Abstracts

In this thesis, we begin our study with the classical notion of symplectic flexibility - the existence on symplectic (X2n,) of a continous family t of symplectic forms along which at least one symplectic harmonic number hk,hr(t) varies. We use this theory, which was put forth by Iba~{n}ez, Rudyak, Tralle, and Ugarte, to analyze the flexibility of the new symplectic Bott-Chern and Aeppli cohomologies invented by Tsai, Tseng, and Yau. We next explore Guillemin and Sternberg's notion of symplectic birational cobordism (especially the symplectic blow-up), where we improve the existing theory on its capacity for shrinking the kernels of the Lefschetz maps cup []k:Hn-kdR(X) Hn+kdR(X), k n. Examples are given of Hard Lefschetz and non-Hard Lefschetz symplectic manifolds that are symplectic birational cobordant. We consider the merit of using this equivalence relation as a classification tool. The problem of classifying which symplectic manifolds are symplectic birational cobordant to a Kahler manifold remains open, but we provide discussion of the problem as well as a first conjecture. Lastly, we review and discuss a new example in the literature of a deformation equivalence between a Kahler and a non-Hard Lefschetz form; the first of its kind. Yunhyung Cho achieved this on the special class of compact six-dimensional simple Hamiltonian-S1 manifolds that have diffeomorphic four-dimensional fixed componenets; we generalize the the result to compact ten-dimensional simple Hamiltonian-S1 manifolds that have diffeomorphic eight-dimensional fixed components. We also discuss the outlook for extending the result to higher dimensions 4k+2 in this class.