400 problems

Mukai's conjecture. One has

Let $M_{n,a}$ be a smooth hypersurface of degree $a\le n+1$ in $\mathbb{CP}^{n+1}$, let $D_{n,a,i}$ be a union of $i\le n+2$ generic hyperplanes, and let…

Let $X$ be a Fano variety and $C$ a smooth projective curve. A Manin component is an irreducible component of the relevant morphism space distinguished from exceptional components.…

Let $d$ be a positive integer and let $\boldsymbol{\mathcal{X}_{d,\varepsilon}}$ denote the class of $\varepsilon$-klt Fano varieties of dimension $d$, where $\varepsilon>0$ is rat…

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.

Let $X\subset \mathbb{P}^N$ be a smooth Fano hypersurface of degree $d$, and let $F(X)$ denote its Fano scheme of lines. The expected dimension is $2N-d-3$. Debarre–de Jong conject…

Let $X$ be a Fano variety. The invariant $b3(X)$ characterizes a notion called Gibbs stability, while $b4(X)$ is the stability threshold in the log Fano setting. Gibbs stability co…

Let $M$ be a compact Kähler manifold with positive first Chern class, so that $M$ is Fano. A Kähler–Einstein metric on $M$ is a Kähler metric whose Ricci curvature is proportional…

Let $\mathrm{OG}$ be the odd quadric considered above, let $c_1=c_1(-K)\in H^2(\mathrm{OG})$ be its anticanonical class, and let $\operatorname{FPdim}(A(c_1))$ denote the Frobenius…

A Fano hypersurface is a hypersurface whose anticanonical class is ample; in the statistical model of random Fano hypersurfaces over the rationals, one asks whether such a hypersur…

Tian's strong stabilization conjecture. If $R(X,L)$ is generated by

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…

Log canonical Fano Mori dream space conjecture. Then $X$ is a Mori dream space.

Pseudo-effectivity conjecture for the second Chern class.

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

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

Let $X$ be a smooth cubic fourfold, let $F(X)$ be its Fano variety of lines, and let $S$ be a K3 surface. Galkin–Shinder's conjecture. The cubic fourfold $X$ is rational if and onl…

Let $X$ be a Fano manifold, so $K_X^{-1}$ is ample, and suppose that its tangent bundle $TX$ is nef. An algebraic cellular decomposition is a chain of Zariski closed subsets whose…

Let $Z$ be a complex projective manifold equipped with a contact structure, namely a holomorphic hyperplane distribution $D\subset TZ$ such that the induced bundle morphism … is ev…

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…

Let $X$ be a Fano manifold with Picard rank one. A Fano manifold is K-semistable when its Donaldson–Futaki invariant is nonnegative for every test configuration. Odaka–Okada's conj…

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 a Fano fibration be a fibration whose fibres are Fano varieties, possibly with singular fibres in a degeneration. Fano-fibration degeneration conjecture. A Fano fibration is po…

Let $X$ be a Fano manifold. A Landau–Ginzburg model for $X$ is a one-parameter family $Y\to\mathbb{A}^1$, with associated Fukaya category $Fuk(X)$, Fukaya–Seidel category $FS(Y)$,…