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

TL;DR

This paper introduces a higher rank analog of stable pairs on Calabi-Yau threefolds, develops a moduli theory for these objects, and applies virtual localization to compute related enumerative invariants.

Contribution

It develops a moduli theory for higher rank stable pairs and constructs a virtual fundamental class, extending Pandharipande-Thomas theory.

Findings

Constructed a zero-dimensional virtual fundamental class for higher rank stable pairs.

Applied virtual localization to compute enumerative invariants for local P^1.

Outlined the main theorems and computational results, with detailed proofs to appear elsewhere.

Abstract

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.