78 problems

Hassett–Tschinkel's conjecture. These are all the extremal rays of the cone of curves.

Hyperkähler SYZ conjecture. The line bundle $L$ is semiample.

Lagrangian conjecture. If

Let $M$ be a projective hyper-Kähler manifold. A filtration on the Chow ring is called a Beauville–Voisin filtration when it is analogous to the filtration on the Chow group of zer…

Let $S$ be a K3 surface, let $H$ be a primitive ample divisor on $S$, and let $m\geq 2$. The linear system $|mH|$ parametrizes curves whose variation in the moduli space of curves…

Let $X$ be an irreducible holomorphic symplectic manifold deformation equivalent to $S^{[n]}$. A monodromy operator is an element of the monodromy group $Mon(X)$ acting on the coho…

Numerical criterion for abelian fibrations. The Hilbert scheme $X$ admits a structure of an abelian fibration if and only if

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…

Let $N$ be a hyper-Kähler manifold of complex dimension $2n$. An isometric deformation of its complex structure is a deformation preserving a chosen Calabi–Yau metric. The SYZ defo…

A compact hyper-Kähler manifold is a compact Kähler manifold with a holomorphic symplectic form; its complex dimension is even. The SYZ finiteness conjecture. In each fixed complex…

Monodromy restriction conjecture. The restriction homomorphism

Mirror Conjecture. The quantum variation of Frobenius algebras of a mirror partner is equivalent to the Yukawa variation of Frobenius algebras of $M$. Equivalently, after shrinking…

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…

Let $X_t$ be a degeneration limit of hyperkähler manifolds. A degeneration limit of hyperkähler manifolds should belong to Fujiki class $\mathcal C$. This conjecture concerns wheth…

Let $X$ be a higher-dimensional hyperkähler manifold, let $\mathcal{F}$ be a simple torsion-free sheaf on $X$, and write $\operatorname{Def}(\mathcal{F})$ for its deformation space…

Weak hyperkähler SYZ conjecture. If $q(M)=0$ and $h^0(X,\mathcal{O}_X(M))\geq n+1$, then $M$ is semiample; that is, some positive multiple of $M$ is globally generated.

Divisibility conjecture. If $q(M)=0$ and $h^0(X,\mathcal{O}_X(M))\geq n+2$, then $M$ is divisible in $\operatorname{Pic}(X)$.

Debarre–Huybrechts–Macrì–Voisin conjecture. Then $a=1$ and $X$ is of $K3^{[n]}$ or OG10 deformation type. This more ambitious conjecture is verified for all known examples of hyper…

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 $M$ be a hyperkähler manifold and let $\ell$ be a cohomology class. Define the nef Teichmüller space by … and define the semiample Teichmüller space by … Here a line bundle is…