Luigi Vezzoni
In this paper we give a generalization of the normal holomorphic frames in the symplectic manifolds and find conditions for the integrability of complex structures.
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We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with non-negative self-intersection, following the strategy of McDuff's classification of rational and ruled symplectic four manifolds.
For a smooth map $f:X^4\toΣ^2$ that is locally modeled by holomorphic maps, the domain is shown to admit a symplectic structure that is symplectic on some regular fiber, if and only if $f^*[Σ]\ne0$. If so, the space of symplectic forms on $X$ that are symplectic on all fibers is nonempty and contractible. The cohomology classes of these forms vary with the maximum possible freedom on the reducible fibers, subject to the obvious constraints. The above results are derived via an analogous theorem for locally holomorphic maps $f:X^{2n}\to Y^{2n-2}$ with $Y$ symplectic.
In this note, we reformulate Donaldson's construction as a compactness result. Approximately holomorphic sections accumulate to "limit holomorphic sections" and uniform transversality properties of the approximately holomorphic sections correspond to transversality properties of their limits.
It is a well known fact that every embedded symplectic surface $Σ$ in a symplectic 4-manifold $(X^4,ω)$ can be made $J$-holomorphic for some almost-complex structure $J$ compatible with $ω$. In this paper we investigate when such a $J$ can be chosen from a generic set of almost-complex structures. As an application we give examples of smooth and non-empty Seiberg-Witten and Gromov-Witten moduli spaces whose associated invariants are zero.
We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant mod…
Let M be a closed symplectic manifold with a compatible almost complex structure J. We prove that for a point p in M and E>0, if v is a non-constant J-holomorphic curve with symplectic area smaller than E, then the number of the pre-images of p is bounded, and the bound is independent of v. We also provide a uniform Hofer's energy bound for J-holomorphic curves in M\p based on the symplectic area. Using these two results we compactify the moduli space of J-holomorphic curves in M by adding holomorphic buildings at the point p.
This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,\om) when J\in \Jj(\Ss), the set of \om-tame J for which a fixed chain \Ss of transversally intersecting embedded spheres of self-intersection \le -2 is J-holomorphic. Extending work by Biran (in Invent. Math. (1999)), it shows that when (M,\om) is the blow up of a rational or ruled symplectic 4-manifold, any homology class A\in H_2(M;\Z), with nonzero Gromov invariant and nonnegative intersection both with the spheres in \Ss and with the exceptional classes other than A, has an embedded J-holomorphic repres…
These notes are based on a course that took place at the Université de Nantes in June 2011 during the "Trimester on Contact and Symplectic Topology". We will explain how holomorphic curves can be used to study symplectic fillings of a given contact manifold. Our main goal consists in showing that certain contact manifolds do not admit any symplectic filling at all.
Holomorphic approximation for Hamiltonian diffeomorphism groups via the Symplectic Density Property
该研究证明全纯灵活Stein流形的余切丛具有辛密度性质，进而推导得出实余切丛的哈密顿微分同胚可由复化余切丛的复辛自同构实现Carleman意义下的全纯逼近。
In this paper we will prove that for a compact, symplectic manifold $(M, ω)$ and for $ω$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to redefine the Floer symplectic homology. In particular, in subsequent papers, we will prove the Arnold conjecture in both degenerate and non-degenerate cases with integer coefficients for general, symplectic manifolds.
We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable - this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori - this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fil…
The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds respectively. Basic properties of such structures are established.
It is shown that any symplectic $2n\times 2n$-matrix, whose entries are complex holomorphic functions on a reduced Stein space, can be decomposed into a finite product of elementary symplectic matrices if and only if it is null-homotopic. Moreover, if this is the case, the number of factors can be bounded by a constant depending only on $n$ and the dimension of the space.
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).
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.
This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This …
Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.
The symplectic holomorphic density property for Calogero-Moser spaces
本文引入辛全纯密度性质等概念及对应理论，证明n粒子Calogero-Moser空间$\mathcal{C}_n$具有该性质并刻画其全纯辛自同构群，为相关空间的自同构研究提供基础。
Let (M,ω) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,ω) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in …
Coarsely Holomorphic Curves and Symplectic Topology
研究具有面积界的几乎复流形与驯服辛结构的关系，证明反之有面积界的流形允许驯服辛结构，还刻画了驯服辛结构锥，证明复二维周期可由粗全纯曲线逼近并给出低能量界，证实格罗莫夫推测。
In this note, genus zero Gromov-Witten invariants are reviewed and then applied in some examples of dimension four and six. It is also proved that the use of genus zero Gromov-Witten invariants in the class of embedded $J$-holomorphic curves to distinguish the deformation types of symplectic structures on a smooth $6$-manifold is restricted in the sense that they can not distinguish the symplectic structures on $X_1\times S^2$ and $X_2\times S^2$ for two minimal, simply connected, symplectic $4$-manifolds $X_1$ and $X_2$ with $b_2^+(X_1)>1$ and $b_2^+(X_2)>1$.
