44 problems

Homological 3d mirror symmetry conjecture. There are 3d mirror pairs $X$ and $X^{!}$ whose A-model 2-category and B-model 2-category are interchanged under mirror symmetry.

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 $(F,g)$ 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 $F$ be a holomorphic symplectic fourfold deformation equivalent to the Hilbert square of a K3 surface, and suppose its Picard lattice contains two nodal classes $\rho_1$ and…

Let $F$ be an irreducible holomorphic symplectic fourfold, and let $\lambda$ be a primitive divisor class of Beauville square zero on the boundary of the closure of the ample cone.…

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…

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

Huybrechts' projectivity and bigness conjecture.

Let $F$ be an irreducible holomorphic symplectic fourfold deformation equivalent to $S^{[2]}$ for a K3 surface $S$, and let the reflection group act on the positive cone of divisor…

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 $X^{\prime}$ be an irreducible holomorphic symplectic manifold containing a non-trivial divisor $D^{\prime}$ with $q_X(D^{\prime})=0$. Let $X$ be a birational model correspondi…

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 $X$ be a K3 surface and let $A$ be an abelian surface. For a connected Jacobi diagram $\Gamma$, write $\beta_\Gamma(X^{[n]},\sigma^{[n]})$ for the associated Rozansky–Witten in…

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 $\pi:M\to X$ be a Lagrangian fibration, and say that its fibers have a reduced component in codimension $1$ when the fiber over a general point of each discriminant divisor is…

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

An irreducible holomorphic symplectic manifold may admit a Lagrangian fibration, meaning a fibration whose fibres are complex tori that are Lagrangian with respect to the holomorph…

An irreducible holomorphic symplectic (IHS) manifold is a simply connected compact Kähler manifold carrying a nondegenerate holomorphic $2$-form. A projective IHS manifold is cover…

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 $X$ be a holomorphic symplectic 4-fold with an ample divisor $omega$. Fix $n\in\mathbb{Z}$, $\beta\in H_2(X,\mathbb{Z})$, and insertions $\gamma_i\in H^{m_i}(X,\mathbb{Z})$. Fo…

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…