Geometry & Topology

Orbifold quantum Riemann–Roch, Lefschetz and Serre

Hsian-Hua Tseng

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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.

Article information

Geom. Topol., Volume 14, Number 1 (2010), 1-81.

Received: 16 July 2006
Revised: 20 May 2009
Accepted: 22 June 2009
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 14C40: Riemann-Roch theorems [See also 19E20, 19L10]

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


Tseng, Hsian-Hua. Orbifold quantum Riemann–Roch, Lefschetz and Serre. Geom. Topol. 14 (2010), no. 1, 1--81. doi:10.2140/gt.2010.14.1.

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