Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

TL;DR

This paper develops a new algebraic approach to relate and compute genus zero Gromov-Witten invariants of very ample hypersurfaces within a smooth projective variety, connecting them to the invariants of the ambient space.

Contribution

It introduces a technique to express relative Gromov-Witten invariants of hypersurfaces in terms of invariants of the ambient variety and proves cycle relations in moduli spaces.

Findings

Establishes an equality of cycles in Chow groups relating relative and absolute invariants.

Provides a method to compute all relative invariants from the Gromov-Witten invariants of X.

Enables calculation of all genus zero Gromov-Witten invariants of Y induced by X.

Abstract

For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y. Given the Gromov-Witten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of Y whose homology and cohomology classes are induced by X.