Intersection theory of punctured pseudoholomorphic curves

Lagrange边界条件下全纯曲线模空间上的相交理论

Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions

在拉格朗日子流形中具有边界的任意亏格的伪全纯曲线的模空间上，基于相交理论定义了一族新的开Gromov-Witten型不变量.我们假设拉格朗日子流形作为反辛对合的不动点而产生，并且具有2或3维。在强半正亏格为0的情况下，新的不变量与实拟全纯曲线的Welschinger不变量计数一致。此外，我们还计算了亏格为0，次数为1的实五次三重数的新不变量为30。

几乎复 4 维流形中相交的正性与驯服性

Positivity of Intersections and Tameness of Almost Complex 4-manifolds

研究几乎复 4 维流形中伪全纯曲线与复 2 - 循环的相交正性，以此给出通用简单证明，说明此类流形存在驯服辛结构，还证明了相交正性具有几何稳定性。

三维流形中嵌入伪全纯曲线的Compactness

Compactness for Embedded Pseudoholomorphic Curves in 3-manifolds

我们证明了一个紧性定理的全纯曲线在四维辛，有嵌入投影的基础3-流形。它通过使用相交理论来加强SFT紧性定理的圆柱情况，以表明这种序列的退化永远不会产生多个覆盖或节点，因此很容易实现横截性。这适用于稳定的有限能量叶理理论，也提出了一种新的方法来定义接触3-流形，或更一般地说，3-流形稳定的哈密顿结构的SFT型不变量。

Relatively Open Gromov-Witten Invariants for Symplectic Manifolds of Lower Dimensions

Let $(X,ω)$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary $(Σ,\partialΣ)$ to the pair $(X,L)$ satisfying Lagrangian boundary conditions and intersecting $V$. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If $L\cap V=\emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there e…

Courbes symplectiques de haute auto-intersection dans les surfaces symplectiques

We first study symplectically embedded curves in symplectic surfaces with high self-intersection numbers compared to their genus. We prove in two different ways that such a curve completely determines both the diffeomorphism type of the surface in which it is embedded and the embedding itself. The first proof uses Seiberg-Witten theory whereas the second one only involves pseudoholomorphic techniques. We deduce from this result that the contact $3$-manifolds naturally associated with those curves admit a unique strong symplectic filling up to diffeomorphism. We next examine symplectic sections…

From Smooth to Almost Complex

This article mainly aims to overview the recent efforts on developing algebraic geometry for an arbitrary compact almost complex manifold. We review the results obtained by the guiding philosophy that a statement for smooth maps between smooth manifolds in terms of René Thom's transversality should also have its counterpart in pseudoholomorphic setting without requiring genericity of the almost complex structures. These include intersection of compact almost complex submanifolds, structure of pseudoholomorphic maps, zero locus of certain harmonic forms, and eigenvalues of Laplacian. In additio…

Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four

We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that fo…

Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds

For each contact diffeomorphism $ϕ: (Q,ξ) \to (Q,ξ)$ of $(Q,ξ)$, we equip its mapping torus $M_ϕ$ with a \emph{locally conformal symplectic} form of Banyaga's type, which we call the \emph{$\text{\rm lcs}$ mapping torus} of contact diffeomorphism $ϕ$. In the present paper, we consider the product $Q \times S^1= M_{id}$ (corresponding to $ϕ= id$) and develop basic analysis of the associated $J$-holomorphic curve equation, which has the form $$ \overline{\partial}^πw = 0, \quad w^*λ\circ j = f^*dθ$$ for the map $u = (w,f): \dot Σ\to Q \times S^1$ for the $λ$-compatible almost complex structure $…

Contact 3-manifolds, holomorphic curves and intersection theory

This is a revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology. The main lectures are aimed primarily at students and require only a minimal background in holomorphic curve theory, as the emphasis is on topological rather than analytical issues. Some of the gaps in the analysis are then filled in by the appendices, which include self-contained proofs of the similarity principle and positivity of intersections, and conclude with a "quick reference" fo…

Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces

Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by Lagrangian submanifolds. The main result proves that the $N^{th}$ coloured Jones polynomial and $N^{th}$ coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in ea…

Unknotted Reeb orbits and nicely embedded holomorphic curves

We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number -1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic cu…

曲面上的曲线与关联代数的模

Curves on surfaces and moduli of associative algebras

研究通过曲面上的曲线作为Fukaya范畴对象的A∞代数计算方法，展示如何利用Gauss字和可见多边形计算自交次数不超过三的浸入体的A∞积，并证明有限维代数可作为某些曲面的Lagrangian浸入的端射影代数。

辛对数柯尔加雷维度为-∞，仿射被覆性和无 cusps 理性曲线

Symplectic log Kodaira dimension $-\infty$, affine-ruledness and unicuspidal rational curves

本文研究了辛对数柯尔加雷维度为-∞的辛对 (X, ω, D) 的性质，引入了辛仿射被覆性的概念，并探讨了无 cusps 理性曲线在该结构中的作用。