44 problems

Let $X$ be a hyper-Kähler variety, and let $\mathrm{R}^{\bullet}(X)\subset \operatorname{CH}^{\bullet}(X)_{\mathbb{Q}}$ be the subalgebra generated by divisor classes. Beauville's…

Voisin's conjecture. If $\phi$ is symplectic, then

Let $X$ be a hyper-Kähler variety. Write $A^\ast(X)$ for its Chow ring with rational coefficients, and let $c_j(T_X)$ denote the Chern classes of its tangent bundle. Beauville–Vois…

Let $\mathcal{F}$ be the moduli stack of a locally complete family of polarised hyperkähler varieties, and let $\mathcal{X}\to\mathcal{F}$ be its universal family. For each…

Let $T$ be a hyperkähler variety, and let $\operatorname{CH}^1(T)$ denote its divisor classes. The cycle class map sends algebraic cycles to rational cohomology. Beauville's diviso…

Let $X$ be a hyperkähler variety of dimension $4$ admitting a Lagrangian fibration, and let $A$ be a general fibre. Write $A^i(X):=CH^i(X)_{\mathbb{Q}}$ for the Chow groups with ra…

Let $X$ be a hyperkähler variety, and let a multiplicative Chow–Künneth decomposition (MCK decomposition) mean a Chow–Künneth decomposition whose projectors respect the multiplicat…

O'Grady's modified diagonal conjecture. If $X$ is a hyperkähler variety of dimension $d=2n$, then there exists a point $p\in X$ such that

Franchetta conjecture. Every locally complete family of hyperkähler varieties satisfies the Franchetta property. The property is known for some Fano varieties, but the source prese…

Let $\mathcal{F}$ be the moduli stack of a locally complete family of polarised hyperkähler varieties, and let $\mathcal{X}\to\mathcal{F}$ be its universal family. Let $T$ be any m…

Anti-symplectic involution conjecture. Then

Let $X$ be a projective hyper-Kähler variety, and let $\mathrm{KS}(X)$ be its associated Kuga–Satake abelian variety. Kuga–Satake motivation conjecture. The variety $X$ is motivate…

Let $X$ be a hyper-Kähler variety. Let $B^\ast(X)$ be the ring of algebraic cycles with rational coefficients modulo algebraic equivalence, and let $c_j(T_X)$ denote the Chern clas…

Let $X$ vary in the moduli space of hyperkähler varieties. A hyperkähler variety is said to admit a Lagrangian fibration if it admits a fibration whose fibers are Lagrangian with r…

Conjecture. The morphism $\phi_{\mathcal{L}}$ is generically finite of degree two over the Plücker variety.

The moduli space $\mathcal{M}_{{\operatorname{OG10}}}^{d,\gamma}$ parametrises polarised irreducible holomorphic symplectic manifolds of OG10 type, of degree $d$ and divisibility…

Motivic control conjecture. The variety $X$ should be motivated by $\mathrm{KS}(X)$, and the motives of $X$ and $Y$ should be isomorphic. These assertions express the expectation t…

Let $(X,h)$ be a $2n$-dimensional polarized hyper-Kähler variety, and let $H^2(X,\mathbb{Q})_{\mathrm{prim}}$ denote the orthogonal complement of the polarization class in…

Let $X$ and $Y$ be hyper-Kähler varieties of $K3^{[n]}$-type. Let $L(X)$ and $L(Y)$ be their Markman–Mukai lattices, with primitive generators $v$ and $w$ of the rank-one orthogona…

Let $Y$ be a smooth projective hyper-Kähler variety, and let $\mathfrak{h}^{\circ}(Y)$ denote its birational motive. Vial's grading conjecture. The birational motive…

Let $X$ be a projective hyper-Kähler variety, and let $H^2_{\mathrm{tr}}(X)$ denote its transcendental second cohomology. Let $\mathrm{KS}(X)$ be the Kuga-Satake variety associated…

Let $X$ be a smooth projective hyper-Kähler $2n$-fold, let $h_X$ be a polarization, and let $Z_{\mathrm{lef}}\in\operatorname{CH}^2(X\times X)$ be a cycle such that its cohomologic…

Let ${\mathcal G}^{[n]}_{an}$ be the groupoid of the relevant cohomological morphisms between projective irreducible holomorphic symplectic manifolds of $K3^{[n]}$-type, and let…

Multiplicativity conjecture. The intersection product preserves the eigenspace grading: products of classes in ${\rm CH}^i(X)_s$ and ${\rm CH}^j(X)_t$ lie in…

Strict grading conjecture. This unital grading is a strict grading. More strongly, the graded co-algebra morphism co-induced by the graded split surjection…