On invariant Einstein metrics on Kähler homogeneous spaces $S{U}_4/T^3$, $G_2/T^2$, $E_6/T^2(A_2{)}^2$, $E_7/T^2{A}_5$, $E_8/T^2{E}_6$, $F_4/T^2{A}_2$

Abstract

We study invariant Einstein metrics on the indicated homogeneous manifolds $M$, the corresponding algebraic Einstein equations $E$, the associated with $M$ and $E$ Newton polytopes $P(M)$, and the integer volumes $\nu =\nu (P(M))$ of it (the Newton numbers). We show that $\nu =80,152,...,152$ respectively. It is claimed that the numbers $\epsilon =\epsilon (M)$ of complex solutions of $E$ equals $\nu -18,\nu -18,\nu ,...,\nu$. The results are consistent with classification of non Kähler invariant Einstein metrics on $G_2/T^2$ obtained recently by Y.Sakane, A. Arvanitoyeorgos, and I. Chrysikos. We present also a short description of all invariant complex Einstein metrics on $S{U}_4/T^3$. We prove existence of Riemannian non Kähler invariant Einstein metrics on $G_2/T^2$-like Kähler homogeneous spaces $E_6/T^2\cdot (A_2{)}^2,E_7/T^2\cdot A_5,E_8/T^2\cdot E_6,F_4/T^2\cdot A_2$, where $T^2\cdot A_5\subset A_2\cdot A_5\subset E_7$ and some other results.

arXiv comment18 pages, 2 figures

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