On the Number of Invariant Einstein Metrics on a Compact Homogeneous Space, Newton Polytopes and Contractions of Lie Algebras

We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope P M which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number [Formula: see text] of isolate invariant holomorphic Einstein metrics (up to homothety) on M ℂ = G ℂ /H ℂ . Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that [Formula: see text], where ν(M) is the integer volume of P M , and give conditions when the defect [Formula: see text]. In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space M γ = G γ /H P which is a contraction of M and is associated with a proper face γ of P M .