Higher rank stable pairs and virtual localization

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2 Scopus citations

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 OX⊕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 languageEnglish (US)
Pages (from-to)139-193
Number of pages55
JournalCommunications in Analysis and Geometry
Volume24
Issue number1
DOIs
StatePublished - 2016
Externally publishedYes

ASJC Scopus subject areas

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

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