On relative Gromov--Witten invariants of projective completions of vector bundles

TL;DR

This paper demonstrates that the relative Gromov--Witten invariants of projective completions of vector bundles are canonically identified when the bundles have the same total Chern classes, extending previous results on absolute invariants.

Contribution

It establishes a canonical identification of relative Gromov--Witten invariants for projective bundle completions based on Chern class equality, generalizing earlier absolute invariant results.

Findings

Relative Gromov--Witten invariants are identified when $c(V_1)=c(V_2)$.

Extension of absolute invariant results to relative invariants.

Provides a new perspective on invariants of projective bundle completions.

Abstract

It was proved by Fan--Lee and Fan that the absolute Gromov--Witten invariants of two projective bundles $\mathbb P(V_i)\rightarrow X$ are identified canonically when the total Chern classes $c(V_1)=c(V_2)$ for two bundles $V_1$ and $V_2$ over a smooth projective variety $X$. In this note we show that for the two projective completions $\mathbb P(V_i\oplus\mathcal O)$ of $V_i$ and their infinity divisors $\mathbb P(V_i)$, the relative Gromov--Witten invariants of $(\mathbb P(V_i\oplus\mathcal O),\mathbb P(V_i))$ are identified canonically when $c(V_1)=c(V_2)$.