20 problems

Let $X$ be a complex hyper-Kähler variety, and let $A^*(X)=\bigoplus_i A^i(X)$ be its Chow ring with $\mathbb{Q}$-coefficients. A subvariety $Z\subset X$ is a constant cycle subvar…

Huybrechts's conjecture. For every hyper-Kähler variety $X$, the divisibility

Beckmann's conjecture. This Yoneda product is skew-symmetric.

Beckmann's conjecture. The element $\operatorname{obs}_E(\eta)$ has the hard Lefschetz property for the Yoneda product on $\operatorname{Ext}^{\bullet}(E,E)$.

Beckmann's conjecture. The object $E$ is $1$-obstructed.

Finiteness conjecture. The fundamental group $\pi_1(M_\mathrm{reg})$ is finite.

Let $X$ be a hyper-Kähler variety, and let $\mathfrak{h}(X)$ denote its motive. For each degree $i$ and prime $ell$, let $\mathbf{G}_{\ell,i}(X)$ be the $ell$-adic algebraic monodr…

Let $X$ and $Y$ be hyper-Kähler varieties of dimension $2n$, let $\mathcal E$ be the twisted vector bundle used in the source, and let $\widetilde{\mathfrak F}$ and…

Let $X$ and $Y$ be hyper-Kähler varieties of dimension $2n$, and let $\mathfrak{F}$ and $\mathfrak{F}^{-1}$ be the Fourier transforms and their inverses considered in the source, w…

Let $X$ and $X'$ be derived equivalent hyper-Kähler varieties, and let $h_{\mathrm{hom}}(X)$ denote the homological motive of $X$, equipped with its cup-product. The homological mu…

Let $X$ and $X'$ be derived equivalent hyper-Kähler varieties. The Chow motive $h(X)$ is equipped with its cup-product structure, written $(h(X),\cup)$. Fu–Vial's multiplicative Or…

Isogenous hyper-Kähler varieties conjecture. If $X$ and $Y$ are isogenous hyper-Kähler varieties, then there exists an isomorphism of rational Chow motives

Let $d$ be an integer and $M$ a deformation type of hyper-Kähler manifolds. For a hyper-Kähler variety $X$ over a number field $K$ with $[K:\mathbb{Q}]\leq d$ and an embedding…

Fix a number field $K$ and a finite set of places $S$. A smooth projective $K$-variety is hyper-Kähler if the associated complex variety is hyper-Kähler. Unpolarized Shafarevich co…

Triviality conjecture for the defect group. The defect group $P(X)$ should be trivial. Equivalently,

Generalized Bloch conjecture. The action of $\iota$ on $\operatorname{CH}_{0}(X)$ should be the identity on

Beauville--Voisin conjecture. The restriction of the cycle class map to $\langle c_i(X),\operatorname{Pic}(X)\rangle$ is injective. Equivalently, any polynomial in line bundles and…

Generic-fiber Franchetta conjecture. The group

Let $X$ be a hyper-Kähler variety of dimension $2n$. For an integer $i$, let $C_i(X)\subset \operatorname{CH}_i(X)=\operatorname{CH}^{2n-i}(X)$ be the $\mathbb{Q}$-vector space gen…

Let $X$ be a hyper-Kähler variety of dimension $2n$. For each $i$, let $C_{2n-i}(X)\subset \operatorname{CH}^i(X)$ be the $\mathbb{Q}$-vector space generated by codimension-$i$ sub…