Volume 14, issue 1 (2010)

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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 $\mathsc{X}$ and an invertible multiplicative characteristic class $c$, we define orbifold Gromov–Witten invariants of $\mathsc{X}$ 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 $\mathsc{X}$ 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