On a generalization of a theorem of McDuff

Frankel's theorem in the symplectic category

We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel theorem. The case of n=2 is the well known theorem of McDuff. From the well known example of McDuff (a six dimensional symplectic non-Hamiltonian circle action with fixed tori) and its products with copies of a two dimensional sphere with the usual rotation, the condition on the dimension of the acting torus is optimal to obtain the result.

The generating function of the embedding capacity for 4-dimensional symplectic ellipsoids

Quite recently, McDuff showed that the existence of a symplectic embedding of one four-dimensional ellipsoid into another can be established by comparing their corresponding sequences of ECH capacities. In this note we show that these sequences can be encoded in a generating function, which gives several new equivalent formulations of McDuff's theorem.

Ehrhart polynomials and symplectic embeddings of ellipsoids

McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that part of the answer is given by an infinite "Fibonacci staircase." Similarly, Frenkel and Müller determined when a four-dimensional ellipsoid can be symplectically embedded into the ellipsoid E(1,2) and found that part of the answer is given by a "Pell staircase." ECH capacities give an obstruction to symplectically embedding one four-dimensional ellipsoid into another, and McDuff showed that this obstruction is sharp. We use this result to give new proofs of the staircases…

具有不可缩轨道的辛环面作用

Symplectic torus actions with non-contractible orbits

研究闭连通2n维辛流形上辛\(T^{n - 1}\)作用，证明其为哈密顿作用的充要条件是轨道可缩，推广相关结果。当轨道迷向时得到更强结论，还证明了此类作用可分解，最后提出关于其拓扑的若干开放性问题。

Holomorphic curves in log-symplectic manifolds

Log-symplectic structures are Poisson structures that are determined by a symplectic form with logarithmic singularities. We construct moduli spaces of curves with values in a log-symplectic manifold. Among the applications, we classify symplectically ruled log-symplectic $4$ manifolds (both orientable and non-orientable), and obstruct the existence of contact boundary components, in analogy with well-known theorems by McDuff. Moreover, we study certain log-symplectically ruled surfaces, using tools from symplectic field theory.

On the symplectic fillings of standard real projective spaces

We prove, in a geometric way, that the standard contact structure on the real projective space of dimension $2n-1$ is not Liouville fillable for $n \ge 3$ and odd. We also prove that, for all $n$, semipositive fillings of those contact structures are simply connected. Finally we give yet another proof of the Eliashberg-Floer-McDuff theorem on the diffeomorphism type of the symplectically aspherical fillings of the standard contact structure on the $(2n-1)$-dimensional sphere.