24 problems

Let $\pi: (X,L)\to \mathbb{D}^*$ be a polarised maximal degeneration of compact Calabi-Yau manifolds, and let $(X_t,\omega_{CY,t},\Omega_t)$ denote its fibre with the Calabi-Yau st…

Self-dual mirror symmetry conjecture. For each point near the large complex structure limit of $X$, there is a corresponding point near the large Kähler structure limit of $Y$ such…

Monodromy-determination conjecture. In general, the allowable compactification of the dual special Lagrangian torus fibration should be determined uniquely by the monodromies.

Let $(X,\Phi)$ be a $\operatorname{Spin}(7)$-manifold, and let a toric Cayley mean a Cayley four-fold of the toric type intended in the source. The Spin(7) SYZ fibration conjecture…

Metric SYZ conjecture. For any $0<\varepsilon\ll1$, there exists $\delta>0$ such that if $|t|<\delta$, then there exists a special Lagrangian $\mathbb{T}^d$-fibration on an open su…

SYZ conjecture. A Calabi–Yau manifold admits a special Lagrangian torus fibration, and its mirror Calabi–Yau manifold is obtained by taking the dual fibration.

Let $X_t$ be a $1$-parameter maximally degenerate family of polarized $d$-dimensional Calabi–Yau manifolds over the punctured disc $\mathbb{D}_t^*$. For a Calabi–Yau metric…

A TCS $G_2$ manifold $X$ is constructed by gluing asymptotically cylindrical Calabi–Yau three-folds, and globally it can be viewed as a $K3$ fibration over $S^3$: … The fiber is a…

Let $\omega_i$ be the sequence of symplectic forms considered in the paper, let $B_0$ be the base of the special Lagrangian fibration, and let $\mathcal{U}_i\subseteq B_0$ be a cha…

Let $Y$ be a toric Fano surface, and let $W(u)$ denote the superpotential of a special Lagrangian torus fibre $L_u$; let $W^{HV}$ be the Hori–Vafa superpotential. Special Lagrangia…

SYZ conjecture. A Calabi–Yau manifold $X$ admits a special Lagrangian fibration whose fibres are diffeomorphic to $T^n$.

Universal metric model conjecture. The 3-dimensional analogues of the Ooguri–Vafa metric should be the universal metric models for neighbourhoods of the most singular fibres in a g…

Let $\\{(X_{\hbar},\omega_{\hbar})\\}_{\hbar>0}$ be a family of Kähler manifolds approaching the large complex structure limit, with base polytope $P$ and a rescaled Kähler metric…

SYZ conjecture. There is an affine manifold with singularities $B$ and maps $\phi:X\rightarrow B$ and $\check{\phi}:\check{X}\rightarrow B$ which are dual special Lagrangian fibrat…

Combinatorial retraction conjecture. The projection of $C^{(i)}_{\lambda_1,\lambda_2,\vec\alpha}$ retracts to the graph so constructed.

Combinatorial discriminant conjecture. After choosing suitable triangulation data of the dual polytope, the image of each component of $C^{(i)}$ retracts to a codimension-one subse…

Main conjecture. If $Z$ is close to $Z_0$, the $n$-torus fibration on $Z_0$ deforms to a fibration $\pi:Z\to B$ by real $n$-tori, with parameter space including singular fibers dif…

Geometric conjecture. The singular locus and its image have precisely properties (i)--(v) above, with positive vertices corresponding to singular points of $C$ and negative vertice…

Gross–Ruan topological conjecture. The discriminant and monodromy data of the fibration satisfy properties (i)--(iv) above, including the specified positive and negative vertex mod…

Let $X$ and $\check{X}$ be Calabi–Yau manifolds which are mirror to each other. A special Lagrangian torus fibration is a fibration with torus fibers calibrated by the special Lagr…

Let $X$ be a Calabi–Yau manifold with a B field $\mathcal{B}$, and let $\check{X}$ be its mirror. SYZ mirror symmetry conjecture. There should exist a possibly singular special Lag…

Let $X$ and $\widehat X$ be mirror pairs of Calabi–Yau $n$-folds, and let $B$ be a base. A fibration means a map to $B$, with fibers denoted by $X_b$ and $\widehat{X_b}$. SYZ conje…

Let $X$ be a Kähler manifold, let $D\subset X$ be a smooth Calabi–Yau hypersurface in the anticanonical linear system, and let $\pi:X\setminus D\to B$ be a reasonable special Lagra…

Let $X$ and $X^\vee$ be mirror pairs of Calabi–Yau manifolds. SYZ conjecture. The manifolds $X$ and $X^\vee$ should carry dual special Lagrangian torus fibrations. This is the geom…