Multiplicity-free Hamiltonian actions and existence

As for toric varieties, with any projective spherical variety is associated a convex polytope, and any facet of this polytope is defined by a prime divisor stable under a Borel subgroup [4]. In this paper we use the moment map to prove, for certain smooth projective spherical varieties, two characterizations of the facets that are defined by divisors stable under the full group action. As a corollary we get a necessary criterion for certain symplectic manifolds with multiplicity-free Hamiltonian group actions to admit invariant compatible Kiihler structures. In cases when the group acting is SO(5), we prove that the criterion is sufficient as well as necessary, and show that the existence of a compatible Kiihler structure invariant under the action of a maximal torus implies that there exists a compatible Kihler structure invariant under the action of SO(5). Thesis Supervisor: Victor Guillemin Title: Professor of Mathematics