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Gromov-Witten theory of complete intersections via nodal invariants.(English)Zbl 1525.14068

The Gromov-Witten invariants (GWIs) of a smooth projective variety \(X\) over \(\mathbb{C}\) are rational numbers defined by intersection theory on the moduli spaces of stable maps to \(X\) using integration over virtual fundamental classes of moduli spaces, evaluation morphisms at marked points and first Chern classes of cotangent line bundles at marked points. Providing an effective algorithm to compute GWIs is generally challenging. For projective spaces, or more generally for homogeneous varieties, they can be computed using a localization formula and the calculation of Hodge integrals on the moduli spaces of curves. For curves they have been computed by Okounkov-Pandharipande using degeneration techniques, monodromy constraints, and Hurwitz theory. The authors provide an inductive algorithm computing GWIs in all genera with arbitrary insertions of all smooth complete intersections in projective space. They also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal GWIs, they introduce the new notion of nodal relative GWIs. They prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov-Witten theory are stated in complete generality and are of independent interest.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Cite

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