Wall-crossings in toric Gromov–Witten theory I: crepant examples

Coates, Tom; Iritani, Hiroshi; Tseng, Hsian-Hua

doi:10.2140/gt.2009.13.2675

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Wall-crossings in toric {G}romov–{W}itten theory {I}: crepant examples

Tom Coates, Hiroshi Iritani and Hsian-Hua Tseng

Geometry & Topology 13 (2009) 2675–2744

DOI: 10.2140/gt.2009.13.2675

Abstract

Let $X$ be a Gorenstein orbifold with projective coarse moduli space $X$ and let $Y$ be a crepant resolution of $X$ . We state a conjecture relating the genus-zero Gromov–Witten invariants of $X$ to those of $Y$ , which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan–Graber, and prove our conjecture when $X = ℙ (1, 1, 2)$ and $X = ℙ (1, 1, 1, 3)$ . As a consequence, we see that the original form of the Bryan–Graber Conjecture holds for $ℙ (1, 1, 2)$ but is probably false for $ℙ (1, 1, 1, 3)$ . Our methods are based on mirror symmetry for toric orbifolds.

Keywords

quantum cohomology, crepant resolution, Gromov–Witten invariants, mirror symmetry, variation of semi-infinite Hodge structure, Crepant Resolution Conjecture

Mathematical Subject Classification 2000

Primary: 53D45

Secondary: 14N35, 83E30

References

Publication

Received: 4 December 2006

Revised: 21 October 2008

Accepted: 25 May 2009

Published: 2 August 2009

Proposed: Jim Bryan

Seconded: Richard Thomas, Lothar Goettsche

Authors

Tom Coates
Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ UK

Hiroshi Iritani
Faculty of Mathematics Kyushu University 6-10-1, Hakozaki Higashiku, Fukuoka, 812-8581 Japan

Hsian-Hua Tseng
Department of Mathematics University of Wisconsin–Madison Van Vleck Hall, 480 Lincoln Drive Madison, WI 53706-1388 USA

Volume 13, issue 5 (2009)