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?