Gromov-Witten Theory of Complete Intersections via Nodal Invariants

TL;DR

This paper develops an inductive algorithm to compute Gromov-Witten invariants for all genera and insertions of smooth complete intersections, and proves these invariants belong to the tautological ring, advancing enumerative geometry.

Contribution

It introduces a novel approach using nodal invariants and degeneration formulas to compute Gromov-Witten invariants for complete intersections in projective space.

Findings

Provides an inductive algorithm for all genera invariants

Proves Gromov-Witten classes are in the tautological ring

Introduces nodal relative Gromov-Witten invariants

Abstract

We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We 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 Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. We then 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.