9 problems

Hypercomplex HKT or bi-HKT conjecture. Any hypercomplex manifold is either HKT or bi-HKT.

Let $(X^{4n},I,J,K)$ be a hypercomplex manifold. A continuous function $h:X\to\mathbb{R}$ is quaternionic plurisubharmonic if $\partial_J\partial h$ is a non-negative generalized s…

Let $\mathrm{SU}(5)$ carry a Joyce hypercomplex structure parametrised by a matrix $\left(\begin{smallmatrix} a_1 & a_3 \\ a_2 & a_4 \end{smallmatrix}\right)$. In the complementary…

Let $(\mathfrak{g}, \{J_{\alpha}\})$ be a nilpotent hypercomplex Lie algebra. Here, $\mathbb{H}$-solvability means that the descending $\mathbb{H}$-central series defined by…

A hypercomplex structure $(X,I,J,K)$ is exotic if $(X,I)$ is holomorphically symplectic, but for no choice of $alpha \in \operatorname{Kahl}(X)$ does the corresponding hyperkähler…

A hypercomplex structure on a real Lie algebra ${\mathfrak g}$ is a triple of complex structures $I$, $J$, and $K$ satisfying the quaternionic relations. A hypercomplex nilpotent L…

Alesker–Verbitsky conjecture. Every HKT $\mathrm{SL}(n,\mathbb{H})$-manifold admits a balanced HKT metric, which need not coincide with its initial HKT metric.

Let $(M,I,J,K)$ be a hypercomplex manifold, and let $g$ be a quaternionic Hermitian metric, meaning that $I$, $J$, and $K$ are orthogonal with respect to $g$. Define the correspond…

Let $T$ be a compact hyperkähler torus, and let $L$ be a generic induced complex structure. All complex subvarieties of $T$ are again tori. The conjecture concerns analogous behavi…