## Abstract

We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs [PT09a] on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_{X}^{⊕r} (-n) _{φ} → 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 deformationobstruction 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 P1 using the Graber-Pandharipande [GP99] virtual localization technique.

Original language | English (US) |
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Pages (from-to) | 139-193 |

Number of pages | 55 |

Journal | Communications in Analysis and Geometry |

Volume | 24 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty