Higher rank stable pairs and virtual localization

摘要

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(circle plus r) (-n) ->(phi) 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 P-1 using the Graber-Pandharipande [GP99] virtual localization technique.