Special lagrangian fibrations on flag variety $F^3$

Classification of Lagrangian fibrations

This paper classifies Lagrangian fibrations over surfaces with compact total spaces up to fiberwise symplectomorphism identical on the base.

Lagrangian torus fibrations and homological mirror symmetry for the conifold

We discuss homological mirror symmetry for the conifold from the point of view of the Strominger-Yau-Zaslow conjecture.

辛通量几何与拉格朗日环面纤维化

Geometry of symplectic flux and Lagrangian torus fibrations

本文通过Fukaya代数定义数值不变量研究辛通量，揭示其在线性通量同伦上的凹性，并推导Fano簇Gelfand-Cetlin纤维的约束及复射影平面纤维空间描述。

Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces I: Fermat quintic case

In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.

关于极大卡拉比 - 丘退化及其他情形的拉格朗日环面纤维化

Fibrations by Lagrangian tori for maximal Calabi-Yau degenerations and beyond

介绍在凯勒流形退化中构造拉格朗日环面纤维化的通用技术，该技术利用阿坎波空间边界及辛联络，证明极大卡拉比 - 丘退化附近纤维一般区域有符合期望渐近性质的拉格朗日环面纤维化。

Special Lagrangian fibrations, wall-crossing, and mirror symmetry

In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.

Lagrangian torus fibration and mirror symmetry of Calabi-Yau hypersurface in toric variety

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).

Twin Lagrangian fibrations in mirror symmetry

A twin Lagrangian fibration, originally introduced by Yau and the first author, is roughly a geometric structure consisting of two Lagrangian fibrations whose fibers intersect with each other cleanly. In this paper, we show the existence of twin Lagrangian fibrations on certain symplectic manifolds whose mirrors are fibered by rigid analytic cycles. Using family Floer theory in the sense of Fukaya and Abouzaid, these twin Lagrangian fibrations are shown to be induced from fibrations by rigid analytic subvarieties on the mirror. As additional evidences, we discuss two simple applications of our…

A Derived Lagrangian Fibration on the Derived Critical Locus

We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb{A}_k^1$, the derived critical locus has a natural Lagrangian fibration $\textbf{Crit}(f) \rightarrow X$. In the case where $f$ is non-degenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.

Decomposition of symplectic vector fields with respect to a fibration in lagrangian tori

Given a fibration of a symplectic manifold by lagrangian tori, we show that each symplectic vector field splits into two parts : the first is Hamiltonian and the second is symplectic and preserves the fibration. We then show an application of this result in the study of the regular deformations of completely integrable systems.

Semi-global invariants of piecewise smooth Lagrangian fibrations

We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle coordinates to these fibrations by encoding the information on the non-smoothness into certain invariants consisting, roughly, of a sequence of closed 1-forms on a torus. The main motivation for this work is given by the piecewise smooth Lagrangian fibrations previously constructed by the authors (in Some piece-wise smooth Lagrangian fibrations, http://seminariomatematico.dm.unito.it/rendiconti/63-3.html), …

Classification of Lagrangian Fibrations over a Klein Bottle

This paper completes the classification of regular Lagrangian fibratiopns over compact surfaces. \cite{misha} classifies regular Lagrangian fibrations over $\mathbb{T}^2$. The main theorem in \cite{hirsch} is used in order to classify integral affine structures on the Klein bottle $K^2$ and, hence, regular Lagrangian fibrations over this space.

Lagrangian torus fibration models of Fano threefolds

Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that the numerical invariants coincide for all 105. Our construction relies on a notion of toric degeneration for affine manifolds with singularities, and the correspondence we obtain between polytopes and Fano manifolds is compatible with that appearing in the work of Coates-Corti-K…

Topological Classification of Lagrangian Fibrations

We define topological invariants of regular Lagrangian fibrations using the integral affine structure on the base space and we show that these coincide with the classes known in the literature. We also classify all symplectic types of Lagrangian fibrations with base $\rpr$ and fixed monodromy representation, generalising a construction due to Bates.

Projective embeddings and Lagrangian fibrations of Kummer varieties

In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the projective spaces.

