Mathematics - Symplectic Geometry Publications (50)

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Mathematics - Symplectic Geometry Publications

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Nemethi. Read More


In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $\mathbb{R}^2$ and the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2 $. Read More


We show that the main theorem of Morse theory holds for a large class of functions on singular spaces. The function must satisfy certain conditions extending the usual requirements on a manifold that Condition C holds and the gradient flow around the critical sets is well-behaved, and the singular space must satisfy a local deformation retract condition. We then show that these conditions are satisfied when the function is the norm-square of a moment map on an affine variety. Read More


We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existence theorem for symmetric Morse-Smale pairs is also proved. Read More


We show that two Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) induce two Lagrangian submanifolds in the one-point blow-up (\widetilde{CP}^2,\widetilde{\omega}_\rho) that are not Hamiltonian isotopic. Furthermore, we show that for any integer k>1 there are k Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) that induce k Lagrangian submanifolds in the one-point blow-up such that no two of them are Hamiltonian isotopic. Read More


The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quotients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form \[\int_X \alpha = Res_{\mathbf{z}=\infty} \mathcal{Z}(\mathbf{z}, \mathbf{t}) \cdot \alpha(\mathbf{t}),\] for a certain residue operation and a map $\mathcal{Z}(\mathbf{z}, \mathbf{t})$, associated to the root system. Read More


Floer theory was originally devised to estimate the number of 1-periodic orbits of Hamiltonian systems. In earlier works, we constructed Floer homology for homoclinic orbits on two dimensional manifolds using combinatorial techniques. In the present paper, we study theoretic aspects of computational complexity of homoclinic Floer homology. Read More


This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by K\'alm\'an, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. Read More


In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of "ghost" obstructions that generate an infinite "ghost staircase" determined by the even index Fibonacci numbers. The ghost obstructions are not visible for the four-dimensional embedding problem because strictly stronger obstructions also exist. We show that in contrast, the embedding constraints associated to the ghost obstructions are sharp for the stabilized problem; moreover, the corresponding optimal embeddings are given by symplectic folding. Read More


We will consider the 3-point blow-up of the manifold $ (S^2 \times S^2, \sigma \oplus \sigma)$ where $\sigma$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, \omega)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Read More


In this work we connect Poisson and near-symplectic geometry by showing that there are two almost regular Poisson structures induced by a near-symplectic $2n$-manifold. The first structure is of maximal rank $2n$ and vanishes on a codimension-2 subspace. The second one is log-f symplectic of maximal rank $2n-2$. Read More


Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. Read More


We prove a homological mirror symmetry equivalence between the $A$-model of the pair of pants, in its guise as wrapped microlocal sheaves, and its mirror Landau-Ginzburg $B$-model, in its guise as matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the independence of wrapped microlocal sheaves along different Lagrangian skeleta arising naturally from tropical geometry. The equivalence proceeds through the construction of a combinatorial realization of the $A$-model via arboreal singularities. Read More


In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting and use it to study the fundamental group of a Lagrangian cobordism $W\subset (\mathbb{C}\times M, \omega_{st}\oplus\omega)$ between two Lagrangian submanifolds $L, L'\subset ( M, \omega)$. We show that under natural conditions the inclusions $L,L'\hookrightarrow W$ induce surjective maps $\pi_{1}(L)\twoheadrightarrow\pi_{1}(W)$, $\pi_{1}(L')\twoheadrightarrow\pi_{1}(W)$ and when the previous maps are injective then $W$ is an h-cobordism. Read More


We further develop the symplectic instanton homology defined in our previous article [Hor16] by investigating its behavior under Dehn surgery. In particular, we prove a exact triangle relating the symplectic instanton homology of $\infty$-, $0$-, and $1$-surgeries along a framed knot in a closed $3$-manifold. More generally, we show that for any framed link in a closed $3$-manifold, there is a spectral sequence with $E^1$-page a direct sum of symplectic instanton homologies of all possible combinations of $0$- and $1$-surgeries on the components of the link converging to the symplectic instanton homology of the ambient $3$-manifold. Read More


We describe a complete list of Casimirs for 2D Euler hydrodynamics on a surface without boundary: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface. We also outline a possible extension of main notions to the boundary case and formulate several open questions in that setting. Read More


This paper determines a condition that is necessary and sufficient for a metaplectic-c prequantizable symplectic manifold with an effective Hamiltonian torus action to admit an equivariant metaplectic-c prequantization. The condition is evaluated at a fixed point of the momentum map, and is shifted from the one that is known for equivariant prequantization line bundles. Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of the system. Read More


