110 problems

Optimal index conjecture. The index $m$ of $X$ is the largest possible index among all terminal Calabi-Yau varieties of dimension $n$, and also among all canonical Calabi-Yau varie…

Let $\{\Delta_1',\ldots,\Delta_r'\}$ and $\{\Delta_1”,\ldots,\Delta_r”\}$ be the translated nef-partitions described in the preceding discussion, with corresponding mirror Calabi–Y…

Let $X_0$ be a normal crossing variety with smoothing $X_t$, and let $X$ denote the associated total space. Write $H^k(-,\mathbb Z)_f$ for the torsion-free quotient of integral coh…

Let $Y$ be a Calabi–Yau variety of dimension $n$. Infinitesimal mirror symmetry conjecture. There exists a Calabi–Yau variety $Y^{\circ}$ and isomorphisms of complex vector spaces…

Let $d\leq 3$, and let $X$ and $Y$ be a maximally generic mirror pair of $d$-dimensional smooth projective Calabi–Yau schemes over the Witt ring $W(\mathbb{F}_q)$. A reduction is g…

Let $X$ and $Y$ be schemes of finite type over $\mathbb{Z}$ whose generic fibres $X\otimes\mathbb{Q}$ and $Y\otimes\mathbb{Q}$ form a usual weak mirror pair of $d$-dimensional Cala…

Let $X$ be a smooth projective variety over a finite field $k$. Assume that $X$ has trivial canonical class and that $X_{\bar{k}}$ has trivial algebraic fundamental group. An algeb…

Toric compactification conjecture. The mirror Calabi–Yau varieties with respect to $V$ are Calabi–Yau compactifications of the complete intersection in ${\bf T}$ defined by

Mirror-family conjecture. The one-parameter family of $d$-dimensional varieties $V'$ yields the mirror family for $V$.

Let $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial and $\Delta$ is an $\mathbb{R}$-boundary. Let $\mathcal{A}^e(X/Z)$ and…

Let $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial. Write $\mathcal{A}^e(X/Z)$ for the effective nef cone and $\mathcal{M}^e(X/Z)$ for th…

Let $S$ be a scheme smooth and dominant over $\operatorname{Spec} \mathbb{Z}$, and let $Y_S\to S$ be a smooth proper morphism whose fibers are Calabi–Yau varieties. For a closed po…

Let $S$ be a scheme smooth and dominant over $\operatorname{Spec} \mathbb{Z}$, and let $Y_S\to S$ be a smooth proper morphism whose fibers are Calabi–Yau varieties. Weak ordinarity…

Let ${\sf S}$ be the affine moduli scheme of pairs $(X,\alpha)$ with universal family ${\sf X}\to{\sf S}$ and universal holomorphic form…

Let $X_z$ be a family of Calabi–Yau $n$-folds defined over a perfect field of characteristic $p$, let $\alpha_z$ be a holomorphic $n$-form, and let…

Ducat's conjecture. Then the pair $(X,B)$ is log rational. The conjecture extends the demonstrated surface and projective three-space examples, where Cremona transformations relate…

Let $f\colon (\mathcal{X},\mathcal{L})\to S$ be a flat family of $n$-dimensional polarized smooth Calabi–Yau manifolds over a quasiprojective curve $S$. Let $\overline{S}$ be a pro…

Jiao's conjecture. There exists a positive integer $m$ with the following property: there is an effective $\mathbb{Q}$-Weil divisor $\Delta$ on $X$ such that $(X,\Delta)$ is klt an…

Let $(X,D_X)$ be a log canonical pair with standard coefficients, where … with each $D_i$ a prime divisor and $m_i\in\mathbb{Z}_{>0}\cup\{\infty\}$. The orbifold fundamental group…

Let $(\Delta,\{\Delta_i\}_{i=1}^{r})$ and $(\nabla,\{\nabla_i\}_{i=1}^{r})$ be dual nef-partitions, and let $X$ and $X^{\vee}$ be smooth maximal projective crepant partial desingul…

Let $(\widetilde{T}_\Delta,\widetilde{T}_{\check{\Delta}})$ be the pair of $d$-dimensional singular Calabi–Yau varieties obtained as branched double covers of the toric varieties a…

Broustet–Gongyo's conjecture. Every normal projective variety admitting a polarized endomorphism is of Calabi–Yau type. This has been proved for surfaces, smooth threefolds, and ra…

Let $X$ be the smooth Calabi–Yau complete intersection whose existence is conjectured to match the $A$-periods and GKZ system of $Y_{\mathrm{nc}}$, and let $X_{\mathrm{pre}}$ be th…

Regularizing fractional-transformation conjecture. All such Calabi–Yau Laurent hypersurfaces are also describable, via suitable fractional transformations, in terms of general-type…

Flip-folded transposition-mirror conjecture. (1) The transposition mirror may be found as the transposed hypersurface $Z_{f^{\mathrm T}}$ in $\widetilde X$. (2) The toric space…