Antoine Gournay
The Runge approximation theorem for holomorphic maps (U -> C) is a fundamental result in complex analysis. The aim of this article is to prove such a result for (pseudo-)holomorphic maps from a compact Riemann surface to a compact (almost-)complex manifold M under certain assumptions. Though the setting is definitively that of pseudo-holomorphic maps it also covers some complex varieties.
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We show that one can achieve transversality for lifts of holomorphic disks to a projectivized vector bundle by locally enlarging the structure group and considering the action of gauge transformations on the almost complex structure, which eases computation considerably. As an application we prove a special case of the Arnold-Givental conjecture.
Let $\mathbb{R} \times (P \times \mathbb{R})$ be the symplectisation of the contactisation of an exact symplectic manifold $P$, and let $\mathbb{R} \times Λ$ be a cylinder over a Legendrian submanifold in the contactisation. We show that a pseudo-holomorphic polygon in $P$ having boundary on the projection of $Λ$ can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on $\mathbb{R} \times Λ$. It follows that Legendrian contact homology may be equivalently defined by counting either of these objects. Using this result, we give a proof of Seidel's isomorphism of the li…
We prove analogs of Thom's transversality theorem and Whitney's theorem on immersions for pseudo-holomorphic discs. We also prove that pseudo-holomorphic discs form a manifold.
Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II
本文提出利用阻碍丛数据在伪全纯圆盘模空间上系统构造边界与角点相容的Kuranishi结构（即树状K-system），简化了早期文献描述，为未来研究提供实用工具并证明了其唯一性。
In \cite{FOinteger}, Fukaya and Ono outlined a way of counting pseudo-holomorphic curves in a general compact symplectic manifold to obtain integer valued invariants. This paper contains the details of Fukaya and Ono's suggested construction for any compact symplectic manifold and a large class of exploded manifolds.
Pseudo-holomorphic triangles and the median quasi-state
本文证明全纯三角形最小面积受拉格朗日谱范数界定，并揭示其与中位数拟态的联系，进而给出泊松非交换性度量的尖锐上界及Calabi拟态亏缺下界。
Reeb flows, pseudo-holomorphic curves and transverse foliations
本文是一篇短篇综述，介绍了基于辛化空间伪全纯曲线理论的Reeb动力学最新研究进展，重点聚焦临界能量曲面附近的横截叶状结构，梳理了该理论对哈密顿动力学及天体力学经典问题的推动作用。
We prove two theorems on the removal of singularities on the boundary of a pseudo-holomorphic curve. In one theorem, we need no apriori assumption on the area of the curve. The proof uses a doubling argument with the goal of converting curves with boundary to curves without boundary. Our method is new and geometric and it does not need Sobolev spaces and PDEs.
Let (M, g, omega) be a compact, almost-Kaehler Einstein 4-manifold of negative star-scalar curvature. Then (M, omega) is a MINIMAL symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected sum N # (-CP_2).
In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a non-degenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element, and hence non-trivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on non-vanishing Gromov-Witten invariants of manifolds admitti…
In this paper, we first explore holomorphic Hamiltonian systems. In particular, we define action functionals for those systems and show that holomorphic trajectories obey an action principle, i.e., that they can be understood - in some sense - as critical points of these action functionals. As an application, we use holomorphic Hamiltonian systems to establish a relation between Lefschetz fibrations and almost toric fibrations. During the investigation of action functionals for holomorphic Hamiltonian systems, we observe that the complex structure $J$ corresponding to a holomorphic Hamiltonian…
Corrigendum of "Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks I, Surveys in Differential Geometry XXII (2018), 133-190"
本文对论文 [FOOO3] 中不正确的引理 9.1 进行勘误，指出用本说明中的引理 2 替换原引理后，原有证明及应用均不受影响。
We study analysis over infinite dimensional manifolds consisted by sequences of almost Kaehler manifolds. We develop moduli theory of pseudo holomorphic curves into such spaces with high symmetry. Many mechanisms of the standard moduli theory over finite dimensional spaces also work over these infinite dimensional spaces, which is based on a simple functional analytic framework.
We prove a variant of the Chance-McDuff conjecture for pseudo-rotations: under certain additional conditions, a closed symplectic manifold which admits a Hamiltonian pseudo-rotation must have deformed quantum product and, in particular, some non-zero Gromov-Witten invariants. The only assumptions on the manifold are that it is weakly monotone and that its minimal Chern number is greater than one. The conditions on the pseudo-rotation are expressed in terms of the linearized flow at one of the fixed points and hypothetically satisfied for most (but not all) pseudo-rotations.
A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the latter can be used to define pseudo-holomorphic submanifolds and subvarieties. The main theorem in this paper asserts that if the 4-manifold has a non zero Seiberg-Witten invariant, then the zero set of any given self-dual harmonic 2-form is the boundary of a pseudo-holomorphic sub…
We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.
