Relative homological projective duality conjecture for genus-one Calabi–Yau threefolds with a 5-section

Let $B$ be a generalized del Pezzo surface with $b_2(X)=b_2(B)+1$, and let

$$\pi:X\rightarrow B$$

be a smooth genus-one fibered Calabi–Yau threefold with a 5-section. Assume that $X$ is the degeneracy locus $D_2(\phi)$ of a generic skew-symmetric map associated to rank-five vector bundles $V$ and $E$ and a line bundle $L$ on $B$. Relative homological projective duality conjecture. There exist rank-five vector bundles $V'$ and $E'$ on $B$ with Chern classes given by the stated bundle involution, and a line bundle $L'$ satisfying

$$c_1(L')=\frac15\left(2c_1(E')+c_1(V')+c_1(B)\right),$$

such that the corresponding genus-one fibered Calabi–Yau threefold

$$\pi^\vee:X^\vee\rightarrow B$$

is the relative homological projective dual of $X$.

The conjecture formalizes the observed exchange between the two genus-one fibrations and is supported in the source by checks against the available examples. Its resolution status is not specified.