Classification of transversal Lagrangian stars
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 \(TL_{\mathrm{reg}}\) of \(TL\) (Theorem 3.3, Remark 3.4). In Section 4 the classifications of the algebraic restrictions to \(L\) of symplectic forms vanishing on \(TL_{\mathrm{reg}}\) with respect to the actions of local symmetries of \(L\) is reduced to the classifications of the algebraic restrictions to \(L\) of symplectic forms of degree \(Z\) with respect to the actions of linear local symmetries of \(L\). By this result, to proof of Theorem 2.3 is reduced problems of linear algebra. They are solved in Section 5, and Theorem 2.3 is proved.de
scientific article; zbMATH DE number 6836788
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