The normal-crossings mirror symmetry conjecture

Let $Y$ be a Calabi–Yau manifold with $h^{2,0}(Y)=0$, and let $\pi:\mathcal{Y}\to S$ be a family of complex structures on $Y$ whose Kodaira–Spencer map is an isomorphism everywhere. Let $S\subset\overline{S}$ be a partial compactification with normal-crossings boundary, and let $p$ be a maximally unipotent normal-crossing boundary point. The normal-crossings mirror symmetry conjecture. Associated to $p$ are a Calabi–Yau manifold $X$ with $h^{2,0}(X)=0$, a finite-index lattice $L\subset H^2(X,\mathbb{Z})/\mathop{\rm torsion}\nolimits$, a relative interior $\sigma\subset H^2(X,\mathbb{R})$ of a rational polyhedral cone generated by a basis of $L$, and a map $\mu$ from a neighborhood of $p$ in $\overline{S}$ to $((H^2(X,\mathbb{R})+i\sigma)/L)^-$, satisfying: (a) $\sigma$ lies in the Kähler cone for some complex structure on $X$; and (b) $\mu$ identifies the formally degenerating geometric variation of Hodge structure at $p$ with the framed A-variation of Hodge structure associated to $X$ with framing $\sigma$. This is the paper’s mathematical formulation of mirror symmetry at maximally unipotent boundary points; it is presented as conjectural.