Consider a moduli space of geodesic tetrahedra in the hyperbolic space $\mathbb{H}^3.$ In the Klein's model the hyperbolic space can be presented as the interior of a unit ball $$ \mathbb{H}^3=\{(x_1,x_2,x_3)\in \mathbb{R}^3| x_1^2+x_2^2+x_3^2<1\} $$ and hyperbolic planes are obtained by intersecting $\mathbb{H}^3$ with Euclidean planes. To every hyperbolic tetrahedron one can associate a configuration $(Q; H_1, H_2, H_3, H_4)$ of a smooth quadric $Q$ in $\mathbb{P}^3_{\mathbb{C}}$ and four projective planes $H_i.$ The quadric is obtained by complexification and projectivization of the ideal boundary $$ \partial \mathbb{H}^3=\{(x_1,x_2,x_3)\in \mathbb{R}^3| x_1^2+x_2^2+x_3^2=1\}. $$ Planes $H_i$ are obtained in a similar way from the faces of the tetrahedron.

In this way a moduli space of hyperbolic tetrahedra can be viewed as a real $6-$dimensional subspace of the complex algebraic moduli space of configurations $(Q; H_1, H_2, H_3, H_4).$

**Question:** Does the moduli space of hyperbolic tetrahedra have an algebraic structure? Does the moduli space of configurations $(Q; H_1, H_2, H_3, H_4)$ have a hyperkähler structure?

An indication of this is the following observation: a moduli space of ideal hyperbolic tetrahedra can be identified with $\mathbb{P}^1_{\mathbb{C}}\setminus\{0, 1, \infty\}.$ For this one views the absolute $$ \partial \mathbb{H}^3=\{(x_1,x_2,x_3)\in \mathbb{R}^3| x_1^2+x_2^2+x_3^2=1\}. $$ as a Riemann sphere and associats to an ideal tetrahedron a cross-ratio of its vertices on $\partial \mathbb{H}^3$.

**Comment:** There are a few different ways to think about the moduli space of configurations $(Q; H_1, H_2, H_3, H_4).$ Probably, the most relevant one is characterized by the fact that for configurations, obtained from hyperbolic tetrahedra, functions $e^{2 l_i}$ and $e^{2\alpha_i}$, where $l_i$ are lengths of edges and $\alpha_i$ are dihedral angles, become algebraic.

For this one considers marked configurations, where by marking I mean the ordering of points, where each line $H_i \cap H_j$ intersects the quadric. Also one fixes the choice of a family of generators of $Q.$ Then the moduli space of configurations can be identified with a quasi-affine six dimensional subset of a torus $\mathbb{C}^6 \times \mathbb{C}^6$ given by Zariski closure of the set $(\ldots e^{2l_i},\ldots ,\ldots e^{2\alpha_i}\ldots ).$ Configurations, corresponding to hyperbolic tetrahedra, form a real 6-dimensional subvariety there.