On invariant Einstein metrics on K\"ahler homogeneous spaces $SU_4/T^3$, $G_2/T^2$, $E_6/T^2(A_2)^2$, $E_7/T^2A_5$, $E_8/T^2E_6$, $F_4/T^2A_2$

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 K\"ahler 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 $ SU_4/T^3 $. We prove existence of Riemannian non K\"ahler invariant Einstein metrics on $G_2/T^2$-like K\"ahler 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.