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Descendant log Gromov-Witten invariants for toric varieties and tropical curves. (English) Zbl 1442.14166
The authors study genus zero, as well as higher genus Gromov-Witten invariants (in non-superabundant situations) of smooth toric varieties with Psi-class conditions. The main result shows that the tropical description of such invariants coincides with the classical one. The technique that is used to show this correspondence builds on the approach of T. Nishinou and B. Siebert [Duke Math. J. 135, No. 1, 1–51 (2006; Zbl 1105.14073)]. In particular, it uses logarithmic Gromov-Witten theory and toric degenerations. The authors also allow incidence conditions in the toric boundary for applications to non-toric situations. Moreover, tropically they study arbitrary tropical cycles as incidence conditions, not just affine linear ones as in Nishinou-Siebert.

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T15 Combinatorial aspects of tropical varieties
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