98 problems

Let $L$ be a graded Lagrangian submanifold of a Calabi–Yau $n$-fold $X$, with the phase of the holomorphic volume form chosen so that the cohomological phase satisfies $\phi(L)=0$.…

Let $X$ be a compact Kähler Calabi-Yau manifold, and let $\alpha\in H^{1,1}_{\mathbb{R}}(X)$ be a class that is nef and big but not Kähler. A smooth $(1,1)$-form representing…

Let $M$ be a Weil–Petersson variety, and let $X$ be a Weil–Petersson subvariety of dimension $q$ in $M$. Let $c_k(\omega_{WP})$ denote the $k$-th elementary polynomial of the curva…

A polarized Calabi–Yau manifold is a pair $(X,\omega)$ consisting of a compact algebraic manifold $X$ with zero first Chern class and a Kähler form $\omega\in H^2(X,\mathbb Z)$. Le…

Let $\pi:{X}^{an}\to B$ be the continuous map from the analytic Calabi–Yau space to its limiting base constructed in the non-archimedean collapse picture. Steinness conjecture. The…

Let a dual pair of Calabi–Yau families have Gromov–Hausdorff limits, and let the smooth parts of these limits carry the induced integral Monge–Ampère structures. A Monge–Ampère man…

Let ${X}_{mer}$ be an algebraic $n$-dimensional Calabi–Yau manifold over the field of meromorphic germs, with a non-vanishing volume form, and let $(X_t,\Omega_t)$ be the associate…

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 $(M,\omega,\Phi)$ be a Calabi–Yau $n$-fold, and let $\eta$ be an $n$-dimensional homology class in $M$. For an oriented exact Lagrangian $L$ in $M$, write … where…

SYZ and integrable-system conjecture. Near the large-complex-structure/large-volume limit: the SYZ picture holds on $\mathcal M_{\mathbb R}$; for a B-field $B'$ on $X'$, $X_{B'}$ a…

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 $({\mathcal Y}\rightarrow T,P)$ be a family of Calabi–Yau manifolds with a complex cusp $P\in\partial T$, let $X$ be the topological mirror of $Y$ with respect to $P$, and assu…

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 the smooth part of the limiting base of a maximal degeneration of a Calabi–Yau family, equipped with its Monge–Ampère metric and integral affine structure, and let…

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 ${{\cal X}}_{mer}$ be a maximally degenerate Calabi–Yau family at $q=0$, let $X_q^{new}$ be the rescaled family, and let $n$ be the complex dimension. Maximal-degeneration limi…

Let $(X,Y)$ be a geometric mirror pair, meaning that there is a parameter space $U$ and, for each $t\in U$, a correspondence $Z_t\subset X\times Y$, sections of the projections, Ri…

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 $\frac{}{}$ be a pair of $\bar d$-dimensional dual reflexive Gorenstein cones $\frac{}{}$ and $\frac{}{}$ of index $r$. The associated generalized Calabi–Yau manifolds arise as…

Let $X$ be a projective Calabi–Yau manifold, meaning that its canonical bundle is torsion and … and let $\mathcal U$ be a universal torsor for $X$. Lift conjecture. The action of…

Let $X$ be a compact Kähler manifold. A class $\alpha \in H^{1,1}(X,\mathbb{R})$ is semi-ample if there exists a holomorphic contraction $\rho:X\to X'$ to a normal analytic space…

Let $(X,\omega,\Omega)$ be a compact Calabi–Yau manifold, and let $L_1,\dots,L_r$ be special Lagrangian submanifolds in $X$ with the same phase that intersect transversely. Let…