65 problems

Strominger–Yau–Zaslow conjecture. The mirror $Y^{\vee}$ is the moduli space of certain A-branes in $Y$, and the mirror functor in homological mirror symmetry is given by a fiberwis…

Let $(M,J,\omega,\Omega)$ be an almost Calabi--Yau $m$-fold, let $X$ be a compact special Lagrangian $m$-fold in $M$ with conical singularities, and let…

Let $X_t$ be the smooth Calabi–Yau fibres of the family … A special Lagrangian fibration is a fibration of $X_t$ by special Lagrangian submanifolds. SYZ conjecture. For sufficientl…

Let $(\mathcal{X}_{t_i},g_i)$ converge to $(X_{\infty},g_{\infty})$. Let $(B_i,d_i)$ be non-empty for large $i$ and converge to $(B_{\infty},d_{\infty})$, where $B_i$ are compactif…

A special Lagrangian fibration is a fibration of a Calabi–Yau manifold by special Lagrangian submanifolds, and a large complex structure limit point is a boundary point in the comp…

A special Lagrangian fibration is a fibration whose fibres are special Lagrangian submanifolds. Joyce's conjecture. Special Lagrangian fibrations should in general be piecewise smo…

A Calabi--Yau threefold is a Calabi--Yau manifold of complex dimension three, and a large complex structure limit is the degenerate limit of complex structures referred to in the s…

Let $(M,J,\omega,\Omega)$ be an (almost) Calabi–Yau $3$-fold with compact underlying $6$-manifold $M$, complex structure $J$, Kähler form $\omega$, and holomorphic volume form…

Singular-locus resemblance conjecture. The singular locus of the generalized special Lagrangian torus fibration should resemble $\tilde{\Gamma}$.

Joyce's classification conjecture. The cone on the Clifford torus, and perhaps also the cone $T_{1,2}$, are the only rigid special Lagrangian $T^2$-cones.

Joyce's transversality conjecture. This transversality condition holds in any almost Calabi–Yau manifold whose Kähler class is sufficiently generic.

Quadratic growth conjecture. The quantities $\operatorname{l-ind}(\Sigma)$, $\operatorname{s-ind}(\Sigma)$ and $\operatorname{Area}(\Sigma)$ grow quadratically with $d$.

Let $X_t$ be a Calabi–Yau hypersurface near the large complex limit, equipped with a Lagrangian torus fibration whose singular locus is denoted by $\tilde{\Gamma}$. Consider the co…

Let $X$ and $\hat X$ be mirror Calabi--Yau $3$-folds. A special Lagrangian torus fibration is a surjective continuous map from a Calabi--Yau $3$-fold to a compact topological $3$-m…

Consider a generic almost Calabi--Yau $3$-fold and a $\mathrm{U}(1)$-invariant local model for codimension-two singularities in a special Lagrangian fibration, in which two codimen…

Suppose $M$ and $\hat M$ are mirror Calabi--Yau $3$-folds. A special Lagrangian torus fibration is a fibration whose generic fibers are special Lagrangian $3$-tori. The SYZ conject…

Let $F$ be the stated $\mathrm U(1)$-invariant model for a codimension-two singularity in a special Lagrangian fibration, in which two codimension-one $T^2$-cones come together and…

Let $F:\mathbb C^3\to\mathbb R\times\mathbb C$ be the piecewise-smooth special Lagrangian fibration whose fibres are $T^2$-cones at $a=0$ and nonsingular $S^1\times\mathbb R^2$ for…

Let $C$ be a rigid special Lagrangian cone in $\mathbb C^m$ with an isolated singularity at $0$ and $k$ ends at infinity, and let $L$ be a weakly asymptotically conical special Lag…

Let $L$ be a weakly asymptotically conical special Lagrangian $m$-fold in $\mathbb C^m$ with cone $C$, and let $k$ be the number of ends of $C$ at infinity. Let $\mathcal M^w_L$ be…

Let $L$ be a strongly asymptotically conical special Lagrangian $m$-fold in $\mathbb C^m$ with cone $C$, and let $k$ be the number of ends of $C$ at infinity. Let $\mathcal M^s_L$…

Let $M$ and $\hat M$ be generic mirror Calabi–Yau $3$-folds. Suppose there exist special Lagrangian fibrations … Let $\Delta_f\subset B$ and $\Delta_{\hat f}\subset\hat B$ be their…

Let $M$ be an almost Calabi–Yau $3$-fold with a smooth special Lagrangian fibration $f:M\rightarrow B$, and let $\tilde M$ be a small generic deformation of $M$. For the smooth fib…

Let $f:M\rightarrow B$ be a generic special Lagrangian fibration of an almost Calabi–Yau $3$-fold, including fibres with singularities, where generic' has the meaning specified in…

An SL fibration is a special Lagrangian fibration, and a singularity has codimension two when its image in the base has codimension two. The fibration $\hat F$ is the local model c…