Gross–Wilson K3 and Borcea–Voisin source interfaces

This is a source and scope interface. Under AC, the separate construction theorem locally reconstructs the K3 rotation, elliptic pencil, real Kähler-chamber realization and quotient duality; the cited named source results are not external proof premises. Gross–Wilson Proposition 1.3 states that for a primitive $M\subset L$ of signature $(1,t)$, an m-admissible isotropic vector in $T=M^{\perp }$, a general marked K3 period in $D_M$, and a general compatible Kähler–Einstein metric, the K3 has a fibration $S\to S^2$ whose general fibres are special-Lagrangian $T^2$'s and which has a special-Lagrangian numerical m-section. The proof uses a decomposition $T\cong U(m)\oplus \check{M}$, a special period choice, hyper-Kähler rotation, an elliptic fibration after reflection in $(-2)$-classes, and an extension from special to general choices. The source invokes K3 period/Torelli and elliptic-fibration results at the cited locations. The local construction replaces those uses by analytic Riemann–Roch, the nef elliptic pencil, and an explicit conjugate-model mirror realization. For its richer numerical m-section conclusion, genuinely general choices exclude the additional countable real walls where another class becomes algebraic in the section rotation; the public fibration proof does not assume that irreducibility follows solely from period hyperplane avoidance.

A numerical m-section here is a surface whose topological intersection number with a general fibration fibre is $m$; it need not be an actual section of degree one.

For the Borcea–Voisin case, the source takes the stated K3 involution and elliptic-curve negation quotient, resolves its $4N$ $A_1$-curves, and restricts to a codimension-$4N$ wall limit of the Kähler cone. The limiting metric is degenerate on the compact resolution and Kähler–Einstein only on the complement of the exceptional locus. The special-Lagrangian $T^3$-fibration is asserted on that complement. Proposition 3.1 identifies the named Borcea–Voisin mirror as a smooth compactification of the topological dual torus fibration over the complement of the discriminant. This is topological duality; the source does not thereby prove an analytic or corrected-complex mirror equivalence.

On the quotient before resolution, the source's product fibration descends through the involutions. Let ${\iota }^{\mathrm{\prime }}$ be the map induced by $\iota$ on $S^2$, and let $j^{\mathrm{\prime }}$ be the map induced by elliptic negation $j$ on the base circle; in circle coordinate $t\in \mathbb{R}\mathrm{/}\mathbb{Z}$, $j^{\mathrm{\prime }}(t)=-t$. The quotient square is

The square commutes because the product fibration $\tilde{\pi }$ is equivariant for the stated involutions, so its composite with $q_B$ is constant on $q_X$-orbits and factors through $\bar{\pi }$. The diagram concerns the source quotient; the resolution and the degenerate metric are handled only on the stated smooth complement.

The Ricci-flat metric is supplied by the locally proved Calabi theorem in the selected actual Kähler class. In the Borcea–Voisin comparison retain the restricted D′/T′ regime and both original- and mirror-period hyperplane exclusions in its data Definition. The chamber realization uses two operations. Put $a=B+E^{\mathrm{\prime }}+cE$, where $c=({\omega }^2-B^2)\mathrm{/}2$, and $\sigma =E^{\mathrm{\prime }}-E$. Take $w=\mathrm{i}\mathrm{d}$ when $(a,\sigma )>0$ and otherwise take the root reflection $w(x)=x+(x,\sigma )\sigma$; the mirror-period exclusion gives $(a,\sigma )=c-1\mathrm{\ne }0$. First replace $a$ by the Kähler class $a^{\mathrm{\prime }}=w(a)$ on the conjugate elliptic model; rotation then constructs a mirror whose marked period is $[a^{\mathrm{\prime }}+i\omega ]$. Re-marking that mirror by $w^{-1}$ gives the prescribed period $[a+i\omega ]$, since $w$ fixes $\omega$ and $\check{\omega }$, and retains the mirror polarization $\check{M}$ and selected Kähler class $\check{\omega }$. The reflection alone does not fix the prescribed marked mirror period: it swaps $E,E^{\mathrm{\prime }}$ and sends $[B+E^{\mathrm{\prime }}+cE+i\omega ]$ to $[B+E+c{E}^{\mathrm{\prime }}+i\omega ]$. The auxiliary elliptic-fibre label consequently may change from $E$ to $E^{\mathrm{\prime }}$. The topological dual bundle and its involutions are unchanged; no fixed fibre-label preservation or corrected analytic mirror equivalence is asserted.

Marked M-polarized K3 data and m-admissible vector

Borcea–Voisin K3 involution and quotient data