Orbifold quantum Riemann–Roch, Lefschetz and Serre

Tseng, Hsian-Hua¹

¹ 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

Geometry & topology, Tome 14 (2010) no. 1, pp. 1-81.

Résumé

Given a vector bundle $F$ on a smooth Deligne–Mumford stack $\mathcal{X}$ and an invertible multiplicative characteristic class $c$, we define orbifold Gromov–Witten invariants of $\mathcal{X}$ 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 $\mathcal{X}$ and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.

Détail

DOI : 10.2140/gt.2010.14.1

Keywords: orbifold Gromov–Witten invariant, Deligne–Mumford stack, Givental's formalism, Grothendieck–Riemann–Roch formula, mirror symmetry

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