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 $BC_{2}$

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 $M_{n_{1},n_{2},n_{3}}=SO_{2(n_{1}+n_{2}+n_{3})+1}/U_{n_{1}}\times U_{n_{2}}\times SO_{2n_{3}+1}$ 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 $P(BC_{2})$ .