Commentarii Mathematici Helvetici

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Volume 89, Issue 2, 2014, pp. 273–297
DOI: 10.4171/CMH/319

Published online: 2014-06-18

Orbifold quantum D-modules associated to weighted projective spaces

Martin A. Guest[1] and Hironori Sakai[2]

(1) Waseda University, Tokyo, Japan
(2) Universität Münster, Germany

We construct in an abstract fashion (without using Gromov–Witten invariants) the orbifold quantum cohomology of weighted projective space, starting from a certain differential operator. We obtain the product, grading, and intersection form by making use of the associated self-adjoint D-module and the Birkhoff factorization procedure. The method extends in principle to the more difficult case of Fano hypersurfaces in weighted projective space, where Gromov–Witten invariants have not yet been computed, and we illustrate this by means of an example originally studied by A. Corti. In contrast to the case of weighted projective space itself or the case of a Fano hypersurface in projective space, a “small cell” of the Birkhoff decomposition plays a role in the calculation.

Keywords: Quantum cohomology, D-module, weighted projective space, Birkhoff decomposition

Guest Martin, Sakai Hironori: Orbifold quantum D-modules associated to weighted projective spaces. Comment. Math. Helv. 89 (2014), 273-297. doi: 10.4171/CMH/319