The rigidity of Dolbeault-type operators and symplectic circle actions

Rigidity and vanishing of basic Dolbeault cohomology of Sasakian manifolds

The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, the dimensions of the basic Dolbeault cohomology groups, only depend on the isomorphism class of the underlying CR structure. Equivalently, we show that the basic Hodge numbers are invariant under deformations of type I. This result reduces their computation to the quasi-regular ca…

Rigidity of elliptic genera: from number theory to geometry and back

In this paper we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to each compact almost complex manifold. In particular, on the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold of dimension $2n$ admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side we prove that from each compact almost complex manifold of index greater than…

关于半自由哈密顿圆周作用的等变上同调刚性

On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions

本文针对六维半自由哈密顿$S^1$-流形，修正了Gonzales的同构型判定结论，用必要的等变上同调全局假设替代非必要的不动点分布条件，并将结果推广至等变拓扑范畴，给出该领域的确定性结论。

四维哈密顿 $\mathbf{S^1}$-流形的等变上同调刚性

Equivariant cohomological rigidity for four-dimensional Hamiltonian $\mathbf{S^1}$-manifolds

本文解决了紧致连通辛四流形上哈密顿圆作用的等变上同调决定等变微分同胚型的问题，证明了等变上同调环同构与等变微分同胚的充要条件及更强的诱导关系。

On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

Let $(M,J)$ be a compact, connected, almost complex manifold of dimension $2n$ endowed with a $J$-preserving circle action with isolated fixed points. In this note we analyse the `geography problem' for such manifolds, deriving equations relating the Chern numbers to the index $k_0$ of $(M,J)$. We study the symmetries and zeros of the Hilbert polynomial associated to $(M,J)$, which imply many rigidity results for the Chern numbers when $k_0\neq 1$. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon, also known…

关于接近极小值的哈密顿量刚性的研究，及其在磁系统和近凯勒流形中的应用

On the Rigidity of Hamiltonians which are Zoll Near a Minimum, with an Application to Magnetic Systems and Almost-Kähler Manifolds

研究接近极小值的哈密顿系统，证明在能量水平序列趋近极小值时，哈密顿量的Hessian与辛结构的限制相容，且在磁系统中，度量与磁形式相容，定义了近凯勒结构，揭示了复空间形式的动态特征。