38 problems

For each fixed complex dimension, consider compact hyper-Kähler manifolds and identify two such manifolds when they are homeomorphic, equivalently when they have the same topologic…

Let $N$ be a hyper-Kähler manifold of complex dimension $2n$, and suppose its Néron–Severi group has rank one and is generated by an isotropic element of $H^{2}(N,\mathbb{Z})$. The…

Monodromy restriction conjecture. The restriction homomorphism

Classification conjecture. The manifold $X$ should be either a complex torus or an irreducible hyperkähler manifold. In particular, $K_X$ should be trivial, and the two cases shoul…

Campana–Demailly–Verbitsky conjecture. Either $X$ has a finite étale cover which is bimeromorphic to a complex torus, or $H^0(X,\Omega_X^2)$ is generated by a holomorphic $2$-form…

Let $X_t$ be a smooth family of compact holomorphically symplectic manifolds over a disk $\Delta$. Assume that $X_t$ is hyperkähler for all $t\neq 0$. Soldatenkov–Verbitsky conject…

Let $S$ be a K3 surface with a Kähler class $\omega$ and a Mukai vector … where $\xi\in \mathrm{NS}(S)$, $r>1$ is prime to $\xi$, and $\omega$ is $v$-generic. Let $M_v(S,\omega)$ b…

Debarre–Huybrechts–Macrì–Voisin conjecture. Then $c_X=(2n-1)!!$ and the Huybrechts–Riemann–Roch polynomial of $X$ is

Rational-curves conjecture for torsion-canonical manifolds. The manifold $X$ is an étale quotient of an Abelian variety.

Let $X$ be a compact hyperkähler manifold, and define … The perverse-Hodge octahedron conjecture. The convex hull of the non-zero $h^{i,k,d}$ is an octahedron. This is stronger tha…

Generalized Fujiki positivity conjecture.

Rozansky--Witten generalized Fujiki relation.

Degree-four generalized Fujiki constant conjecture.

Chern-character positivity conjecture. All signed Chern character numbers $(-1)^n\operatorname{Ch}_{2\lambda}$, where $|\lambda|=n$, of irreducible hyper-Kähler manifolds are posit…

The known irreducible hyper-Kähler manifolds include Hilbert schemes of points on K3 surfaces, generalized Kummer varieties, and two further examples in dimensions $6$ and $10$. A…

Let $X$ be a projective irreducible holomorphic symplectic manifold, and let $\widetilde{H}(X,\mathbb{Q})$ be its extended rational cohomology. A line $\ell$ in…

Non-vanishing conjecture. One has $\kappa(L)\geq 0$.

Let $X$ be a hyperkähler manifold, and let $\mathcal{M}_{rat}(\mathbf{C})$ denote the category of rational motives over $\mathbf{C}$. A motive is of Abelian type if it belongs to t…

Let $X$ be a projective hyperkähler manifold. The Lunts equality conjecture. There is an equality of Lie subalgebras of $\operatorname{End}(H^\bullet(X,\mathbb{Q}))$: … This conjec…

Let $M$ be a compact hyperkähler manifold admitting a holomorphic Lagrangian torus fibration $\ell:M\to B$, and let $\beta$ be the pullback of an ample divisor on $B$. Let $M_1$ be…

Let $2n$ be an even natural number. Let $(X,H)$ be a polarised hyperkähler manifold, where $X$ has dimension $2n$ and $H$ is an ample Cartier divisor. Write … for the projectivised…

Fix a connected moduli space $M$ of polarized $2n$-dimensional irreducible holomorphic symplectic manifolds $(X,L)$ with second-cohomology lattice $\Lambda$, and write the associat…

Let ${\rm NL}^\ast_{\rm hom}(\mathcal{F}_h)$ be the image of the Noether–Lefschetz ring in $H^\ast(\mathcal{F}_h,\mathbb{Q})$, and let $\mathrm{R}^\ast_{\rm hom}(\mathcal{F}_h)$ be…

Let $\mathcal{F}_h$ be the moduli space under consideration of hyperkähler manifolds, and let ${\rm NL}^\ast(\mathcal{F}_h)$ be the ring generated by Noether–Lefschetz cycles. Let…

Let $X$ be a compact hyperkähler manifold of dimension $2n$, and let $a(X)$ denote its algebraic dimension. Algebraic-dimension values conjecture. Then … The source reports that th…