## Algebraic Geometry Seminar: Artan Sheshmani (UBC)

- Date: 09/19/2011

University of British Columbia

**Abstract:**

We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_X^{r}(-n)-->F where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique.

In a sequel to this project (arXiv:1101.2251), we show how to compute similar invariants associated to frozen triples using Kontsevich Soibelman, Joyce-Song wall-crossing techniques.

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