This article is cited in
2 papers
The number of invariant Einstein metrics on a homogeneous space, Newton polytopes and contractions of Lie algebras
M. M. Graev Scientific Research Institute for System Studies of RAS
Abstract:
To every homogeneous space
$M=G/H$ of a Lie group
$G$ with a compact isotropy group
$H$, where the isotropy representation consists of
$d$ irreducible components of multiplicity
$1$, we assign a compact convex polytope
$P=P_M$ in
$\mathbb R^{d-1}$, namely, the Newton polytope of the rational function
$s(t)$ defined to be the scalar curvature of the invariant metric
$t$ on
$M$. If
$G$ is a compact semisimple group, then the ratio of the volume of
$P$ to the volume of the standard
$(d-1)$-simplex is a positive integer
$\nu(M)>0$. We note that in many cases,
$\nu(M)$ coincides with the number
$\mathcal E(M)$ of isolated invariant holomorphic Einstein metrics (up to homothety) on
$M^{\mathbb C}=G^{\mathbb C}/H^{\mathbb C}$. We deduce from results of Kushnirenko and Bernshtein that in all cases,
$\delta_M=\nu(M)-\mathcal E(M)\geqslant0$. To every proper face
$\gamma$ of
$P$ we assign a non-compact homogeneous space
$M_\gamma=G_\gamma/H_P$ with Newton polytope
$\gamma$ that is a contraction of
$M$. The appearance of a “defect”
$\delta_M>0$ is explained by the fact that there is a Ricci-flat holomorphic invariant metric on the complexification of at least one of the
$M_\gamma$.
UDC:
515.16
MSC: 53C25,
53C30 Received: 12.09.2005
Revised: 20.09.2006
DOI:
10.4213/im569