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

Speaker: Aleksei Ivanov

Title: Complex invariant Einstein metrics on flag manifolds with $T$-root system $B{C}_2$

Abstract:
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 $M_{n_1,n_2,n_3}=S{O}_{2(n_1+n_2+n_3)+1}/U_{n_1}\times U_{n_2}\times S{O}_{2n_3+1}$ with $T$-root system $B{C}_2$ will be considered. The faces of corresponding 5-dimensional Newton polytope $P(B{C}_2)$ 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 $P(B{C}_2)$.

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.