An introduction to the theory of Higher rank stable pairs and Virtual localization

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This article is an attempt to briefly introduce some of the results from arXiv:1011.6342 on development of 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 highly frozen triples given by the data O^r-->F for r>1 where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory. Instead, we build a zero-dimensional virtual fundamental class by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. We briefly include the results of calculations in this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique. We emphasize that in this article we merely include the statement of our theorems and illustrate the final result of some of the computations. The proofs and more detailed calculations in arXiv:1011.6342 will appear elsewhere.
Original languageEnglish
Title of host publicationProceedings of Symposia in Pure Mathematics
Volume2012
Publication year15 Oct 2012
Edition85
ISBN (print)0082-0717
Publication statusPublished - 15 Oct 2012
Externally publishedYes
SeriesProceedings of Symposia in Pure Mathematics

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11 pages

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