Witten's Gauged Linear $\sigma$-Model (GLSM) unifies the Gromov-Witten theory and the Landau-Ginzburg theory, and provides a global perspective on mirror symmetry. In this article, we summarize a mathematically rigorous construction of the GLSM in the geometric phase using methods from symplectic geometry. Read More


We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in\mathbb{R}$, $q\in\mathbb{R}^n$, $a\colon\mathbb{R}\to\mathbb{R}$ is a continuous positive bounded function, $G\colon\mathbb{R}^n\to\mathbb{R}$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb{R}\to\mathbb{R}^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations. Read More


We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Read More


We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the generalised Monge-Amp`ere equation is conditioned on a conjecture from algebraic geometry. Read More


We develop a new homological invariant for the dynamics of the bounded solutions to the travelling wave PDE \[ \left\{ \begin{array}{l l} \partial_t^2 u - c \partial_t u + \Delta u + f(x,u) = 0 \qquad & t \in \mathbf{R},\; x \in \Omega, \newline B(u) = 0 & t \in \mathbf{R},\; x \in \partial \Omega, \end{array} \right. \] where $c \neq 0$, $\Omega \subset \mathbf{R}^d$ is a bounded domain, $\Delta$ is the Laplacian on $\Omega$, and $B$ denotes Dirichlet, Neumann, or periodic boundary data. Restrictions on the nonlinearity $f$ are kept to a minimum, for instance, any nonlinearity exhibiting polynomial growth in $u$ can be considered. Read More


We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. Read More


We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Read More


We shall explain how the idea of microlocal analysis of the seventies has been reformulated in the framework of sheaf theory in the eighties and then applied to various branches of mathematics, such as linear partial differential equations or symplectic topology. Read More


This paper is intended both an introduction to the algebraic geometry of holomorphic Poisson brackets, and as a survey of results on the classification of projective Poisson manifolds that have been obtained in the past twenty years. It is based on the lecture series delivered by the author at the Poisson 2016 Summer School in Geneva. The paper begins with a detailed treatment of Poisson surfaces, including adjunction, ruled surfaces and blowups, and leading to a statement of the full birational classification. Read More


Consider a $2n$-dimensional symplectic vector space $E$ over an arbitrary field $\mathbb{F}$. Given a contraction map $f: \wedge^n E \rightarrow \wedge^{n-2} E$ such that the Lagrangian--Grassmannian $L(n,2n)=G(n,2n)\cap{\mathbb P}(\ker f)$, where $\wedge^r E$ denotes the $r$-th exterior power of $E$ and ${\mathbb P}(\ker f)$ is the projectivization of $\ker f$. In this paper, for a symplectic vector space $E$ of dimension $n=6$, we prove that the surjectivity of the contraction map $f:\wedge^{6} E \rightarrow \wedge^{4} E$ depends on the characteristic of the base field and we calculate the codimension of the linear section ${\mathbb P}(\ker f)\subseteq {\mathbb P}(\wedge^{6}E)$ for any characteristic. Read More


Let L be a Legendrian knot in the standard contact 3-sphere. If L bounds an orientable exact Lagrangian surface S in the standard symplectic 4-ball, then the genus of S is equal to the slice genus of (the smooth knot underlying) L, the sum of the Thurston-Bennequin number of L and the Euler characteristic of S is zero as well as the rotation number of L, and moreover, the linearized contact homology of L with respect to the augmentation induced by S is isomorphic to the (singular) homology of S. It was asked in arXiv:1212. Read More


In 2007, Alekseev-Meinrenken proved that there exists a Ginzburg-Weinstein diffeomorphism from the dual Lie algebra ${\rm u}(n)^*$ to the dual Poisson Lie group $U(n)^*$ compatible with the Gelfand-Zeitlin integrable systems. In this paper, we explicitly construct such diffeomorphisms via Stokes phenomenon and Boalch's dual exponential maps. Then we introduce a relative version of the Ginzburg-Weinstein linearization motivated by irregular Riemann-Hilbert correspondence, and generalize the results of Enriquez-Etingof-Marshall to this relative setting. Read More


In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider constraints coming from the ambient space and Deligne-Mumford moduli, called primary insertions, as well as intrinsic classes such as $\psi$-classes and Hodge classes. Read More


We introduce the notion of a compatible spherical functor. This is an additional structure on a spherical functor with Calabi-Yau target. Our first result is that this structure coincides with the structure of a weak relative Calabi-Yau structure on the functor. Read More


