188 problems

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 $X$ be a Calabi–Yau variety in one of the cases under consideration, and let $\operatorname{Aut}(X)$ act on its Kähler cone. A Kawamata–Morrison conjecture. The action of…

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 $f:X\to Y$ be an algebraic fiber space with $X$ and $Y$ normal projective varieties. Consider a pair $(X,\Delta)$ such that … and suppose that $\Delta$ has no horizontal diviso…

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…

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

Let $X$ be a canonical Calabi–Yau variety. The index of a Calabi–Yau pair $(X,B)$ is the smallest integer $m\geq 1$ such that $m(K_X+B)\sim 0$. Index conjecture. There exists…

Let $X$ and $Y$ be Calabi--Yau varieties that are related by a one-to-one correspondence, rather than merely belonging to more general paired mirror families. For a finite field…

Let $P$ and $P^*$ be a pair of polar dual reflexive polyhedra, and let the associated Calabi–Yau hypersurfaces lie in Gorenstein toric Fano varieties. Batyrev's polar-duality conje…

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. Weak ordinarity…

Multisection-index conjecture. One has

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…

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

Let $V$ and $W$ be $n$-dimensional, possibly singular, complex Calabi–Yau varieties. Their stringy invariants are denoted by $E_{\operatorname{st}}(V;u,v)$ and…

Mirror Symmetry Conjecture. There exist mirror pairs of Calabi–Yau varieties with these compatibilities; in particular, the displayed Hodge-number relations should hold when the re…

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 $X$ and $Y$ be a maximally generic mirror pair of $3$-dimensional Calabi–Yau manifolds defined over $\mathbb{F}_q$. Let $S_p(X,u,T)$ denote the $p$-adic slope zeta function. Sl…

Let $X$ and $Y$ be a strong mirror pair of Calabi–Yau manifolds defined over $\mathbb{F}_q$. For a closed point count, write $\#X(\mathbb{F}_{q^k})$ for the number of…

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…

The Chen–Ruan orbifold cohomology ring assigns an obstruction bundle to each relevant sector of a complex orbifold. Riemann-bilinear-relations conjecture. The Riemann bilinear rela…