We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve are improved. We introduce a natural holomorphic structure on the pulled back tangent bundle, in which the differential of a ps.-hol. map in an analytic morphism and describe cusps of ps.-hol. curves using Bennequin index.
n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of Maurer-Cartan equation of the filtered A infinity algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant. This invariant depends on the choice of almost complex structure. The way…
Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuous across the fold. The boundary values on the fold are mediated by tunneling maps which are punctured H-holomorphic maps into the folding hypersurface with prescribed asymptotics on closed characteristics. Our main result is that the linearized operator of this boundary value pro…
This article is the second part of the article we promised to write at the end of Section 1 of [FOOO15] (arXiv:1209.4410). (Part I appeared in [Part I] (arXiv:1503.07631).) We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version that appeared in [FOn] and also in Section A1, Section 7.5 [FOOO4], Section 12 [FOOO7], [Fu2]. This article is independent of our earlier writing [FOOO15]. We also do not assume that the readers have any knowledge on the pseudo-holomorphic curve. In this second part, we consider a system of spaces with Kuranishi structures…
Metrisability of projective surfaces and pseudo-holomorphic curves
本文证明了定向射影曲面的可度量化性等价于某个圆盘丛上伪全纯曲线的存在性，从而将射影结构的度量实现问题转化为伪全纯曲线问题。
This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, ω, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \…
Pseudo-holomorphic curves with a fixed complex structure in positive symplectic manifolds
本研究证明正辛流形中高次固定定义域Gromov-Witten不变量为伪全纯曲线的带符号计数，验证了Lian-Pandharipande猜想的辛版本成立，同时正面回应了Ruan与Tian的相关问题。
First, we provide another proof that the signed count of the real $J$-holomorphic spheres (or $J$-holomorphic discs) passing through a generic real configuration of $k$ points is independent of the choice of the real configuration and the choice of $J$, if the dimension of the Lagrangian submanifold $L$ (fixed points set of the involution) is two or three, and also if we assume $L$ is orientable and relatively spin, and $M$ is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the degree of evaluation maps …
This is the first part of the article we promised at the end of [FOOO13, Section 1]. We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version appeared in [FOn] and also in [FOOO4, Section A1, Section 7.5], [FOOO7, Section 12], [Fu2]. In Part 1, we focus on the construction of the virtual fundamental chain on a single space with Kuranishi structure. We mainly discuss the de Rham version and so work over $\mathbb R$-coefficients, but we also include a self-contained account of the way how to work over $\mathbb Q$-coefficients in case the dimension of…
We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from "counting" pseudo-holomorphic curves. We introduce the notion of an *implicit atlas* on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudo-holomorphic…
Chain level loop bracket and pseudo-holomorphic disks
本文基于弦拓扑链模型与 Kuranishi 结构理论，详细论证了如何利用伪全纯圆盘模空间的虚拟基本链在链级构造环路括号的 Maurer-Cartan 元素，从而落实 Fukaya 在辛拓扑中的重要观察。
In a previous paper [FT1], for any logarithmic symplectic pair (X,D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural set up for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove…
This is the second part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, I_F. (See math.DG/0111313 for part I). Having constructed I_F and outlined a proof of its invariance based on bifurcation analysis in part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in part I. These gluing theorems are different from (and much harder than) the more conventional versions in that they deal with broken trajectories or broken orbits connected at degenerate rest points. The issues of orientation and signs are also se…
Existence of pseudo-holomorphic disks via non-archimedean disk potentials
通过引入非阿基米德解析势函数，证明若分次单调拉格朗日子流形具有非零圆盘势，则其任何光滑同痕中的拉格朗日子流形在任意驯服近复结构下都界一个Maslov指标为2的J-全纯圆盘。
Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I
本文是系列论文首篇，基于已有指数衰减估计等成果，完成单个伪全纯圆盘模空间的Kuranishi结构构造，是相关研究的改进版本，第2版修正了两处表述的排版错误。
Pseudo-holomorphic dynamics in the three-body problem: fillability and convexity
本文推广了arXiv:2011.06568的方法，证明在接触5-流形具有辛填充时，其伪全纯曲线模空间为4维球，且严格凸性保证Reeb动力学动态凸，应用于圆型限制性三体问题。
We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher dimensional analogue is established. We also show the dimension of closed $J$-anti-invariant $2$-forms on an almost complex $4$-manifold is a birational invariant, in the sense that it is invariant under degree one pseudoholomorphic maps.
We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for $J$-holomorphic curves in almost complex $4$-manifolds to higher dimensions. As an application, we discuss pseudoholomorphic sections of a complex line bundle. We introduce a method to produce $J$-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset o…
In this article, we identify the 5-dimensional analogue of the finite energy foliations introduced by Hofer--Wysocki--Zehnder for the study of 3-dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-…
A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the integral over the variety of the given self-dual 2-form is finite. This article proves a regularity theorem for such finite energy subvarieties when the metric is particularly simple near the form's zero set. To be more precise, this article's main result asserts the following: A…
Pseudo-Holomorphic Hamiltonian Systems and Kähler Duality of Complex Coadjoint Orbits
本文研究伪全纯哈密顿系统并引入PHHS以在近复流形上描述经典力学，同时证明复余伴随轨道具有超凯勒与全纯凯勒结构的凯勒对偶性，并猜想其源于双余切丛。
This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, (I_F). (See math.DG/0505013 for part II). The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance--the Reidemeister torsion (τ_F) of the Floer-Novikov complex of (possibly non-hamiltonian) symplectomorphisms. (τ_F) turns out NOT to be invariant under hamiltonian isotopies, but this failure may be fixed by introducing certain ``correction term'': We define…
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