Multiplicity-free Hamiltonian actions need not be K\"ahler

Abstract

We show that Tolman's example (of a six dimensional Hamiltonian $T^2$-space with isolated fixed points and no compatible K\"{a}hler structure) can be constructed from the flag variety $U(3)/U(1)^3$ by $U(2)$-equivariant symplectic surgery. This implies that Tolman's space has a ``transversal multiplicity-free'' action of $U(2)$ and that Delzant's theorem ``every compact multiplicity-free torus action is K\"{a}hler'' \cite{D1} does not generalize to non-abelian actions.