simplex in hyperbolic space and quadrics in projective space

Question

I am trying to understand Goncharov's paper "Volumes of hyperbolic manifolds and mixed Tate motives", http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00293-3/S0894-0347-99-00293-3.pdf

In order to construct these motives, he looks at geodesic simplexes in the Klein model $\mathcal{H}^m$ of the hyperbolic space. Recall that this is given by $$ \{(x_1, \ldots, x_n) \in \mathbb{R}^n \ | \ \sum x_i^2 <1 \}. $$ In this model, geodesics are Euclidean lines, so a geodesic simplex is just a usual Euclidean simplex inside the unit ball.

On page 573, he says "After complexification and compactification we get $\mathbb{CP}^m$ together with a quadric $Q$ and a collection of hyperplanes $M=(M_1, \ldots, M_{m+1})$ provided by the faces of the simple"

Can someone explain where this quadric comes from?

Answer 1

Answer in the projective version of the Klein model, cf. the Wikipedia article: hyperbolic $n$-space $\mathbb{H}^n$ is realized as the domain $U^n\subseteq \mathbb{RP}^n$ given by $U^n=\{[x_0:\cdots:x_n]\in\mathbb{RP}^n\mid Q(x_0,\dots,x_n)>0\}$ with $Q(x_0,\dots,x_n)=x_0^2-x_1^2-\cdots-x_n^2$. Geodesics are induced from straight lines in the projective space, and so faces of simplices are given by real hyperplanes in $\mathbb{RP}^n$.

Now complexification gives you: the ambient $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the quadric $Q(x_0,\dots,x_n)=0$ which is the boundary of $U^n$ becomes the quadric $Q$ mentioned by Goncharov, and the real hyperplanes giving the faces of the simplex become the complex hyperplanes.

Answer in the ball version of the Klein model: hyperbolic $n$-space is realized as $\{(x_1,\dots,x_n)\in \mathbb{R}^n\mid \sum x_i^2<1\}$. This can be compactified via $\mathbb{R}^n\subseteq\mathbb{RP}^n$. Complexification gives you the same thing as above: $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the boundary sphere $S^n\subseteq\mathbb{R}^n$ becomes the complex quadric hypersurface defined by $\sum x_i^2-1=0$, and the faces of the simplices give you complex hyperplanes.

Both versions are related by compactification and coordinate change.

Maybe I should also point out that the quadric in the complex case is relevant data to remember: from the quadric and the hyperplanes you can recover the vertices of the simplex.