An important question with a rich history is the extent to which the symplectic category is larger than the Kähler category. Many interesting examples of non-Kähler symplectic manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be Kähler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first symplectic manifold with large non-trivial symmetries which does not admit an invariant Kähler structure. The proof that it is not Kähler is based on the Atiyah-Guillemin-Sternberg convexity theorem [At] [GS]. Using the ideas of this paper, C. Woodward shows that even the symplectic analogue of spherical varieties need not be Kähler [W].
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