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

It was proved by Fan–Lee and Fan that the absolute Gromov–Witten invariants of two projective bundles $\mathbb{P}(V_i)\to X$ 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 $c(V_1)=c(V_2)$.