429 problems

Let $X$ be a smooth cubic fourfold. Write … where $\mathcal{A}_X$ is the Kuznetsov component. An associated K3 surface is a polarized K3 surface $S$ whose primitive degree-two Hodg…

Shafarevich's conjecture. For every positive integer $e$, only finitely many lattices occur as $\operatorname{Pic}(\overline{X})$ for K3 surfaces $X$ over number fields $K$ with…

Skorobogatov's conjecture. The Brauer–Manin obstruction is the only obstruction to the Hasse principle for K3 surfaces.

Strange duality conjecture. For any $g\geq6$, this map is an isomorphism.

A K3 surface is a smooth, simply connected projective surface with trivial canonical bundle. For a variety $X$ over a number field $k$, write $X(\mathbb{A}_k)$ for its adelic point…

O'Grady's conjecture. For every ${\mathcal{E}}\in D^b(S)$,

Let $S$ be a smooth projective surface defined over $\bar{\mathbb Q}$. Denote by $\operatorname{CH}_0(S)^0$ the Chow group of zero-cycles of degree zero, by $\operatorname{Alb}(S)$…

Let $S$ be a K3 surface, let $beta eq 0$ be a curve class, and let $n{g,beta}(S)$ denote the corresponding BPS invariant. The class $beta$ has square $beta^2$ and a divisibility in…

Projective-normality and quadratic-generation conjecture. If $g\geq8$ (equivalently, $q(L_2-2\delta)\geq6$), then this embedding is projectively normal; and if $g\geq10$ (equivalen…

Kaledin–Lehn formality conjecture. The differential graded algebra $\operatorname{RHom}(\mathcal{F},\mathcal{F})$ is formal, namely, it is quasi-isomorphic to its cohomology algebr…

Donagi–Morrison conjecture. There exists a line bundle $M\in\operatorname{Pic}(S)$ adapted to $|H|$ such that $|A|$ is contained in the restriction of $|M|$ to $C$ and

Várilly-Alvarado's conjecture. The order

Let $X$ be a $K3$ surface over a field of positive characteristic. A $K3$ surface is called supersingular if it has maximal possible Picard rank, namely $22$, and it is unirational…

Mirror-family conjecture. Based on numerical evidence, the mirror of the K3 family is given by a certain family of double covers over a del Pezzo surface of degree $6$.

Multiple cover formula.

Let $(S,H)$ be a polarized K3 surface, and suppose that the complete linear system $|H|$ is base point free. For smooth curves $C$ in $|H|$, the gonality is the least integer $k$ f…

Let $K_{2l}$ be the moduli space of quasi-polarized K3 surfaces of degree $2l$, and let the Noether–Lefschetz divisors be the divisors parametrizing K3 surfaces acquiring additiona…

The constructions under discussion arise from toric tops and their associated elliptically fibered Calabi–Yau geometries. In particular, K3 compactifications and the corresponding…

Let $X$ be a $K3$ surface. The group ${\operatorname{Aut}}_s(X)$ of symplectic automorphisms, namely automorphisms acting trivially on a nonzero holomorphic $2$-form, acts on Bloch…

Let $\mathcal{M}^{r}_{g,d}$ denote the Brill–Noether locus of smooth genus-$g$ curves possessing a linear series $g^r_d$, and let $\rho(g,r,d)=g-(r+1)(g-d+r)$ be the Brill–Noether…

Let $L_{K3}$ denote the K3 lattice, and let $N\hookrightarrow L_{K3}$ be a lattice of signature $(1,r)$. Let ${\mathcal F}_N$ be the moduli space of $N$-quasi-polarised K3 surfaces…

Kuga--Satake Hodge conjecture. The embedding $\iota_V$ is induced by a correspondence in $A^\ast(X\times KS(V)\times KS(V))$. This is a special case of the Hodge conjecture; the pa…

Let $(X,\alpha)$ and $(Y,\beta)$ be twisted K3 surfaces, with $\alpha$ and $\beta$ killed by $p$. Let $B$ and $B'$ be B-field lifts of $\alpha$ and $\beta$, respectively, and let…

Let $X$ be a supersingular K3 surface over an algebraically closed field. Artin's conjecture. The surface $X$ is unirational. The paper proves this conjecture, resolving Artin's qu…

Artin's conjecture. Every Artin-supersingular K3 surface $X$ is supersingular.