Ivanov A. Complex invariant Einstein metrics on flag manifolds with T-root system BC 2 (abstract)

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Speaker: Aleksei Ivanov

Title: Complex invariant Einstein metrics on flag manifolds with T-root system BC_2

Complex G-invariant metrics on flag manifold M=G/H are parameterised by algebraic torus (\mathbb{C}^{*})^n and corresponding Einstein equations have form of Laurent polynomial equations. So we can consider Newton polytope P of Einstein system which depends only on T-root system of manifold M. Hence, according to Bernstein-Kushnirenko theorem, the number E(M) of isolated complex invariant Einstein metrics (up to multiplication on complex number) on M is no greater than the normalized volume Vol(P) of Newton polytope P. Moreover, the equality E(M)=Vol(P) holds only when Einstein system restricted on every face has no solutions in algebraic torus (also it implies that all solutions are isolated). On the other hand, if there exists a face of polytope P such that Einstein system restricted on it has solution then this solution can be interpreted as a complex Ricci-flat invariant metric on some non-compact homogeneous manifold (called Inonu-Wigner contraction) associated with this face. In the talk the series of flag manifolds M_{n_1,n_2,n_3}=SO_{2(n_1+n_2+n_3)+1}/U_{n_1} \times U_{n_2} \times SO_{2n_3+1} with T-root system BC_2 will be considered. The faces of corresponding 5-dimensional Newton polytope P(BC_2) were described by M. M. Graev. Using this description it will be shown that the number of complex invariant Einstein metrics on M_{n_1,n_2,n_3} is equal to exactly Vol(P)=132 unless the parameters n_1, n_2, n_3 satisfy one of some algebraic equations which will be provided explicitly. Moreover, the family of (real) Ricci-flat Lorentzian manifolds will be constructed as Inonu-Wigner contractions of M_{n_1,n_2,n_3} by certain 2-dimensional faces of P(BC_2).

Slides: IvanovTrieste2018slides.pdf

Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.