284 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…

Every projectively induced Kähler–Einstein metric on a smooth compact toric manifold is, up to automorphisms, a product of Fubini–Study metrics on projective spaces with matched mu…

McKernan's conjecture. Fix a positive integer $r$ and a real number $0<\epsilon\leq 1$. There exists $\delta>0$ depending only on $r,\epsilon$ such that, whenever…

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…

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 $X\to\mathbf{P}_{\Delta}$ 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…

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.

Toric 2-Fano classification conjecture. The only toric 2-Fano manifolds are projective spaces.

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…

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…

Let $X(\Delta_{n-1})$ be the toric variety associated with a reflexive toric diagram $\Delta_{n-1}$, and let $\widetilde{X(\Delta_{n-1})}$ denote the relevant resolution. Let…

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…

Donaldson's conjecture. There exists a solution of the modified Abreu equation in $\mathcal{S}(\Delta, {\bf L})$ if and only if $(\Delta, {\bf L}, f)$ is K-polystable.

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…

Toric modified-Calabi-flow conjecture. The modified Calabi flow converges exponentially fast to an extremal metric in $c_1(L)$.

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 $P$ be a reflexive polytope. Let $\Sigma$ be a simplicial fan whose ray primitive generators are the vertices of $P$, and let $\mathcal{V}$ be the fan of the canonical bundle o…

Neatness conjecture. All smooth lattice polytopes are neat.

Picard number conjecture. Then $S$ has Picard number one.

Let $\Sigma_1$ and $\Sigma_2$ be smooth fans with the same support. A smooth star subdivision is the subdivision of a fan obtained by inserting the ray through the sum of the primi…

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 semi-Fano toric manifold, meaning that its anti-canonical divisor $-K_X$ is nef. Let $W^{\text{HV}}$ and $W^{\text{LF}}$ denote its Hori–Vafa and Lagrangian Floer supe…

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…