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Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties. (English. Russian original) Zbl 1207.14059
Sb. Math. 198, No. 9, 1325-1340 (2007); translation from Mat. Sb. 198, No. 9, 107-122 (2007).
According to Givental’s mirror theorem genus zero Gromov-Witten potential (\(I\)-series) of complete intersections in projective toric varieties can be found from a solution (\(J\)-series) to an explicit differential equation, the Riemann-Roch equation. In this note the author generalizes the mirror theorem to complete intersections in weighted projective spaces and singular toric varieties. As a consequence, he verifies that the remaining \(3\) families of Fano threefolds with Picard group \(\mathbb{Z}\) satisfy the Golyshev conjecture, i.e. solutions to their so-called counting equations are modular. For the other \(14\) families this follows from an old result of Beauville.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J45 Fano varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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