We set up a topological framework for degenerations of symplectic manifolds into singular spaces paying a special attention to the behavior of Lagrangian manifolds and their (holomorphic) membranes. We show that degenerations into singular toric varieties provide a source of exotic Lagrangian tori.
We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.
Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a well-defined Gromov-Witten invariant for $J$-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds. The Gromov-Witten axioms, as originally described by Kontsevich and Manin, give algebraic relationships between the Gromov-Witten invariants. In this paper, we prove the Gromov-Witten axioms for the polyfold Gromov-Witten invariants.
We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties.
Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, π)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic s…
The spaces of Kähler and holomorphically tamed symplectic forms on closed 4-manifolds
研究闭 4 维流形上凯勒形式空间的相关性质，受猜想启发还研究全纯驯服辛形式，提出平行问题并关联其与凯勒型辛形式问题，探讨唯一性、连通性和开放性等。
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…
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 define a Gaussian measure on the space $H^0_J(M, L^N)$ of almost holomorphic sections of powers of an ample line bundle $L$ over a symplectic manifold $(M, ω)$, and calculate the joint probability densities of sections taking prescribed values and covariant derivatives at a finite number of points. We prove that they have a universal scaling limit as $N \to \infty$. This result completes our proof (with P. Bleher) that correlations between zeros of sections in the almost-holomorphic setting have the same universal scaling limit as in the complex case (see Universality and scaling of zeros o…
In this paper, we provide a bound for the generalized Hofer energy of punctured $J$-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends. As an application, we prove a version of Gromov's Monotonicity Theorem with multiplicity. Namely, for a closed symplectic manifold $(M,ω)$ with a compatible almost complex structure $J$ and a ball $B$ in $M,$ there exists a constant $\hbar>0,$ such that any $J$-holomorphic curve $\tilde{u}$ passing through the center of $B$ for $k$ times (counted with multiplicity) with boundary mapped to $\partial B$ has symplectic area $\int_…
We study the asymptotics of almost holomorphic sections $s \in H^0_J(M, ω)$ of an ample line bundle $L \to M$ over an almost complex symplectic manifold in the sense of Boutet de Monvel-Guillemin. Such sections are defined as the kernel of a complex which is analogous to the $\bar{\partial}$ complex for a positive line bundle over a complex manifold. Our main result is the scaling limit asymptotics of the Szego projectors $Π_N$ of powers $L^N$. The Kodaira embedding theorem and Tian almost isometry theorem are almost immediate consequences of the scaling limit. We also relate such almost holom…
For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that $W(X)\leq 1$, and the equality is reached if and only if the subvariety $X\subset M$ is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_1 \arrow X_2 \arrow ... X_n\arrow M$ of immersions of simple holomorphic symplectic manifolds, we show that $W(X_1) \leq W(X_2) \leq >... \leq W(X_n)$.
Given a symplectic three-fold $(M,ω)$ we show that for a generic almost complex structure $J$ which is compatible with $ω$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class $β$ in $\H_2(M,\Z)$ with $c_1(M).β=0$, provided that the divisibility of $β$ is at most 4 (i.e. if $β=nα$ with $α\in H_2(M,\Z)$ and $n\in \Z$ then $n\leq 4$). Moreover, each such curve is embedded and 4-rigid.
There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (stan…
We give an inequality of the displacement energy for exact Lagrangian immersions and the symplectic area of punctured holomorphic discs. Our approach is based on Floer homology for Lagrangian immersions and Chekanov's homotopy technique of continuations.
Slices for reductive group actions in algebraic and holomorphic symplectic geometry
本文在复形几何和辛几何中推广了辛切片定理，研究复可约代数群作用的局部结构，并利用Slodowy切片和分解类进行深入分析。
We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors. This can be regarded as fixing a "complex analogue of the holonomy" of a connection along a "complex analogue of the boundary" in analogy with the real case.
