21 problems

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let…

Huybrechts' projectivity and bigness conjecture.

Let $X^{2n}$ be an irreducible holomorphic symplectic manifold equipped with an abelian fibration whose fibres have polarization type $(1,\ldots,1,n+1)$. Assume the fibration admit…

Let $X^{2n}$ be an irreducible holomorphic symplectic manifold equipped with an abelian fibration whose fibres are isomorphic to Jacobians of curves. A multisection is a subvariety…

Let $X$ be an irreducible holomorphic symplectic $4$-fold, let $E$ be a nef line bundle on $X$, and let $L^*$ denote the map in the sheaf Lefschetz action. The sheaf Lefschetz inje…

Let $X$ be an irreducible holomorphic symplectic manifold, and let $q_X$ denote its Beauville–Bogomolov quadratic form. A divisor is nef if it has nonnegative intersection with eve…

Let $\mathfrak{L}$ be a Solomon–Verbitsky collection of Lagrangian submanifolds in a holomorphic symplectic manifold $(\mathrm{X},\sigma)$. Let $\widehat{\mathcal{A}}_\mathfrak{L}$…

Let $(\mathrm{X},\sigma)$ be a holomorphic symplectic manifold and let $\mathfrak{L}$ be a countable collection of orientable compact Lagrangian submanifolds such that the weight f…

Let $M$ be a holomorphic symplectic manifold with complex structure $I$ and holomorphic symplectic form $\Omega_I$. For $I$-holomorphic Lagrangian submanifolds $L_1,L_2\subset M$,…

Let $X$ be a holomorphic symplectic 4-fold and let $\beta\in H_2(X,\mathbb{Z})$ be an irreducible curve class. Let $P_n(X,\beta)$ be the moduli space of stable pairs and…

Let ${\mathcal H}_g(r)$ be the moduli space of the vector-bundle and connection data considered in the paper, equipped with the holomorphic symplectic form from Corollary $(1)$. Th…

Let $X$ be an irreducible holomorphic symplectic manifold of dimension $2n$, and let $Y$ be a smooth ample hypersurface in $X$. A subvariety $Z\subset Y$ is coisotropic when its ta…

Let $M$ be a smooth projective holomorphic symplectic variety endowed with a faithful symplectic action of a finite group $G$. Let $X:=M/G$ be the quotient, and suppose that $X$ ha…

Boundedness conjecture. Let $Y$ be an irreducible holomorphic symplectic manifold birational to $X$. The set

Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type, meaning that it is deformation equivalent to a Hilbert scheme of $n$ points on a $K3$ surface. Let…

Let $X$ be an irreducible holomorphic symplectic variety with polarization $H$, and let $N_1(X,\mathbb{Z})$ be its group of curve classes modulo homological equivalence. Write…

Let $(X,g)$ be a polarized variety deformation equivalent to a four-dimensional generalized Kummer variety. Let $(R,R)$ denote the Beauville–Bogomolov form on homology. The general…

Holomorphic Weinstein conjecture. The generic leaf of the characteristic foliation on $Y$ is a rational curve. In particular, if $Y$ is smooth, then it is a $\mathbb{P}^1$-bundle o…

Hwang's conjecture. The characteristic foliation on $Y$ cannot have all of its leaves algebraic.

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to $S^{[2]}$, and let $E^*_{\mathrm{nod}}$ be the classes corresponding to the extrema…

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to $S^{[2]}$, where $S$ is a K3 surface. Let $\mathrm{N}^1_+(F,g)$ be the positive hal…