Consider a fibration of symplectic manifolds, with an induced fibration of Lagrangians. We develop a new version of Lagrangian Floer theory that is well defined when the fiber Lagrangian is monotone and the base is rational and unobstructed. Then, we write down a Leray-Serre type spectral sequence that computes the Floer cohomology of the total Lagrangian from the Floer complexes of the base and fiber. Read More


In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem. Read More


In this article, we study convexity issues of the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doubly-covered elliptic coordinates provide a 2-to-1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have the equal mass, but does not holds for the bounded component near the heavier body. Read More


In their previous work, S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra (i. Read More


We study the noncommutative Poincar\'e duality between the Poisson homology and cohomology of a unimodular quadratic Poisson algebra and its Koszul dual, and that between the Hochschild homology and cohomology of their deformation quantizations. We show that Kontsevich's deformation quantization preserves the corresponding Poincar\'e duality, and as a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are all isomorphic. Read More


In this article, we first introduce the notion of a {\it continuous cover} of a manifold parametrised by any compact manifold endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the usual sum is replaced by integrals. We then generalize Polterovich's notion of Poisson non-commutativity to such a context in order to get a richer definition of non-commutativity and to be in a position where one can compare various invariants of symplectic manifolds, for instance the relation between critical values of phase transitions of symplectic balls and eventual critical values of the Poisson non-commutativity. Read More


We prove a {\Gamma}-equivariant version of the algebraic index theorem, where {\Gamma} is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Read More


We extend Floer theory for monotone Lagrangians to allow coefficients in local systems of arbitrary rank. Unlike the rank 1 case, this is often obstructed by Maslov 2 discs. We study exactly what the obstruction is and define some natural unobstructed subcomplexes. Read More


Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also $\mathcal{Z}_{\omega}(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,\omega)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $\omega$ and $f$. We construct a canonical map $\varphi: \mathcal{Z}_{\omega}(f) \to \mathcal{S}_{\mathrm{id}}(f,\omega)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. Read More


We introduce diffusively coupled networks where the dynamical system at each vertex is planar Hamiltonian. The problems we address are synchronisation and an analogue of diffusion-driven Turing instability for time-dependent homogeneous states. As a consequence of the underlying Hamiltonian structure there exist unusual behaviours compared with networks of coupled limit cycle oscillators or activator-inhibitor systems. Read More


In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the deformation theory of {\it unstable} marked curves using the language of Lie groupoid (which is {\it not} necessary etale) and the Riemannnian center of mass technique. This proof is actually similar to [FOn,Sections 13 and 15] except the usage of the language of Lie groupoid makes the argument more transparent. Read More


We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham operator. As applications we give a characterization of generalized complex manifolds that are locally the product of generalized complex factors and discuss infinitesimal deformations of generalized CRF-structures. Read More


We consider a hyperk\"ahler reduction and describe it via frame bundles. Tracing the connection through the various reductions, we recover the results of Gocho and Nakajima. In addition, we show that the fibers of such a reduction are necessarily totally geodesic. Read More


We describe the quantization of a four-dimensional locally non-geometric M-theory background dual to a twisted three-torus by deriving a phase space star product for deformation quantization of quasi-Poisson brackets related to the nonassociative algebra of octonions. The construction is based on a choice of $G_2$-structure which defines a nonassociative deformation of the addition law on the seven-dimensional vector space of Fourier momenta. We demonstrate explicitly that this star product reduces to that of the three-dimensional parabolic constant $R$-flux model in the contraction of M-theory to string theory, and use it to derive quantum phase space uncertainty relations as well as triproducts for the nonassociative geometry of the four-dimensional configuration space. Read More


In this note we present a brief introduction to Lagrangian Floer homology and its relation with the solution of Arnol'd conjecture, on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism. We start with the basic definition of critical point on smooth manifolds, in oder to sketch some aspects of Morse theory. Introduction to the basics concepts of symplectic geometry are also included, with the idea of understanding the statement of Arnol'd Conjecture and how is related to the intersection of Lagrangian submanifolds. Read More


We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits. Read More


In the mapping class group of a $k$-holed torus with $0 \leq k \leq 9$, one can factorize the boundary multi-twist (or the identity when $k=0$) as the product of twelve right-handed Dehn twists. Such factorizations were explicitly given by Korkmaz and Ozbagci for each $k \leq 9$ and an alternative one for $k=8$ by Tanaka. In this note, we simplify their expressions for the $k$-holed torus relations. Read More