1011.6342 (Artan Sheshmani)

Higher rank stable pairs and virtual localization    [PDF]

Artan Sheshmani

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 $O^r(-n)\rightarrow F$ 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 $\mathbb{P}^1$ using the Graber-Pandharipande virtual localization technique. In a sequel to this paper, we show how to compute similar invariants associated to frozen triples using Kontsevich-Soibelman, Joyce-Song wall-crossing techniques.

View original: http://arxiv.org/abs/1011.6342