On symplectic lifts of actions for complete Lagrangian fibrations

In this note we discuss symplectic lifts of actions for a complete Lagrangian fibration. Firstly, we describe the symplectic cotangent lifts of a G-action on a manifold Q in terms of 1-cocycles in the cohomology of G induced by the action with values in the space of closed 1-forms on Q. After this, we consider the general case of complete Lagrangian fibrations.

Symplectic invariants of some families of Lagrangian T^3 fibrations

In this paper we construct and classify Lagrangian T^3-fibrations on non compact symplectic manifolds with singular fibres of prescribed topological type. This contributes to the understanding of the structure of the singular fibres that may appear on a mirror pair of T^3-fibred Calabi-Yau 3-folds.

Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces II: Technical results on gradient flow construction

This paper provides the technical details of gradient flow construction and related problems, which are essential for our construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces.

Lagrangian cobordisms, Lefschetz fibrations and quantum invariants

We study Lagrangian cobordisms with the tools provided by Lagrangian quantum homology. In particular, we develop the theory for the setting of Lagrangian cobordisms or Lagrangians with cylindrical ends in a Lefschetz fibration, and put the different versions of the quantum homology groups into relation by a long exact sequence. We prove various practical relations of maps in this long exact sequence and we extract invariants that generalize the notion of discriminants to Lagrangian cobordisms in Lefschetz fibrations. We prove results on the relation of the discriminants of the ends of a cobord…

Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $(\mathbb{C}^*)^n$, and study the singular fibers.

Lagrangian fibrations and theta functions

In this thesis we study asymptotic behavior of projective embeddings of abelian varieties and their amoebas. The projective embeddings are given by theta functions. It is known that a Lagrangian fibration of the abelian variety determines a basis of theta functions. After reviewing the relation from the viewpoint of geometric quantization and mirror symmetry, we prove that the Lagrangian fibration of the abelian variety can be approximated by the moment maps of natural torus action in a suitable sense. In the final section we discuss a part of the result in more general setting.

Lagrangian Torus Fibrations and Mirror Symmetry of Calabi-Yau Manifolds

In this paper we summarize our recent work in the construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties and the symplectic Strominger-Yau-Zaslow conjecture, together with some new development. It is submittded to the Proceedings of the Conference in Symplectic Geometry and Mirror Symmetry held in KIAS, Seoul, Korea. The paper was finished in Jan. 2001, with revisions and added reference.

关于奇异拉格朗日纤维化及其在辛嵌入中的应用 I

On singular Lagrangian fibrations and applications to symplectic embeddings I

本文在 4 维和 6 维圆盘余切丛上构造奇异拉格朗日纤维化，可用于获取环面域，恢复了相关结果，还能用于研究辛嵌入问题。

Deformed Hamiltonian vector fields and Lagrangian fibrations

Certain dissipative physical systems closely resemble Hamiltonian systems in $\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in each conjugate pair rescaled by a real parameter. To generalise these dynamical systems to symplectic manifolds in this paper we introduce and study the properties of deformed Hamiltonian vector fields on Lagrangian fibrations. We describe why these objects have some interesting applications to symplectic geometry and discuss how their physical interpretation motivates new problems in mathematics.

Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which…

Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers

The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special Lagrangian tori in X-D and weighted counts of holomorphic discs in X can be used to build a Landau-Ginzburg model mirror to X. In the second part we turn to more speculative considerations about Calabi-Yau manifolds with holomorphic involutions and their quotients. Namely, given a hypersurface H representing twice the anticanonical class in a Kahler manifold X, …

Lefschetz 纤维化中的正则 Lagrangian 子流形

Regular Lagrangians in Lefschetz fibrations

本文刻画 Weinstein Lefschetz 纤维化中 Lagrangian 子流形的正则性，证明 Giroux–Pardon 猜想，并将其用于构造使精确填充及其突变映为底弧的纤维化。

Lefschetz fibrations on adjoint orbits

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.