Symplectic Field Theory studies J-holomorphic curves in almost complex manifolds with cylindrical ends. One natural generalization is to replace 'cylindrical' by 'asymptotically cylindrical'. In this article, we generalize the asymptotic results about the behavior of J-holomorphic curves near infinity to the asymptotically cylindrical setting. We also sketch how these asymptotic results allow the main compactness theorems in Symplectic Field Theory proved by Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder to be extended to the asymptotically cylindrical case.
Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids
本文将 Feix-Kaledin 定理推广到复拉格朗日子流形邻域，把超凯勒结构的存在性归结为全纯辛结构的形变问题，并证明凯勒型全纯泊松曲面上的全纯辛群胚在单位截面附近具有超凯勒结构。
This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G with finite center, such that G/K is a Hermitian symmetric space, where K is a maximal compact subgroup of G. Holomorphic coadjoint orbits are a generalization of the Hermitian symmetric space G/K. In this thesis, we prove that the McDuff's symplectomorphism on Hermitian symmetric…
Lefschetz fibration is the symplectic analogue of stable holomorphic fibration in complex geometry. A 4-dimensional stable holomorphic fibration satisfies the famous Parshin-Arakelov inequality. In this note we present an analogous inequality for a 4-dimensional Lefschetz fibration.
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 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。
We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologically trivial vanishing spheres in these pencils. Moreover we show some basic topological properties of the space of asymptotically holomorphic transverse sections. These properties allow us to define a new set of symplectic invariants of the original symplectic structure.
We outline a program for incorporating holomorphic curves with Lagrangian boundary conditions into symplectic field theory, with an emphasis on ideas, geometric intuition, and a description of the resulting algebraic structures.
We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an adjunction inequality for singularities is derived. Moreover, we show that for a coordinate class of a monotone Lagrangian split torus generically the number of non-immersed holomorphic discs is even.
A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence simple unified proofs are obtained for formulas of Poincare polynomials of toric hyperkahler varieties, Poincare polynomials of Hilbert schemes of points and twisted ADHM spaces of instantons on C^2 and Poincare polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable represent…
This short note considers varieties of the form $G\times S_{\text{reg}}$, where $G$ is a complex semisimple group and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. Such varieties arise naturally in hyperkähler geometry, theoretical physics, and in the theory of abstract integrable systems developed by Fernandes, Laurent-Gengoux, and Vanhaecke. In particular, previous work of the author and Rayan uses a Hamiltonian $G$-action to endow $G\times S_{\text{reg}}$ with a canonical abstract integrable system. One might therefore wish to understand, in some sense, all examples…
Constructions of symplectic forms on 4-manifolds
本文修改Lambert-Cole等的构造，证明带有理辛形式的辛4维流形可存在在其2-骨架邻域为凯勒形式的上同调等价辛形式，还定义了全纯2-骨架上全纯线丛的皮卡群。
We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational $J$-holomorphic curves in a given homology class passing through a given real configuration of points.
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…
This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from readers, I've decided to make a presentable half-finished version available here. Further chapters will be added in future updates.
Untriangular factorization of holomorpic symplectic matrices
本文证明全纯辛矩阵可按反对角辛形式作单位三角分解，并将相关矩阵进一步分解为至多7个标准辛形式下的因子，从而解决公开问题并给出因子数估计与更简洁证明。
We study the moduli space of twisted quasimaps from a fixed smooth projective curve to a Nakajima's quiver variety and the moduli space of $δ$-stable framed twisted quiver bundles with moment map relations. We show that they carry symmetric obstruction theories and when $δ$ is large enough, they exactly coincide. These results generalize works of D.E. Diaconescu about the ADHM quiver, in the framework of the quasimap theory of I. Ciocan-Fontanine, D. Maulik and the author.
We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces.
On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification
本文证明当接触3流形具有平面页的统一脊柱开书分解时，其(弱、强和/或精确)辛和Stein填充可由Lefschetz纤维化的微分同胚类分类。进一步结果包括新的ECH接触不变量和SFT中代数扭力消失准则，非定向圈 bundles的填充分类，以及关于四维实数中Stein结构变形类的
Symplectic double groupoids and the generalized Kähler potential
本文解决了广义Kähler几何的基本自由度描述问题，证明其全纯结构为双辛广群间的全纯辛Morita双双模，度量由拉格朗日子流形决定，正面验证了广义Kähler势存在性猜想，并以紧偶维半单李群为例构造验证。
We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several ex…
We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some …
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