Volume 13, issue 5 (2009)

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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
Abstract

Let $\mathsc{X}$ 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 $\mathsc{X}$ to those of $Y$, which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan–Graber, and prove our conjecture when $\mathsc{X}=ℙ\left(1,1,2\right)$ and $\mathsc{X}=ℙ\left(1,1,1,3\right)$. As a consequence, we see that the original form of the Bryan–Graber Conjecture holds for $ℙ\left(1,1,2\right)$ but is probably false for $ℙ\left(1,1,1,3\right)$. 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
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