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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.

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  • $\begingroup$ A manifold of real dimension 6 can't have a hyperkahler structure. $\endgroup$ – Sasha Jan 29 at 7:08
  • $\begingroup$ The space of configurations has complex dimension six. $\endgroup$ – Daniil Rudenko Jan 29 at 8:37
  • $\begingroup$ what are the equations defining "the space of configurations"? It is not clear to me to which subvariety of $\mathbb{P}(S^2 (\mathbb{C}^4)^*) \times ((\mathbb{P}^{3})^*)^4$ you refer to. $\endgroup$ – Libli Jan 29 at 8:54
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    $\begingroup$ @DaniilRudenko: Then it would be better if you correct the question (now it is written it has real dimension 6). $\endgroup$ – Sasha Jan 29 at 10:01
  • $\begingroup$ @Sasha : I am not sure I understand everything in this question, but it seems that the OP considers two different moduli spaces. One (which is not defined) is the moduli of configurations, which should have complex dimension 6. The other, the moduli space of "tetrahedra", is a topological subspace of the former and should have real dimension $6$. Apparently, it is not even clear that this last space has an algebraic structure. Of course, they are difficult to distinguish since none of them is defined by the OP in a precise way. $\endgroup$ – Libli Jan 29 at 10:17

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