A Note On Lagrangian Vanishing Sphere Bundles

A classical way to construct a Lagrangian in a symplectic manifold $Σ$ is to let $Σ$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $Σ$ and these spheres, in turn, are representatives of the corresponding vanishing cycles in the homology of $Σ$. In this paper our aim is twofold: The first is to describe a generalization of the above mentioned construction to the "Morse-Bott" case. This leads, whenever such a degeneration exists, to the existence of Lagrangian sphere bundles rather than just spheres. In the …

Lagrangian cobordism in Lefschetz fibrations

Given a symplectic manifold $(M^{2n},ω)$ we study Lagrangian cobordisms $V\subset E$ where $E$ is the total space of a Lefschetz fibration having $M$ as generic fiber. We prove a generation result for these cobordisms in the appropriate derived Fukaya category. As a corollary, we analyze the relations among the Lagrangian submanifolds $L\subset M$ that are induced by these cobordisms. This leads to a unified treatment - and a generalization - of the two types of relations among Lagrangian submanifolds of $M$ that were previously identified in the literature: those associated to Dehn twists tha…

Monotone Lagrangians in flag varieties

In this paper, we give a formula for the Maslov index of a gradient holomorphic disc, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian non-toric fibers of Gelfand-Cetlin systems on partial flag manifolds.

Almost Lagrangian Obstruction

The aim of this paper is to describe the obstruction for an almost Lagrangian fibration to be Lagrangian, a problem which is central to the classification of Lagrangian fibrations and, more generally, to understanding the obstructions to carry out surgery of integrable systems, an idea introduced by Zung. It is shown that this obstruction (namely, the homomorphism of Dazord and Delzant) is related to the cup product in cohomology with local coefficients on the base space B of the fibration. The map is described explicitly and some examples are calculated, thus providing the first examples of n…

家族 Floer 纲领与非阿基米德 SYZ 镜像构造

Family Floer program and non-archimedean SYZ mirror construction

本文基于家族 Floer 理论，用全纯圆盘计数、非阿基米德分析和 $A_\infty$ 同调代数构造含刚性解析空间与超势的镜像 Landau-Ginzburg 模型，并将假设减弱到半正 Lagrangian 纤维化。

On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces

Given any smooth cubic curve $E\subseteq \mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $\mathbb{P}^2\setminus E$ constructed by Collins--Jacob--Lin arXiv:1904.08363 coincides with the affine structure used in Carl--Pomperla--Siebert for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl--Pomperla--Siebert.

Generalized Weinstein tubular neighbourhood & Space of Lagrangian submersions

We present a generalized Weinstein Tubular Neighbourhood theorem for Lagrangian subbundles of Symplectic fibrations. This is then used to study the space of Lagrangian fibrations of symplectic manifolds.

Kähler度量与toric拉格朗日纤维化

Kähler metrics and toric Lagrangian fibrations

论文扩展了Abreu-Guillemin理论，将不变Kähler度量从toric辛流形推广到具有toric作用的辛流形，揭示了这些流形恰好是具有椭圆奇点的拉格朗日纤维化，并建立了不变Kähler度量与椭圆连接和混合b度量的对应关系。

近 Lefschetz 纤维化、Stein 结构与正则 Lagrangian 子流形

Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians

研究近Lefschetz纤维化上的Stein结构，证明四维Weinstein域中具Legendrian边界的Lagrangian圆盘均为正则，解决邻近猜想与Slice-Ribbon猜想的Lagrangian类比，并给出构造Weinstein Kirby图的算法。

关于两步旗流形中的不可位移拉格朗日子流形

On non-displaceable Lagrangian submanifolds in two-step flag varieties

本文证明了两步旗流形 $\mathcal{F}\ell(1,n;n+1)$ 中存在不可位移且非单调的拉格朗日 Gelfand--Zeitlin 纤维，并在 $n > 2$ 时发现了一族连续的不可位移环面纤维。

