Biran decomposition and relative symplectic cohomology

Abstract

Let $(M,ω)$ be a closed monotone integral symplectic manifold with monotonicity constant $τ>0$, and let $D\subset M$ be a symplectic hyperplane section of degree $0<k<τ$. By Biran&#39;s decomposition theorem, $M$ is the union of a symplectic disk bundle over $D$ and the skeleton of the complement $M\setminus D$ equipped with its Liouville structure. In this paper, we compute the relative symplectic cohomology of circle subbundles of this disk bundle over $\mathbb{Z}$ in terms of the quantum cohomology of $D$. This relative symplectic cohomology depends on the radius of the circle bundle relative to a certain critical radius. For a broad class of pairs $(M,D)$, we prove that the relative symplectic cohomology over $\mathbb{Z}$ does not vanish for supercritical radii. Therefore, circle bundles of supercritical radii are heavy subsets, which yields infinite dimensional quasi-flats in the Hamiltonian diffeomorphism group with respect to both the Hofer metric and the spectral metric over $\mathbb{Z}$. We also establish that if the degree $k$ of $D$ is greater than one, the skeleton of $M\setminus D$ is SH-full over $\mathbb{Z}/k\mathbb{Z}$, and hence non-displaceable in $M$ by any symplectomorphism.