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## Relative stable pairs and a non-Calabi-Yau wall crossing

##### Abstract:

For complex smooth threefolds, there are 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 by work of Bridgeland, Toda, 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 contraction maps $Y\rightarrow X$. 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.

Host: Dragos Oprea

### Pre-talk at 3:30 PMContact Samir Canning (srcannin@ucsd.edu) for zoom access.

Research Areas

Algebraic Geometry

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