247 problems

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Let $X$ be a Fano variety with at most Kawamata log terminal singularities. An anti-canonical polar cylinder means a cylinder polarized by $-K_X$. The anti-canonical cylinder conje…

Injectivity conjecture. With the same notation as in the quantum Lefschetz proposition, there is a natural injective morphism of algebras

Let $k$ be a number field and let $X$ be a smooth Fano variety over $k$, so that $-K_X$ is ample. Fix an adelic metrization of the anticanonical bundle $comathcal O(-K_X)$, with as…

Mukai's conjecture. One has

Let $X$ be a smooth Fano variety over $\mathbb C$ with Brauer group $Br(X)$. Let $C$ be a smooth connected genus $g$ curve. A component of $\operatorname{Mor}(C,X,\alpha)$ is under…

Let $X$ be a smooth Fano variety. Denote by $\rho_X$ its Picard number, by $\iota_X$ its pseudo-index, and by $\dim X$ its dimension. The generalised Mukai conjecture. … with equal…

BAB conjecture. The set of $\epsilon$-lc Fano varieties of dimension $d$ forms a bounded family.

Borisov–Alexeev–Borisov conjecture. There is a positive number $\delta(n)$ such that

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 smooth Fano variety of complex dimension $r$, let $g$ be a genus, and let $A$ be a curve class. The fixed-domain Gromov–Witten invariants are the virtual counts associ…

Let $X$ be a smooth Fano variety. The Dubrovin conjecture. The derived category $\mathrm{D}^b(X)$ admits a full exceptional collection if and only if the big quantum cohomology rin…

Let $X$ be an $n$-dimensional smooth Fano variety, with index … Let $\rho_X$ be the Picard number of $X$. Mukai's conjecture. One has … with equality if and only if $X$ is isomorph…

Pseudo-effectivity conjecture for the second Chern class.

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…

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

Katzarkov–Kontsevich–Pantev conjecture. One should have

Let $X$ be a smooth Fano variety. A toric Landau–Ginzburg model existence conjecture asserts that $X$ has a Laurent polynomial satisfying the period, Calabi–Yau, and toric conditio…

The multiplicative Chow–Künneth conjecture. The variety $X$ has a multiplicative Chow–Künneth decomposition. This is presented as a partial answer to the problem of describing vari…

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 $X\subset \mathbb{P}^n$ be a smooth hypersurface of degree $d$, and let $\mathcal{F}(X)$ denote its Fano variety of lines. A smooth hypersurface is Fano when $d\leq n$. The exp…

Finiteness of Cox ring. The ring

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

Gongyo's Mukai-type conjecture.

Let $X$ be an $n$-dimensional smooth Fano variety, and let its pseudoindex be … Let $\rho_X$ be the Picard number of $X$. Generalized Mukai conjecture. One has … with equality if a…