Symplectic Homogenization

The symplectic mapping class group of $\CC P^2 n{\bar{\CC P^2}}$ with $n\leq4$

In this paper we prove that the Torelli part of the symplectomorphism groups of the $n$-point ($n\leq 4$) blow-ups of the projective plane is trivial. Consequently, we determine the symplectic mapping class group. It is generated by reflections on $K_ω- $spherical class with zero $ω$ area.

On Symplectic Coverings of the Projective Plane

We prove that a resolution of singularities of any finite covering of the projective plane branched along a Hurwitz curve $\bar H$ and, maybe, along a line "at infinity" can be embedded as a symplectic submanifold into some projective algebraic manifold equipped with an integer Kähler symplectic form (assuming that if $\bar H$ has negative nodes, then the covering is non-singular over them). For cyclic coverings we can realize this embeddings into a rational algebraic 3--fold. Properties of the Alexander polynomial of $\bar{H}$ are investigated and applied to the calculation of the first Betti…

通过辛上同调构造库仑分支代数

Coulomb branch algebras via symplectic cohomology

本文通过辛上同调构造了库仑分支代数，并利用该方法将Braverman-Finkelberg-Nakajima的库仑分支用equivariant辛上同调进行刻画。

Barcode of a pair of compact exact Lagrangians in a punctured exact two-dimensional symplectic manifold

In this article, we modify the classical Floer complex $CF(L_0,L_1)$ of a pair of two compact exact Lagrangian submanifolds $L_0,L_1$ of an exact symplectic 2-manifold $M$ into a $\mathbb{Z}_2[T]$-complex $CF_h(L_0,L_1)$, whose differential keeps track of how many times a pseudo-holomorphic strip passes through a distinguished point $h\in M$. We show that this complex is invariant under Hamiltonian isotopy, and we prove that its barcode, if it exists, is the same as both barcodes $\mathcal{B}(CF(L_0,L_1;M))$ and $\mathcal{B}(CF(L_0,L_1;M\setminus \{h \}))$. This allows us to extend a conjectur…

The action spectrum and C^0 symplectic topology

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorp…

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.

Classification of Group Actions Extended to Symplectic Deformation Quantizations

Consider a group $Γ$ acting on a formal (Fedosov) deformation quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,ω)$. This canonically induces an action of $Γ$ by symplectomorphisms on $M$. We examine the reverse problem of extending group actions by symplectomorphisms to the deformation quantization. To do this we first define a notion of extension that does not impose restrictions on the Fedosov connection realizing $\mathbb{A}_\hbar(M)$ in its gauge equivalence class by considering composition of pull-back with certain inner automorphisms of sections of the Weyl bundle. We refo…

一个辛Hilbert-Smith猜想

A symplectic Hilbert-Smith conjecture

本文证明了辛拓扑中自然同胚作用下的Hilbert-Smith猜想新情形，指出p-进整数群在辛非球面流形上无非平凡Hamilton同胚作用，并利用Floer上同调的条形码与幂运算推导出相关局部紧群必为Lie群。

Generalized Conley-Zehnder index

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,\barΩ)$, having chosen a given reference Lagrangian $V$. Paths of symplectic endomorphisms …

临界可忽略集的瞬时 Hamilton 可移位性与任意辛可挤压性

Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets

本文利用接近恒等映射的 Hamilton 微分同胚和折叠技术，证明临界维数下可矩形可忽略集的可移位性与任意辛可挤压性，并指出结果在参数上尖锐。

Liouville极化与4维Lagrangian骨架的刚性

Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$

本文提出了一种适用于开辛流形的新极化类型，并探讨了其在辛嵌入、Lagrangian相交和Legendrian障碍中的应用。

From double Lie groupoids to local Lie 2-groupoids

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid.