189 problems

Gaiotto–Moore–Neitzke's conjecture. As the radial variable $t\to\infty$ on the Hitchin base, the Hitchin metric on $\mathcal{M}$ is asymptotic to the semi-flat metric constructed f…

Beauville's conjecture. The cycle class map is injective on the subalgebra of $A^{*}(X)$ generated by divisors.

Let $M$ be a hyperkähler manifold of dimension $2n$ and let $f:M\longrightarrow B$ be a Lagrangian fibration, where $B$ is its normal projective base of dimension $n$. In the known…

Oguiso's conjecture. $X$ contains a rational curve.

Hodge conjecture for the Kuga–Satake correspondence. There is a cycle $\Gamma\in {\rm CH}^{2n}(X\times KS(X))_{\mathbb{Q}}$ such that $[\Gamma]=\alpha$, and hence

Let a gravitational instanton be a complete hyperKähler manifold of ALG or ALG$^*$ type, and let a Hitchin moduli space be a moduli space of solutions to a Hitchin system equipped…

Let $X$ be a hyperkähler manifold of complex dimension $2n$, and let $f:X\to B$ be a Lagrangian fibration, where $B$ is a normal analytic variety. Matsushita's conjecture. The base…

Flow-closedness conjecture. If $\mu_c$ is a complex moment map associated to a linear action by a compact group $G$ on a vector space, then $|\mu_c|^2$ is flow-closed.

Let $X\to B$ and $X^\phi\to B$ be projective Lagrangian fibrations related by a Tate-Shafarevich twist, where a Tate-Shafarevich twist is obtained by modifying the transition funct…

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…

A projective hyperkähler manifold is a projective hyperkähler manifold, and a Lagrangian covering family is a flat projective family of Lagrangian subvarieties whose members cover…

Ooguri–Vafa local model conjecture. Near the discriminant locus of the Hitchin base, the hyperkähler metric on a Hitchin moduli space should be described locally by the Ooguri–Vafa…

Let $[(X,h)]\in {\mathscr U}^{il}_e(2n)$, where ${\mathscr U}^{il}_e(2n)$ is the open subset of the moduli space of polarized hyperkähler varieties of type $K3^{[n]}$ for which the…

ALE hyperkähler conjecture. Any ALE Ricci-flat $4$-manifold must be hyperkähler.

The cotangent bundle T^*\bm\hat{\mathcal{T}}^n is obtained by Hamiltonian reduction from the hyperkähler punctual Hilbert scheme, with its given symplectic structure near the zer…

Melted-volume conjecture.

Campana's simple-manifold conjecture. The manifold $X$ is either bimeromorphic to a quotient of a compact complex torus by a finite group of automorphisms, or has even complex dime…

Let $({\overline{\partial}}_E,\varphi)$ be a Higgs bundle in the regular locus $\mathcal M'$, and let $g_{L^2}$ and $g_{\mathrm{sf}}$ denote Hitchin's $L^2$-metric and the semiflat…

Hyperkähler Kirwan surjectivity conjecture. The hyperkähler Kirwan map is surjective.

Cherkis–Kapustin classification conjecture. Every ALF-$D_k$ instanton is exactly the metric constructed by Cherkis and Kapustin.

Let $X$ be a hyper-Kähler manifold and consider the Hilbert spaces of states arising from Rozansky–Witten theory on a manifold with boundary. Rozansky–Witten Hilbert-space conjectu…

Let … and let $\iota$ be the involution induced functorially on the Hilbert scheme $({\mathbb{C}}^2)^{[n]}$. The Hilbert scheme carries the natural hyperkähler structure constructe…

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

Asymptotic preservation conjecture. For general hyperkähler manifolds, generic asymptotically conical or asymptotically locally conical behaviour will be preserved under modificati…

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…