24 problems

Let $X$ and $Y$ be “mirror” Calabi-Yau manifolds. There should exist a base manifold $B$ and surjective continuous maps … with fibers $L_q=\pi^{-1}(q)$ and $F_q=f^{-1}(q)$, and a c…

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 $\mathcal I_{X'}$ and…

Let $(X^4,\omega_1,\omega_2,\omega_3)$ be a hyperkähler $K3$ surface, and let $\alpha_1,\alpha_2,\alpha_3$ be irreducible $(-2)$-classes in $H^2(X^4;\mathbb{Z})$ satisfying…

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…

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…

Suppose $M$ and $\hat M$ are mirror Calabi--Yau $3$-folds. The conjecture asserts the existence, under some additional conditions, of a compact topological $3$-manifold $B$ and sur…

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 $f:M\rightarrow B$ 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…

An SL fibration is a special Lagrangian fibration, and a singularity has codimension one when its image in the base has codimension one. The fibrations $F$ and $F'$ are the local m…

Let $Y$ be a Calabi--Yau manifold with a special Lagrangian $T^n$-fibration, and let ${\cal M}_D(T^n,Y)$ be the family of dual tori over ${\cal M}_{sL}(T^n,Y)$. The dual torus fami…

Let $Y$ be a Calabi--Yau manifold with a special Lagrangian $T^n$-fibration, and let ${\cal M}_D(T^n,Y)$ be the family of dual tori over ${\cal M}_{sL}(T^n,Y)$. The SYZ compactific…

Let $Y$ be an exact special Lagrangian submanifold in $\mathbf C^{n+1}$ with tangent cone $C\times\mathbf R$ at $0$. Assume that $C$ is stable, meaning that the only linear and qua…

Let $f\colon\mathcal{X}^*\to\Delta^*=\{t\in\mathbb{C}\mid 0<|t|<1\}$ be a flat proper family that is a maximal degeneration, meaning that its monodromy has maximal unipotency index…

Let $(X_t,g_t,J_t,\omega_t,\Omega_t)$ be a $1$-parameter maximally degenerate family of polarized $n$-dimensional Calabi–Yau manifolds of holonomy $SU(n)$ over the punctured disc…

Let $X$ be the rational elliptic surface and let $H=D_+\cup D_-$ be the union of two smooth fibers of its elliptic fibration. Rational elliptic surface deformation conjecture. The…

Let $(X,\omega,J)$ be a compact Kähler manifold, let $H$ represent twice the anticanonical class, and let $Y$ be the Calabi–Yau double cover of $X$ branched along $H$. Let $\bar B$…

Let $X$ be a rational elliptic surface obtained by blowing up $\mathbb{CP}^2$ at the nine base points of a pencil of cubics, and let $\hat D$ be a smooth elliptic fiber. Rational e…

Let $D\subset\mathbb{CP}^2$ be a smooth elliptic curve, and equip $\mathbb{CP}^2\setminus D$ with the holomorphic volume form having poles along $D$. Special Lagrangian torus fibra…

The SYZ mirror identification conjecture. The subset $M_D$ is the Strominger–Yau–Zaslow mirror of $D$. This identifies the mirror of the anticanonical divisor with the locus where…

Let $X$ be a compact Kähler manifold, let $D$ be an anticanonical divisor, let $U$ be a neighborhood of $D$, and let $M$ be the moduli space of pairs $(L,\nabla)$ consisting of a s…

Let $(X,\omega,J)$ be a compact Kähler manifold, let $D$ be an anticanonical divisor in $X$, and let $\Omega$ be a holomorphic volume form on $X\setminus D$. A complexified moduli…