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Consider the Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We first establish an equivariant version of the Merkulov-Guillemin d -lemma, and an improved version of the Kirwan-Ginzburg equivariant formality theorem, which says that every cohomology class has a canonical equivariant extension. We then proceed to extend the equivariant d -lemma to equivariant differential forms with generalized coefficients. Finally we investigate the subtle differences between an equivariant Kaehler manifold and a Hamiltonian symplectic manifold with the strong Lefscehtz property. Among other things, we construct six-dimensional compact non-Kaehler Hamiltonian circle manifolds which each satisfy the Hard Lefschetz property, but nevertheless each have a symplectic quotient which does not satisfy the strong Lefschetz property. As an aside we prove that the strong Lefschetz property, unlike that of equivariant Kaehler condition, does not guarantee the Duistermaat-Heckman function to be log-concave.
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Yi Lin received his Ph.D degree in Mathematics in 2004 from Cornell University. He had a visiting assistant professor and a postdoctoral position with the University of Illinois at Urbana-Champaign and the University of Toronto respectively. Since August 2008, he has been an assistant professor at Georgia Southern University.
