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

TL;DR

This paper investigates invariant Einstein metrics on specific Kähler homogeneous spaces, analyzing algebraic Einstein equations, Newton polytopes, and solution counts, revealing new existence results and classifications for these complex geometric structures.

Contribution

It introduces the computation of Newton numbers for Einstein equations on these spaces and establishes the existence of non Kähler invariant Einstein metrics, expanding understanding of their geometric properties.

Findings

Newton numbers computed as 80, 152, ...

Number of complex solutions related to Newton numbers minus 18

Existence of non Kähler Einstein metrics on certain spaces confirmed

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.