Abstract
We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope PM 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 Mℂ = Gℂ/Hℂ. 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 PM, 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 Mγ = Gγ/HP which is a contraction of M and is associated with a proper face γ of PM.
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