246 problems

Noncommutative minimal model conjecture. For any smooth Fano variety $X$, there exist $\tau\in {\mathrm H}^\bullet(X)$, a sector $\mathscr{S}\subset\mathbf{C}^*$, $\epsilon>0$, and…

Let $X$ be a Fano variety of index $m$ over an algebraically closed field $\boldsymbol{\Bbbk}$ of characteristic zero, and assume that the big quantum cohomology…

Hamilton–Tian conjecture. For any global solution $\omega(t)$ as above, any sequence $(X,\omega(t))$ along the Kähler–Ricci flow contains a subsequence converging to a $\mathbb{Q}$…

Let $X$ be a general hypersurface of degree $d$ in $\mathbb{P}^n$. Let $\overline{\mathcal{M}}_{0,0}(X,e)$ denote the Kontsevich space of degree $e$ stable rational maps to $X$. Di…

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 $X$ 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 $\mathbf P^{d}$.

Fano ampleness conjecture. If $Y$ is Fano with $\rho(Y)=1$, then ${\cal E}$ is ample, and is spanned in most cases. This is presented as the guideline for the paper’s section on co…

The negative-canonical-bundle converse. If the canonical bundle $K_X$ of $X$ is negative, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$…

Global threshold conjecture. For some such divisor $D$, one has

Let $V$ be a smooth Fano variety of dimension $\dim V\geq4$ whose Picard group is generated by the canonical class $K_V$. Pukhlikov's conjecture. If $\dim V\geq4$, then $V$ is bira…

Let $Y$ be a Fano complex contact manifold with second Betti number $b_2=1$. For a simple Lie group $G$, let $\mathfrak{g}$ denote its Lie algebra, and let $G$ act on the projectiv…

Pseudo-effectivity conjecture for the second Chern class.

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.

Vertex bound conjecture.

Harris–Tschinkel conjecture. Rational points are potentially dense on

Finiteness of Cox ring. The ring

Index boundedness conjecture. The family of $n$-dimensional log terminal Fano varieties of index $I$ is bounded.

Let $X$ be a $\mathbb Q$-Fano threefold, and let $H\in|-K_X|$ be a generic section of the anticanonical divisor. Reid's general elephant conjecture. The surface $H$ has at worst ca…

Let $\varphi:X\rightarrow Z$ be a Fano–Mori contraction of a manifold of dimension at most $4$. Andreatta–Wiśniewski's conjecture. All fibers are normal, with the exception describ…

Let $X$ be an $n$-dimensional toric Fano variety, and let $\rho$ denote its Picard number. The preceding discussion suggests that varieties with maximal Picard number are $S_3$-bun…

Universal recursion conjecture. There are universal recursive polynomial laws expressing

Canonical ${\bf Q}$-Fano contraction conjecture. There exist a resolution of singularities

Boundedness-of-index conjecture. The set of possible values of $r(W)$ for canonical ${\bf Q}$-Fano varieties $W$ of dimension $n$ is finite.