AI 中文总结
本文利用Biran分解，通过D的量子上同调计算圆子丛的相对辛上同调，证明超临界半径圆丛为重子集并产生无穷维拟平坦，且当k>1时骨架SH-完全不可位移。
AI 中文摘要
设$(M,\omega)$为闭单调积分辛流形，其单调常数$\tau>0$，并设$D\subset M$为次数$0<k<\tau$的辛超平面截面。根据Biran分解定理，$M$是$D$上的辛圆盘丛与补集$M\setminus D$的骨架（配备Liouville结构）之并。本文中，我们利用$D$的量子上同调，计算了该圆盘丛的圆子丛在$\mathbb{Z}$上的相对辛上同调。该相对辛上同调依赖于圆丛半径相对于某一临界半径的大小。对于一大类配对$(M,D)$，我们证明了在超临界半径下，$\mathbb{Z}$上的相对辛上同调不消失。因此，超临界半径的圆丛是重子集，这导致Hamilton微分同胚群中关于Hofer度量与$\mathbb{Z}$上谱度量的无穷维拟平坦子集。我们还证明，若$D$的次数$k>1$，则$M\setminus D$的骨架在$\mathbb{Z}/k\mathbb{Z}$上是SH-完全的，从而在$M$中不能被任何辛同胚位移。
英文摘要
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'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.
Comments55 pages, 1 figure, comments welcome
