110 problems

Let $d$ be a natural number and let $\mathfrak{R}\subset [0,1]$ be a finite set of rational numbers. For a finite set $\mathfrak{R}$, write … A pair $(X,B)$ is semi-log canonical i…

Calabi–Yau type conjecture. Then $X$ is of Calabi–Yau type.

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 $G/P$ be a homogeneous roof, and $\mathcal{Z}$ a homogeneous roof bundle of type $G/P$ with projective bundle structures … Given a general section…

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 $X\to\mathbf{P}_{\Delta}$ and $X^{\vee}\to\mathbf{P}_{\nabla}$ be smooth MPCP desingularizations admitting the stated nef-partitions, and let $Y$ and $Y^{\vee}$ be the associat…

Let $X$ be a smooth projective variety with $c_1(X)=0$ in $H^2(X,\mathbb{R})$, and let $L$ be a nef and big line bundle on $X$. Kawamata's trivial-first-Chern-class conjecture. The…

Semiampleness conjecture on klt Calabi--Yau pairs. The divisor $L$ is num-semiample; that is, there exists a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L'$ on $X$ such that

Integral cohomology and non-isomorphism conjecture. One has an isomorphism of integral cohomology groups

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 $\frX \to \AA^1$ be a Tyurin degeneration to $\frZ_0 \cup_{\frD} \frZ_1$, with mirror Calabi-Yau $\widecheck{\frX_{gen}}$ and mirror divisor $\frD$. A Tyurin degeneration is a…

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 $M$ be a moduli space of polarized log-terminal Calabi–Yau varieties satisfying the assumptions specified in the source. Satake–Baily–Borel compactification conjecture. There s…

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…

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 $S$ 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…