16 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…

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 $2n$, 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…

Let $X$ be a projective hyper-Kähler manifold. Consider the cycle class map from the Chow ring $CH^*(X)$ to cohomology. Beauville's weak splitting conjecture. The cycle class map i…

A hyper-Kähler variety is a smooth projective variety with the usual hyper-Kähler structure; a multiplicative Chow–Künneth decomposition is a Chow–Künneth decomposition compatible…

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 $2n$, and let $S_i\operatorname{CH}_0(X)$ denote the subgroup generated by classes of points whose rational-equivalence orbits have d…

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…

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. For $i\leq n$, call a degree-$2i$ cohomology class coisotropic if it satisfies the coisotropic condition defined by…

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. Let $S_\bullet$ be the filtration on zero-cycles induced by the loci $S_iX$, and let $F'_{BB}$ denote the even leve…

Let $X$ be a projective hyper-Kähler manifold, and let $\operatorname{CH}^*(X)$ denote its Chow ring with $\mathbb{Q}$-coefficients. Beauville's conjecture. The cycle class map is…