563 problems

Let the Calabi–Yau threefolds under consideration be those represented within the Kreuzer–Skarke database. Reid's fantasy. All Calabi–Yau threefolds in this database are birational…

King's conjecture. Every smooth toric variety has a tilting bundle consisting of line bundles.

Oda's conjectures. The following two statements are true: (1) every smooth polytope has the IDP; and (2) if $P,Q\subset\mathbb{R}^n$ are smooth polytopes and the normal fan of $P$…

Let $\Delta$ and $\Delta^*$ be dual reflexive polytopes. Consider anticanonical hypersurfaces in the corresponding toric varieties, together with their crepant resolutions when nee…

Regular-sequence conjecture. The elements $e^0,e^1_{m,n,k},\ldots,e^{mn}_{m,n,k}$ form a regular sequence in $K[(\Delta_m)^{\times n}]$ of maximal length.

Let $R$ be a Gorenstein toric algebra, meaning the coordinate ring associated to a Gorenstein cone. An NCCR of $R$ is an algebra of the form $\Lambda=\operatorname{End}_R(M)$ for…

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…

Mirror-family conjecture. Based on numerical evidence, the mirror of the K3 family is given by a certain family of double covers over a del Pezzo surface of degree $6$.

Let $X$ and $Y$ be toric Fano manifolds, and let $H^*(X)$ and $H^*(Y)$ denote their cohomology rings. A cohomology ring isomorphism preserves their first Chern classes when it maps…

Let $\Sigma$ be an $n$-dimensional complete nonsingular fan, let $G(\Sigma)$ be the set of generators of its rays, and let $\ell$ be the Picard number of $\Sigma$. A primitive coll…

Let $X$ be a normal $\mathbb{Q}$-factorial algebraic variety with a $\mathbb{Q}$-boundary divisor $D=\sum d_iD_i$, where $0\leq d_i\leq 1$ and the $D_i$ are prime Weil divisors. As…

Let $V_d=\{f_{1,d}=\cdots=f_{r,d}=0\}\subset(\mathbb C^\ast)^n$ be the complete intersection under consideration, and let $\Delta_{i,d}$ be the Newton polytope of $f_{i,d}$. Set…

Let $P$ be a reflexive polytope of dimension $d$, and write $\lvert V(P)\rvert$ for its number of vertices. Nill's conjecture. The number of vertices satisfies … This is a proposed…

Let $X$ be a smooth complex projective variety that is the image of a projective toric variety under a morphism. Occhetta–Wiśniewski conjecture. The variety $X$ is automatically to…

Q-Gorenstein–mutation conjecture. $\mathbb{Q}$-Gorenstein equivalence classes of projective Fano varieties with mild singularities correspond to mutation equivalence classes of rig…

Let $P$ be a $G$-invariant lattice polytope, and let $Y$ be the corresponding toric variety with torus-invariant line bundle $L$. The equivariant $H^*$-series is the series…

Toric Fano extremal metric conjecture. Every toric Fano manifold admits an extremal Kähler metric in its first Chern class.

Recall that a rational polytope is smooth if it is simple and every normal cone is unimodular, and reflexive if it is a lattice polytope with the origin in its interior and its pol…

Donald's conjecture. The distance satisfies

Let $\Delta\subset\mathbb{R}^d$ be a polytope, let $\Phi$ be its dual fan, and let $\overline{M}_\Phi$ be the quotient combinatorial intersection cohomology module defined from the…

Let $\Delta$ be a $d$-dimensional polytope, and let $Gh_k(\Delta)$ denote the components of Stanley's generalized $h$-vector. For a polytope that is integral, these numbers agree w…

In an ${\mathcal N}=1$ supersymmetric quiver gauge theory, suppose that different gauge theories have the same infrared moduli space, and that this moduli space is toric. Toric dua…

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 $(M,g)$ be a Hermitian gravitational instanton with either ALF or AF asymptotics. Aksteiner–Andersson toricity conjecture. Every such gravitational instanton is toric. The conj…

Let $X$ be a smooth Calabi–Yau threefold arising from the Kreuzer–Skarke database, and let a non-trivial Hilbert basis element be a divisor class in the Hilbert basis under conside…