Volume 14, issue 1 (2010)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Orbifold quantum Riemann–Roch, Lefschetz and Serre

Hsian-Hua Tseng

Geometry & Topology 14 (2010) 1–81
Abstract

Given a vector bundle F on a smooth Deligne–Mumford stack X and an invertible multiplicative characteristic class c, we define orbifold Gromov–Witten invariants of X twisted by F and c. We prove a “quantum Riemann–Roch theorem” which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus–0 orbifold Gromov–Witten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.

Keywords
orbifold Gromov–Witten invariant, Deligne–Mumford stack, Givental's formalism, Grothendieck–Riemann–Roch formula, mirror symmetry
Mathematical Subject Classification 2000
Primary: 14N35
Secondary: 53D45, 14C40
References
Publication
Received: 16 July 2006
Revised: 20 May 2009
Accepted: 22 June 2009
Preview posted: 10 October 2009
Published: 2 January 2010
Proposed: Jim Bryan
Seconded: Richard Thomas, Frances Kirwan
Authors
Hsian-Hua Tseng
Department of Mathematics
Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210-1174
USA
Department of Mathematics
University of Wisconsin-Madison
Van Vleck Hall
480 Lincoln Drive
Madison, WI 53706-1388
USA