Yi Lin

Visiting assistant professor, Department of Mathematics, University of Illinois at Urbana-Champaign

Equivariant Symplectic Hodge Theory and Strong Lefschetz Manifolds

Reyer Sjamaar

Symplectic Geometry

Abstract: Consider a Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We establish an equivariant version of the Merkulov-Guillemin d\delta-lemma, namely the d_G, \delta-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 examine the equivariant cohomology of a compact strong Lefschetz Hamiltonian manifold (M, \omega) with generalized coefficients and establish a version of the d_G, \delta-lemma for equivariant differential forms with generalized coefficients.

Consider a compact Hamiltonian circle manifold with the strong Lefschetz property. We constructed a family of 6 dimensional compact Hamiltonian S^1 manifold each of which satisfies the strong Lefschetz property itself but has a non-Lefschetz symplectic quotient. As an aside we showed an interesting compact Hamiltonian circle manifold constructed by Karshon has the strong Lefschetz property.

PhD

PhD