Lectures on Lagrangian torus fibrations

This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can use to "read off" a lot of interesting geometry of the total space. Topics covered include: action-angle coordinates, symplectic reduction, toric manifolds, visible and tropical Lagrangians, almost toric systems, Milnor fibres of cyclic quotient singularities, mutation of polygons, non-toric blow-up, an almost toric view on Lisca's classification of fillings…

SYZ transforms for immersed Lagrangian multi-sections

In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multi-sections. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold $(M, ω)$, under which the immersed Floer cohomology is invariant; in particular, this p…

Floer homology of fibrations I: Representing flow lines in Moore path spaces

In their previous work, Barraud and Cornea enriched the Lagrangian Floer complex by adding cubical chains in the based loop space of the Lagrangian, and recovered the Leray-Serre spectral sequence of the based path space fibration, assuming that the Lagrangian is weakly exact and simply connected. In the present article, we remove the simple connectivity assumption and adapt the construction to any Hurewicz fibration. To this end, we introduce a stronger notion of local systems than the classical one, as a topological functor from the free path space of a manifold to the category of topologica…

代数锥光滑化上的奇异拉格朗日环面纤维化

Singular Lagrangian torus fibrations on the smoothing of algebraic cones

本文在代数锥的光滑变形上利用 Minkowski 分解构造奇异拉格朗日环面纤维化，得到凸底图表示，并通过穿墙公式恢复 Lau 的势函数结果，进而讨论单调拉格朗日环面的不可位移性。

Lagrangian matching invariants for fibred four-manifolds: I

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz fibrations. The `Lagrangian matching invariants' are designed to be comparable with the Seiberg-Witten invariants of the underlying four-manifold. They fit into a field theory which assigns Floer homology groups to fibred 3-manifolds. The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres,…

Displacing Lagrangians in the manifolds of full flags in C^3

In symplectic geometry a question of great importance is whether a (Lagrangian) submanifold is displaceable, that is, if it can be made disjoint from itself by the means of a Hamiltonian isotopy. In these notes we analyze the coadjoint orbits of SU(n) and their Lagrangian submanifolds that are fibers of the Gelfand-Tsetlin map. We use the coadjoint action to displace a large collection of these fibers. Then we concentrate on the case n=3 and apply McDuff's method of probes to show that "most" of the generic Gelfand-Tsetlin fibers are displaceable. "Most" means "all but one" in the non-monotone…

The wrapped Fukaya category for semi-toric SYZ fibrations

We introduce the wrapped Donaldson-Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi-Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori-Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an…

Infinitesimally Tight Lagrangian Orbits

We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples.

A Leray-Serre spectral sequence for Lagrangian Floer theory

We consider symplectic fibrations as in Guillemin-Lerman-Sternberg, and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a rational base. We show if the Floer cohomology with field coefficients of the fiber Lagrangian vanishes, then the Floer cohomology with field coefficients of the total Lagrangian also vanishes. We give an application to certain non-torus fibers of the Gelfand-Cetlin system in Flag manifolds, and show that their Floer cohomology vanishes.

完备 Lagrange 纤维化的全局 Darboux 坐标及其在亏格为一的 $\mathbb{R}\mathbb{P}^2$ 结构形变空间上的应用

Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one

本文研究非紧纤维完备 Lagrange 纤维化，证明运动积分可作为作用量坐标，并据此构造环面真凸 $\mathbb{R}\mathbb{P}^2$ 结构形变空间上辛形式的全局 Darboux 坐标及等距表达式。

Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds

In this paper the relations between the existence of Lagrangian fibration of Hyper-Kähler manifolds and the existence of the Large Radius Limit is established. It is proved that if the the rank of the second homology group of a Hyper-Kähler manifold N of complex dimension $2n\geq4$ is at least 5, then there exists an unipotent element T in the mapping class group $Γ$(N) such that its action on the second cohomology group satisfies $(T-id)^{2}\neq0$ and $(T-id)^{3}=0.$ A Theorem of Verbitsky implies that the symmetric power $S^{n}(T)$ acts on $H^{2n}$ and it satisfies $(S^{n}% (T)-id)^{2n}\neq0…

Local torus actions modeled on the standard representation

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which is a generalization of the topological class…