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Moduli spaces of stable quotients and wall-crossing phenomena

Part of:CurvesProjective and enumerative geometry

Published online by Cambridge University Press:  31 May 2011

Yukinobu Toda
Yukinobu Toda*
Affiliation:
Institute for the Physics and Mathematics of the Universe, University of Tokyo, Japan (email: toda-914@pj9.so-net.ne.jp, yukinobu.toda@ipmu.jp)
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Abstract

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The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

Keywords

Quot schemeGromov–Witten invariant

MSC classification

Secondary:14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants14H60: Vector bundles on curves and their moduli

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011, pp. 1479 - 1518
Copyright
Copyright © Foundation Compositio Mathematica 2011

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