# 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 -root system
**Abstract**:

Complex -invariant metrics on flag manifold are parameterised by algebraic torus and corresponding Einstein equations have form of Laurent polynomial equations. So we can consider Newton polytope of Einstein system which depends only on -root system of manifold . Hence, according to Bernstein-Kushnirenko theorem, the number of isolated complex invariant Einstein metrics (up to multiplication on complex number) on is no greater than the normalized volume of Newton polytope . Moreover, the equality 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 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 with -root system will be considered. The faces of corresponding 5-dimensional Newton polytope were described by M. M. Graev. Using this description it will be shown that the number of complex invariant Einstein metrics on is equal to exactly unless the parameters 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 by certain 2-dimensional faces of .

**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.*