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.   Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible Kähler structures, and are therefore toric varieties. In this note we show that Delzant's result does not generalize to the non-abelian case. Our examples are constructed by applying U(2)-equivariant symplectic surgery to the flag variety U(3)/T  3. We then show that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible Kähler structure.

Multiplicity-free Hamiltonian actions need not be Kähler