Einstein homogeneous manifolds and geometric invariant theory

JORGE LAURET

RIMS, Kyoto University

Congreso; RIMS International Conference on "Geometry related to Integrable Systems"; 2007

RIMS

The only known examples until now of noncompact homogeneous Einstein manifolds are standard solvmanifolds: solvable Lie groups endowed with a left invariant metric such that if $\sg$ is the Lie algebra, $\ngo:=[\sg,\sg]$ and $\sg=\ag\oplus\ngo$ is the orthogonal decomposition then $[\ag,\ag]=0$.  This is a very natural algebraic condition which has played an important role in many aspects of homogeneousRiemannian geometry. The aim of this note is to give an idea of the proof, and mainly of the tools used in it, of the fact that any Einstein solvmanifold must be standard.  The proof of the theorem involves a somewhat extensive study of the natural $\G$-action on the vector space $V=\Lambda^2(\RR^n)^*\otimes\RR^n$, from a geometric invariant theory point of view.  We had to adapt a stratification for reductive groups actions on projective algebraic varieties introduced by F. Kirwan, to get a $\G$-invariant stratification of $V$ satisfying many nice properties which are relevant to our problem.