23 problems

Equivariant torus-fibration mirror-symmetry conjecture. There should be equivalences

Let $\mathcal X\rightarrow S$ be a family of Calabi–Yau manifolds, let $X=\mathcal X_t\xrightarrow{f}B$ be a torus fibration with a section $\delta_0$, and let $X^\sharp$ be the co…

Let $M \to B$ and $\check{M} \to B$ be dual torus fibrations arising in the SYZ picture, and let a Lagrangian section of $M \to B$ be a Lagrangian submanifold mapping isomorphicall…

Let $({\cal X}_{mer},vol)=(X_q,vol_q)$ be a one-parameter family of maximally degenerate Calabi–Yau manifolds, let $X_q^{new}$ be the rescaled family, and let…

Let $X$ and $X^{\vee}$ be dual Calabi–Yau manifolds in degenerating families, and let $\overline{Y}$ denote a limiting base. Torus-fibration degeneration conjecture. In the limit,…

Let $(M^{2n},\omega)$ be a Kähler manifold with a nowhere-vanishing holomorphic $(n,0)$-form $\varphi$. A calibrated fibration is a surjective map from $M$ to a topological space w…

Consider mirror symmetry between geometric $N=2$ SCFTs arising from target spaces equipped with a special torus fibration. SYZ conjecture. Mirror symmetry corresponds to T-duality…

Let a $G_2$ manifold admit a coassociative torus fibration, and let its associated Calabi–Yau three-fold be obtained by brane reduction. Conversely, consider the Kaluza–Klein lift…

Let $X\to Y$ be a rank-one algebraic torus fibration with discriminant divisor $D\subset Y$, and let $s$ denote the invertible variable associated with the $S^1$ action on $X$. Let…

Consider a rank-$r$ algebraic torus fibration $X\to Y$ with simple normal-crossing discriminant divisor $D=\bigcup_{i=1}^rD_i$, and let $\lambda\in(\mathbb{C}^*)^r$ be a spectral p…

Let $X\to Y$ be an algebraic torus fibration, with discriminant divisor $D\subset Y$, and let $\mathcal{W}_{T}(X)_{-\mathbf{1}}$ denote the spectral component of its equivariant wr…

Strominger–Yau–Zaslow conjecture. Mirror symmetry should be a torus duality between symplectic geometry on the A-side and complex geometry on the B-side, interchanged by implementi…

Let $F\hookrightarrow E\to B$ be a quasi-nilpotent fibration, with fibre $F$ equal to a torus $T^r$, and suppose that the rational cohomology of $B$ is finite dimensional. Generali…

Let $X$ be a twisted connected sum ($TCS$) $G_2$ manifold, and consider its mirror-symmetry description in analogy with special Lagrangian torus fibrations. A $T^4$-fibration is a…

Let $M_\mathrm{B}$ be a Betti moduli space with compactification $\overline{M_\mathrm{B}}$. The associated Betti Hitchin map is a differentiable torus fibration on a neighborhood o…

Let $X$ be a variety with a smooth anticanonical divisor $D$, set $U=X\setminus D$, and let $\pi:U\to B$ be a special Lagrangian torus fibration, meaning that its smooth fibers are…

Let $B$ be a closed $\mathbb{Z}$-affine manifold, let $M=T^*B/T^*_{\mathbb{Z}}B$ be the associated symplectic torus fibration, and let $\mathcal{O}_B$ be its natural sheaf of…

Degeneration conjecture. There exist non-flat families of subvarieties $\mathcal C_{jk}\subset\mathcal X$ with special fibers $(C_{jk})_{s_0}=X_j\cap X_k$, while $(C_{jk})_s$ is a…

Joyce's holomorphic-disk boundary conjecture. The region $\mathcal H$ provides the region whose complement retracts to $\Delta$ and across whose boundary the fibration fails to be…

Smoothness-away-from-the-discriminant conjecture. The set $\mathcal H$ provides a region on whose complement the fibration is smooth, with the complement of that region retracting…

Gross's geometric conjecture. The fibration is piecewise smooth; $\Sigma$ is a complex one-dimensional subvariety whose singularities are transverse triple points locally given by…

Joyce's singular-fiber conjecture. The discriminant locus has codimension one and is locally contained, in affine coordinates, in the plane corresponding to the monodromy-invariant…

Gross–Ruan topology conjecture. The discriminant locus $\Gamma$ is a trivalent graph; the topology near its edges is modeled by the product of a cylinder with the $k=1$ semistable…