Einstein equations for invariant metrics on flag spaces and their Newton polytopes

Abstract

This paper deals with the number of complex invariant Einstein metrics on flag spaces in the case when the isotropy representation has a simple spectrum. The author has previously showed that this number does not exceed the volume of the Newton polytope of the Einstein equation (in this case, this is a rational system of equations), which coincides with the Newton polytope of the scalar curvature function. The equality is attained precisely when that function has no singular points on the faces of the polytope, which is the case for "pyramidal faces". This paper studies non-pyramidal faces. They are classified with the aid of ternary symmetric relations (which determine the Newton polytope) in the T -root system (the restriction of the root system of the Lie algebra of the group to the center of the isotropy subalgebra). The classification is mainly done by computer-assisted calculations for classical and exceptional groups in the case when the number of irreducible components does not exceed 10 (and, in some cases, 15).2010 Mathematics Subject Classification. Primary 14M15, 14M17, 14M25; Secondary 53C25, 53C30. Key words and phrases. Homogeneous space, flag space, Einstein manifold, Newton polytope, pyramidal and non-pyramidal faces. c 2014 M. M. Graev 13 Licensed to New York Univ, Courant Inst. Prepared on Sun Aug 2 09:44:00 EDT 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 14 M. M. GRAEV § 6. On properties of non-pyramidal faces 42 § 7. Further properties of non-pyramidal faces 46 § 8. Properties of non-pyramidal faces of Type 1 49 Part 3. Tables of faces 54 § 9. Tables of facets 54 § 10. Tables of non-pyramidal faces 58 10.1. The polytope P = P (A 2 ); dim(P ) = 2 59 10.2. The polytope P = P (B 2 ); dim(P ) = 3 59 10.3. The polytope P = P (BC 21 ); dim(P ) = 4 59 10.4. The polytope P = P (BC 2 ); dim(P ) = 5 60 10.5. The polytope P = P (G 2 ); dim(P ) = 5 61 10.6. The polytope P = P (A 3 ); dim(P ) = 5 61 10.7. The polytope P = P (C 31 ); dim(P ) = 6 62 10.8. The polytope P = P (C 32 ); dim(P ) = 7 63 10.9. The polytope P = P (C 3 ); dim(P ) = 8 64 10.10. The polytope P = P (B 3 ); dim(P ) = 8 65 10.11. The polytope P = P (A 4 ); dim(P ) = 9 67 References 67Licensed to New York Univ, Courant Inst. Prepared on Sun Aug 2 09:44:00 EDT 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use EINSTEIN EQUATIONS FOR INVARIANT METRICS 15Modern computer programs allow us to find explicit solutions to that system when d is not too large, but they cannot tell if all the solutions have been found. Therefore it becomes important to estimate the number of solutions. In a prior paper, the author applied the results of A. G. Kushnirenko and D. N. Bernstein to estimate the number of solutions to EE. He showed that the number ε(M ) of complex isolated solutions to EE (considered up to homothety) does not exceed the normalized volume...

Complex invariant Einstein metrics on $SO_{2(n_1+n_2+n_3)+1}/U_{n_1} \times U_{n_2} \times SO_{2n_3+1}$ and Ricci-flat manifolds

“…Now we have to consider Newton polytope ∆ of this function or Newton polytope ∆(BC 2 ) of T -root system BC 2 that is the same and find out conditions when scalar curvature restricted on every face Γ ⊂ ∆ has critical points in algebraic torus (C * ) 6 . In fact, this polytope ∆ is well described by M. Graev in the paper [2]. More precisely, it is known that ν(∆) = 132, so we have the following estimation on the number E(M n 1 ,n 2 ,n 3 ) of isolated Einstein metrics…”

“…Hence, the invariant of de Siebenthal of the flag manifold M n 1 ,n 2 ,n 3 is the following (2,3,4), (3,5,6), (4,5,6), (1,5,5), (2,6,6)}.…”

“…We prove that this number is equal to exactly 132 for general parameters (n 1 , n 2 , n 3 ) and provide algebraic equations on these parameters when it is not true. The proof is based on the technique of Newton polytope that firstly was applied in this setting by M. Graev (see [1,2,3]). Also we give new examples of Ricci-flat Lorentzian manifolds that appear as Inonu-Wigner contractions of the manifold SO 2(n 1 +n 2 +n 3 )+1 /U n 1 × U n 2 × SO 2n 3 +1 by certain two-dimensional faces of the polytope.…”

Progress on homogeneous Einstein manifolds and some open probrems

“…Also, the recent works by M.M. Graev [Gr1], [Gr2] and [Gr3] are important contributions towards the understanding of the number of complex Einstein metrics on flag manifolds.…”