Existence of a common mirror family for the Calabi–Yau pair

Construct a family \(\mathcal{Z}\) of Calabi–Yau threefolds with two maximal unipotent monodromy points such that the degree-33 Picard-rank-one Calabi–Yau threefolds \(X\) arising from \(G(2,6)\) and the degree-21 arithmetically Gorenstein Picard-rank-one Calabi–Yau threefolds \(Y\subset\mathbb{P}^8\) are Hodge-theoretic mirrors to \(\mathcal{Z}\) at the two respective points.

Background

The paper proves that the families XX and YY have matching numerical and geometric properties, including equal Hodge numbers, L-equivalence, and Picard–Fuchs operators related by the coordinate transformation z↦1/zz\mapsto 1/z. These calculations suggest that the two families correspond to distinct large-complex-structure limits of a single mirror moduli space.

The conjectural common mirror is expected to be fiber-wise birational to the Laurent-polynomial pencil {V(G1G2G3−t)⊂(C∗)4}t∈C\{V(G_1G_2G_3-t)\subset(\mathbb{C}^*)^4\}_{t\in\mathbb{C}}, to have Hodge numbers h1,1=52h^{1,1}=52 and h1,2=1h^{1,2}=1, and to exhibit two MUM points and three conifold points. Establishing an actual family with these properties remains beyond the results proved in the paper.

References

Finally, in the light of above results and the discussion in the introduction, we are motivated to make the following conjecture. We state and discuss it in a more precise language in \Cref{sec:mirrorEquivalence}. There exists a family $\mathcal{Z}$ of Calabi-Yau threefolds having two MUM points, with $X$ and $Y$ being Hodge theoretic mirrors to $\mathcal{Z}$ at these points, respectively.

— The Last Picard Rank 1 Double-Mirror Calabi-Yau Pair?  (2609.20704 - Kapustka et al., 17 Sep 2026) in Section 1, Results; Conjecture \(\ref{conj:doubleMirror}\) / Section 4, Conjecture \(\ref{conj:doubleMirror}\)

Disregarding loops around the singularities, there are four homotopy classes of paths from $ X$ to $ Y$ in the conjectured common SKMS of $X$ and $Y$ (figure \ref{fig:SKMS}). So, by mirror symmetry, we expect there to exist four "fundamental" windows, giving in total four "fundamental" equivalences $D(X)\simeq D(Y)$, including the one constructed in \Cref{thm:derivedEquivalence}. We will leave the discovery of the three other windows and the uncovering of their precise relationship to the SKMS, to future work.

— The Last Picard Rank 1 Double-Mirror Calabi-Yau Pair?  (2609.20704 - Kapustka et al., 17 Sep 2026) in Remark \(\ref{rem:4Windows}\), Section 6