Symplectic Flexibility and the Hard Lefschetz Property

Abstract

In this thesis, we begin our study with the classical notion of symplectic flexibility - the existence on symplectic $(X^{2n},\omega)$ of a continous family $\omega_{t}$ of symplectic forms along which at least one symplectic harmonic number $h_{k,hr}(\omega_{t})$ 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 $\cup [\omega]^{k}:H^{n-k}_{dR}(X)\rightarrow H^{n+k}_{dR}(X)$, $k\leq n$. 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 K\"{a}hler 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 K\"{a}hler 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-$S^{1}$ manifolds that have diffeomorphic four-dimensional fixed componenets; we generalize the the result to compact ten-dimensional simple Hamiltonian-$S^{1}$ 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.