283 problems

For every integer $d\geq 2$, any two nonsingular projective toric $d$-dimensional weak Fano varieties are connected by a finite zigzag of projective toric birational morphisms, eac…

Non-toric NCCR extension conjecture. The statement of the theorem holds also if $\Lambda'$ is a non-toric NCCR.

Let $\omega$ be the metric on $\mathbb C^k$ determined by Lemma, and let $\omega_p=iD_0\partial\bar{\partial}s^p$, where $s=|z|^2$ on $\mathbb C^k$. Say that a metric is asymptotic…

Ambient-monodromy realization conjecture. An isomorphism $\Phi\in\operatorname{Aut}H_2(X,T(x))$ can be realized as the ambient monodromy of a Hamiltonian diffeomorphism mapping…

Shokurov's toric-model conjecture. Every maximal log Calabi–Yau pair $(X,\Delta_X)$ whose underlying variety $X$ is a rational $3$-fold has a toric model.

Trichotomy conjecture. For every vertex $i\neq 0$, there is a unique $k(i)\in\\{-1,0\\}$ such that

Let $Q$ be a connected component of the coarse moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ such that $X$ is a connected smooth projective complex variety,…

Simple generalized permutahedron conjecture. There is a natural map

Let $\Delta$ be a smooth fibre-like polytope of dimension $d$, and let $X(\Delta)$ denote the associated smooth toric Fano variety. Assume that $d$ is an odd prime number. Classifi…

Let $X$ be a smooth complete toric variety. Suppose that $X$ is an $F$-fibration over $Y$, where $Y$ and $F$ are smooth toric varieties, and both $Y$ and $F$ have a full strongly e…

Let $A$ be a configuration, let $f$ be a rational $A$-hypergeometric function in variables $x_1,\dots,x_s$, and let a facial subset be a subset of $A$ arising as a face of the conf…

Let $\Delta_k$ denote the $k$-simplex, and let … where $k_0\geq\cdots\geq k_r$. The configuration $A$ is GKZ-rational when its associated GKZ hypergeometric system has the rational…

Let $A_0,A_1,\dots,A_r$ be vector configurations in $\mathbb{Z}^r$, and let $A$ be their Cayley configuration … where $e_0,\dots,e_r$ is the standard basis of $\mathbb{Z}^{r+1}$. T…

Let $X$ be a nonsingular toric weak Fano $d$-fold. Two such varieties are weakly-F-equivalent if they are connected by a sequence of equivariant blow-ups, blow-downs, and flops thr…

Let $X$ and $Y$ be nonsingular toric Fano $d$-folds, and let $\varphi:Y\longrightarrow X$ be an equivariant morphism. An equivariant blow-up is the blow-up along a $T_N$-invariant…

Pseudo-symmetry or F-equivalence conjecture. Any nonsingular toric Fano $d$-fold is either pseudo-symmetric or F-equivalent to the $d$-dimensional projective space $\mathbf P^{d}$.

MSV sheaf conjecture. For every variety $X$ with only Gorenstein toroidal singularities, there exists a local sheaf $\mathcal{MSV}(X)$ locally isomorphic to the product of…

Master Family degeneration conjecture. Vertex algebras arising from Mirror Symmetry for hypersurfaces defined by $\Delta$ and $\Delta^*$ are degenerations of the Master Family of v…

GPSS conjecture. All such cyclic Gorenstein quotient singularities admit torus-equivariant projective, crepant, full resolutions. The theorem immediately preceding the conjecture p…

Toric Fano mirror conjecture. There should exist a primitive form for $f$ given by

Let $G$ be a finite abelian subgroup of ${\operatorname{GL}}(n,\Bbbk)$, and let ${G}\operatorname{-Hilb}$ denote the associated $G$-Hilbert scheme. Disconnectedness conjecture. The…

Let $\mathbb P(\mathcal O\oplus\mathcal O(k)\oplus\mathcal O(-2-k))$ and $\mathbb P(\mathcal O\oplus\mathcal O(-1)\oplus\mathcal O(-1))$ denote the corresponding projective bundles…

Kleiman–Mori cone conjecture.

Let $X$ be a Gorenstein Fano variety with canonical singularities. The source's Theorem A' gives, for $d=2$, $(-K_X)^2\leq 9$ with equality if and only if $X\cong\mathbb{P}^2$; for…

Simplicial vertex bound conjecture.