The projective bundle mirror theorem conjecture

Let $X$ be a projective manifold and let

$$\pi:{\mathbb P}(V)={\mathbb P}(\bigoplus_{j=0}^{n}L_j)\to X$$

be the projective bundle of hyperplanes of a vector bundle $V$, with $L_0=\mathcal O_X$. Assume that $L_i$, for $i=1,\ldots,n$, are nef line bundles and that $-K_X-c_1(V)$ is ample. Let $J({\mathbb P}(V))$ be the genus-zero one-point Gromov–Witten generating series, and let $I({\mathbb P}(V))$ be the twisted hypergeometric series defined from the generators $J_\beta$ of $X$ by the twisting factors associated with the classes $(\nu,\beta)$ of curves in ${\mathbb P}(V)$.

Projective bundle mirror theorem conjecture. Under these assumptions,

$$J({\mathbb P}(V))=I({\mathbb P}(V)).$$

This conjecture extends the equality between the $J$- and hypergeometric $I$-series known for Fano toric varieties to projective bundles. If true, it determines the relevant one-point Gromov–Witten invariants and gravitational descendants of ${\mathbb P}(V)$ from the explicit twisted series.