The moduli space of a rational map is Carathéodory hyperbolic

Let f be a rational map of degree d≥ 2 . The moduli space Mf , introduced by McMullen and Sullivan, is a complex analytic space consisting of all quasiconformal conjugacy classes of f . For f that is not flexible Lattès, we show that there is a normal affine variety Xf of dimension $2d-2$ and a holomorphic injection i:Mf→ Xf such that i(Mf) is precompact in Xf . In particular, Mf is Carathéodory hyperbolic (that is, bounded holomorphic functions separate points in Mf ), provided that f is not flexible Lattès. This solves a conjecture of McMullen. When d≥ 4 , we give a concrete construction of Xf as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.