18 problems

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. Write $\operatorname{CH}^1(T)$ for the divisor classes, $c_i(T)$ for the Chern classes of $T$, and $\mathrm{cl}$ for the cycle class map. Beauvill…

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 $X$ 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 of dimension $2n$. Define … where $\mathrm{S}_{s}{\rm CH}_0(Y)$ is the Beauville–Voisin filtration piece generated by points who…

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…

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…

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…

Let $X$ be a smooth projective variety and let $i$ be a positive integer. Assume that $\mathrm{H}^{*}_{\mathrm{tr}}(X)$ is generated by $\mathrm{H}^i_{\mathrm{tr}}(X)$. Bloch–Beili…

A birational Chow–Künneth decomposition of a smooth projective variety is co-multiplicative when the induced grading on its zero-cycles is compatible with the diagonal coproduct. C…

A hyper-Kähler variety $X$ of dimension $2n$ is called surface decomposable if there exist a smooth projective variety $\Gamma$, smooth projective surfaces $S_1,\ldots,S_n$, and ge…

Let $X$ be a hyperkähler variety of dimension $2m$, and let $A^i(X):=CH^i(X)_{\mathbb{Q}}$ denote its codimension-$i$ Chow group with rational coefficients. Assume the Chow ring ha…

Let $\mathcal C$ be the class of varieties whose Chow rings admit the multiplicative bigrading described in Beauville's weak splitting problem. Let $X$ be an EPW sextic, meaning a…

Bloch's conjecture. The sum of the actions of the powers of $\sigma$ annihilates homologically trivial zero-cycles. This prediction follows in the source from the correspondence fo…

Let $X$ be a smooth projective variety with $c_1(X)=0$ in $H^2(X,\mathbb{R})$, and let $L$ be a nef and big line bundle on $X$. Kawamata's trivial-first-Chern-class conjecture. The…