Nowhere vanishing primitive of a symplectic form

辛平坦性与扭曲原子上同调

Symplectic Flatness and Twisted Primitive Cohomology

本文引入了辛流形上联络和纤维丛的辛平坦性概念，提出了基于$A_\infty$-代数的扭曲微分复形的平坦性条件，并证明辛平坦联络是Yang-Mills联络的特殊子类，同时研究了其上同调及消没定理。

Primitive cohomology of degree 2 on compact symplectic manifolds

In this paper, we define the generalized Lejmi's $P_J$ operator on a compact almost Kähler $2n$-manifold. We get that $J$ is $C^\infty$-pure and full if $\dim\ker P_J=b^2-1$. Additionally, we investigate the relationship between $J$-anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic $4$-manifold.

Cohomology and Hodge Theory on Symplectic Manifolds: I

We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagrangians and more generally coisotropic submanifolds with elements of a primitive cohomology, which dualizes to a homology on coisotropic chains.

辛杨-米尔斯理论

Symplectic Yang-Mills Theory

本文在辛流形上通过分解曲率二形式，将杨-米尔斯泛函分裂为原始与迹两部分，并研究其临界解性质，证明PYM泛函的椭圆性、平坦解分类及单调性公式，为刻画解模空间奠定基础。

The contact property for nowhere vanishing magnetic fields on the two-sphere

In this paper we give some positive and negative results about the contact property for the energy levels $Σ_c$ of a symplectic magnetic field on $S^2$. In the first part we focus on the case of the area form on a surface of revolution. We state a sufficient condition for an energy level to be of contact type and give an example where the contact property fails. If the magnetic curvature is positive, the dynamics and the action of invariant measures can be numerically computed. This hints at the conjecture that an energy level of a symplectic magnetic field with positive magnetic curvature sho…

原始上同调中的横越

Transgression in the primitive cohomology

本文研究了辛流形原始上同调的Chern-Weil理论，证明了横越公式并引入了原始特征类的概念。

Counting Curves in Elliptic Surfaces by Symplectic Methods

We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from \cite{ip3}. In particular, as in \cite{bl1}, these calculations also confirm the well-known Yau-Zaslow Conjecture \cite{yz} for primitive classes in $K3$ surfaces.

辛半特征

Symplectic semi-characteristics

本文针对闭4n维辛流形定义辛半特征，利用具有非退化零点的向量场证明计数公式，并得到消失性质及定义与辛形式选择无关的结论。

Cohomology theories on locally conformal symplectic manifolds

In this note we introduce primitive cohomology groups of locally conformal symplectic manifolds $(M^{2n}, ω, θ)$. We study the relation between the primitive cohomology groups and the Lichnerowicz-Novikov cohomology groups of $(M^{2n}, ω, θ)$, using and extending the technique of spectral sequences developed by Di Pietro and Vinogradov for symplectic manifolds. We discuss related results by many peoples, e.g. Bouche, Lychagin, Rumin, Tseng-Yau, in light of our spectral sequences. We calculate the primitive cohomology groups of a $(2n+2)$-dimensional locally conformal symplectic nilmanifold as …

Symplectic Harmonic theory and the Federer-Fleming deformation theorem

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fundamental property on symplectic Harmonic forms. We show that on a closed symplectic manifold, every real primitive cohomology class of positive degrees admits a symplectic Harmonic representative not supported on the entire mani- fold. As an application, we use it to investigate …

On Special Calibrated Almost Complex Structures and Moduli Space

An \emph{$ω$-admissible almost complex structure} on a $2n$-dimensional symplectic manifold $(M,ω)$ is a $ω$-calibrated almost complex structure $J$ admitting a nowhere vanishing $\bar{\partial}_J$-closed $(n,0)$-form $ψ$. After giving some examples we consider the moduli space of admissible almost complex structures and we study infinitesimal deformations. As special case, we write down explicit computations for the complex torus.

Symplectic Tori in Homotopy E(1)'s

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connected symplectic 4-manifolds known to exhibit such property.

在辛6流形中计数嵌入曲线

Counting embedded curves in symplectic 6-manifolds

本文通过直接证明方法，展示了在辛6流形中嵌入曲线计数的不变性，并证明了高 genus 时计数为零，从而验证了Gopakumar-Vafa有限性猜想。

Homologous Non-isotopic Symplectic Tori in Homotopy Rational Elliptic Surfaces

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive homology class in E(1)_K when K is any nontrivial fibred knot in S^3. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.

Haefliger structures and symplectic/contact structures

For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromov's h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension n. This step works on all manifolds, even closed. The second step, which works only on open manifolds and for all geometries, regularizes the intermediate Haefliger structure and produces a genuine geometric structure. Both steps admit relative parametric versions. The proofs borrow ideas from W. Thurston, like jiggling and inflation. A…

旋转木马与时间依赖的辛形式

Merry-go-round and time-dependent symplectic forms

本文通过引入时间依赖的辛形式及其原始形式，辛几何地处理了加速旋转木马中的欧拉力。

Categorical primitive forms of Calabi-Yau $A_\infty$-categories with semi-simple cohomology

We study categorical primitive forms for Calabi--Yau $A_\infty$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category ${\sf Fuk}(M)$ of a symplectic manifold $M$ has semi-simple Hochschild cohomology, then its genus zero Gromov--Witten invariants may be recovered from the $A_\infty$-category ${\sf Fuk}(M)$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enu…

缩放对称性的相对平衡与中心配置

Relative Equilibria for Scaling Symmetries and Central Configurations

本文探讨了缩放对称性在辛几何框架下的作用，引入了共形辛映射和共形动量映射，推导了缩放对称性的相对平衡方程，并应用于牛顿n体问题。

Lagrangian 嵌入到辛环面与 K3 曲面的刚性

Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces

本文证明配备 Kähler 型形式的环面或 K3 曲面中，Maslov 零 Lagrangian 环面的同调类必非零且本原，并在 K3 情形借助 Shah 的 Ratner 定理算术版本推广已有结果。

Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model

We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin-Li pairing of the mirror Landau-Ginzburg model under localized mirror functor. They are conformally equivalent with an interesting conformal factor $(vol^{Floer}/vol)^2$, which can be described as a ratio of Lagrangian Floer volume class and classical volume class. For this purpose, we introduce $B$-invariant of Lagrangian Floer cohomology with values in Jacobian ring of the mirror potential function. And we prove what we call a multi-crescent Cardy identity under certain conditions, which is a generalize…

关于正本原形式乘积的正性

On the Positivity of the Products of Positive Primitive Forms

该研究探讨正本原n-形式乘积的正性问题，证明维数≤6时乘积正性成立，高维时存在反例，回答了文献[1]的问题2.11，证明内容由ChatGPT 5.6生成并经作者验证。

Lagrangian Surfaces in a Fixed Homology Class: Existence of Knotted Lagrangian Tori

In this paper we show that there exist simply connected symplectic 4-manifolds which contain infinitely many knotted lagrangian tori, i.e. lagrangian embeddings of tori that are homotopic but not isotopic. Moreover, the homology class they represent can be assumed to be nontrivial and primitive. This answers a question of Eliashberg and Polterovich.

Fujiki C类中复辛流形的Beauville--Bogomolov--Fujiki正性与Moishezon性

Beauville--Bogomolov--Fujiki positivity and Moishezonness of complex symplectic manifolds in Fujiki class $\mathscr C$

本文研究Fujiki C类中超Fujiki流形的BBF正性与Moishezon性的关系，证明Moishezon超Fujiki流形的相关性质，给出超Fujiki 4-fold等的对应判据并应用于光滑族的Moishezon轨迹描述。