Classification of transversal Lagrangian stars
type: Item
instance of: scholarly article
title: Classification of transversal Lagrangian stars
zbMATH Open document ID: 1385.53071
published in: Topology and its Applications
publication date: 2018-02-09T00:00:00Z
review text: A Lagrangian \(s\)-star at a point \(p\) of a symplectic manifold \(M\) is a germ of \(s\) Lagrangian submanifolds \(L_1,\ldots,L_s\) intersecting at \(p\), see [\textit{S. Janeczko}, J. Phys. A, Math. Gen. 31, No. 16, 3677--3685 (1998; Zbl 0907.58016)]. The natural invariant of Lagrangian 3-stars is the Maslov index [\textit{G. Lion} and \textit{M. Vergne}, The Weil representation, Maslov index and Theta series. Boston, etc.: Birkhäuser (1980; Zbl 0444.22005)]. In this paper, by using the method of algebraic restriction [the second author et al., J. Reine Angew. Math. 618, 197--235 (2008; Zbl 1157.58016)], the following theorem is proved: Theorem 2.3. In the analytic category, a transversal Lagrangian 3-star in \((\mathbb{R}^{2n},\sum_{i=1}^n dx_i\wedge dy_i)\) is symplectically equivalent to one and only one of \(E^s=(\{L_1,L_2,L_3^s\},0)\), where \[ \begin{gathered} L_1=\{x_1=\cdots=x_n=0\}, \quad L_2=\{y_1=\cdots=y_n=0\} \\ L_3^s=\{x_1-y_1=\cdots=x_s-y_s=x_{s+1}+y_{s+1}=\cdots=x_n+y_n=0\}. \end{gathered} \] The definition of Lagrangian \(s\)-stars with related notions and necessity of the use algebraic restrictions in the study of transversal Lagrangian 3-stars are explained in Section 2. Section 3 deals with the method of algebraic restrictions and its relation to the study of Lagrangian 3-stars. Let \(\theta\) be a \(k\)-form on \(M\), and let \(N\) be a subset of \(M\). Then, the algebraic restriction \([\theta]_N\) of \(\theta\) to \(N\) is the equivalence class of \(\theta\) by the equivalence relation \(\theta\cong \theta'\) if and only if \(\theta-\theta'=\alpha+d\beta\), where both \(\alpha\) and \(\beta\) vanish on \(N\) (Definition 3.1). Results in [Domritz, loc. cit.] show that the classification of transversal Lagrangian stars diffeomorphic to a transversal quasi-homogeneous Lagrangian 3-star \(E=(\{L_1,L_2,L_3\},0)\) reduces to the classification of algebraic restrictions of symplectic forms to \(L\) vanishing on the set of smooth points \(TL_{\math…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 58A10
zbMATH DE Number: 6836788
zbMATH Keywords: Lagrangian stars
zbMATH Keywords: symplectic singularities
zbMATH Keywords: algebraic restrictions
author: Fausto Assunção de Brito Lira
author: Wojciech Domitrz
author: R. G. Wik Atique
MaRDI profile type: Publication
full work available at URL: j.topol.2017.11.022
OpenAlex ID: W2769246711
cites work: Dynamical systems. IV. Symplectic geometry and its applications. Transl. from the Russian by G. Wassermann.
cites work: Local symplectic algebra of quasi-homogeneous curves
cites work: Zero-dimensional symplectic isolated complete intersection singularities
cites work: Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety
cites work: Symplectic singularities of varieties: The method of algebraic restrictions
cites work: Generic singularities of symplectic and quasi-symplectic immersions
cites work: Symplectic 𝑆_{𝜇} Singularities
cites work: Symplectic T7, T8 singularities and Lagrangian tangency orders
cites work: Classification of Lagrangian stars and their symplectic reductions
cites work: The Weil representation, Maslov index and theta series
cites work: Q3445918
cites work: Symplectic W_8 and W_9 singularities
cites work: Symplectic \(U_7\), \(U_8\) and \(U_9\) singularities
DOI: 10.1016/J.TOPOL.2017.11.022
description: scientific article; zbMATH DE number 6836788