72 problems

Let $\mathcal{T}'$ be a $\Gamma$-labelled tree. In the Gaiotto–Moore–Neitzke asymptotic expansion, consider the total contribution to wall crossing obtained by choosing every possi…

Ginzburg's conjecture. The quotient variety $X$ can be naturally desingularized, and this desingularization is holomorphically symplectic and admits a hyperkähler structure. The co…

Let $X$ be an irreducible holomorphic symplectic manifold of dimension $2g$, and let $q_X$ denote its Beauville–Bogomolov form. A line bundle $L$ on $X$ is called non-trivial if it…

Hyperkähler Nahm transform conjecture. Then $M$ is diffeomorphic to $N$, and the hyperkähler Nahm transform is invertible.

Let $X$ be a smooth projective symplectic variety of dimension $2n$, and let … be a birational contraction whose degenerate locus $B$ is contracted to isolated points. Projective-s…

Let $G$ be a compact Lie group with Lie algebra $\mathfrak g$, let $G^c$ be its complexification, and let $\Omega$ be the data defining the examples associated to $G$. Denote the r…

Let $C$ be a hyperkähler cone, and let $B$ be the graded H-algebra defined in Definition 3.3. SGH-algebra conjecture. The graded H-algebra $B$ is an SGH-algebra. This is the algebr…

Let $M$ be a hyperkähler manifold that is AC of order $l\geq 1$, and let $k\geq -1$ be an integer. Let $x:M\rightarrow\mathbb{H}$ be a smooth function such that $D(x)$ is a $1$-for…

Let $\mu_0\in\mathfrak g$ be nonzero, let $\mu:[0,\delta]\to\mathfrak g$ be a regular deformation of $\mu_0$, meaning that $\mu(t)$ is regular for all $t>0$, and set…

Let $A$ be an abelian surface over $\mathbb{C}$, let $A^{[\ell]}$ be its Hilbert scheme of length-$\ell$ subschemes, and let $A^{\langle\ell-1\rangle}$ be the fiber over $0$ of the…

Twisted cotangent conjecture. Under these assumptions, there exists a natural isomorphism of complex manifolds

Hyperkähler cone conjecture. Any conical symplectic singularity is a hyperkähler cone.

Let $G$ be a semisimple group over $\a9\a9$ and let $G^{\mathrm{sc}}$ be its simply-connected cover. Let $\mathcal{M}^{\circ}_G$ denote the neutral component of the affine Toda mod…

Donaldson's conjecture. For every such $M$ and $\underline{\omega}$, there exists a family with these properties.

Let $(\mathrm{X},\mathrm{I},\mathrm{J},\mathrm{K},g)$ be a hyperkähler manifold, and let $\mathfrak{L}$ be a collection of compact spin $\mathrm{I}$-holomorphic Lagrangian submanif…

Let $(\mathrm{X},\mathrm{I},\mathrm{J},\mathrm{K})$ be a hyperkähler manifold with symplectic form $\omega_\mathrm{J}$, and set … Let $\mathcal{DF}(\mathrm{X},\omega_\mathrm{J})$ b…

Let $X$ be a hyperkähler variety of K3$^{[n]}$-type with $n\geq 2$. A K3 category is a category of K3 type, and let $\mathcal A_X^{[n]}$ denote its Hilbert-scheme-type construction…

Flapan–Macrì–O'Grady–Saccà conjecture. The Fano eightfold $F$ admits the semiorthogonal decomposition

O'Grady's conjecture. There is a positive integer $a\geq 1$ and a family of Lagrangian subvarieties $\pi:\mathcal{Z}_S\to X_n$ over a base $S$ such that $\mathcal{Z}_S$ covers…

Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathcal{CM}_c=\operatorname{Spec}Z(H^{\mathrm{rat}}_{c,0})$ be its Calogero–Moser space, with $c$ as in the source. Let…

A simply-connected Ricci-flat ALE instanton is a complete Riemannian manifold with vanishing Ricci curvature and asymptotically locally Euclidean geometry. Gibbons–Bando-Kasue-Naka…

L-equivalence conjecture for hyperkähler manifolds. If $X$ and $Y$ are $L$-equivalent, then their bounded derived categories of coherent sheaves are equivalent:

Donald's conjecture. The structure $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to the triple of Kähler forms arising from a hyperkähler met…

Cohomological conjecture. Among its factors, the vector bundle

Let $(X^4,\omega_1,\omega_2,\omega_3)$ be a hyperkähler $K3$ surface, and let $\alpha_1,\alpha_2,\alpha_3$ be irreducible $(-2)$-classes in $H^2(X^4;\mathbb{Z})$ satisfying…