Residues of Chern-Maslov classes
type: Item
instance of: scholarly article
title: Residues of Chern-Maslov classes
zbMATH Open document ID: 1072.53028
author: Takeshi Izawa
author: Katsunori Nakajima
published in: Journal of the Mathematical Society of Japan
publication date: 2005-05-03T00:00:00Z
review text: Localization of the Maslov class associated with two Lagrangian subbundles in a real symplectic vector bundle is described and a definition of the residue of the Maslov class is given. Then, explicit computation of the residue of the first Maslov class is given in the case the non-transversal sets of the two Lagrangian subbundles have codimension 1. For this purpose, Čech-de Rham cohomology and description of Alexander duality in terms of Čech-de Rham cohomology for a smooth oriented \(m\)-dimensional manifold and its compact subset \(S\) admitting a regular neighborhood \(U\) are reviewed, provided \(M\) and \(S\) are stratified by the strata \(\Sigma_0= M\setminus S\), \(\Sigma_1,\dots, \Sigma_q\), \(S= \Sigma_1\cup\cdots\cup\Sigma_q\) (\S2). Then, according to \textit{R. Bott} [Lect. Notes Math. 279, 1--94 (1972; Zbl 0241.57010)], the \(h\)-th Chern-Maslov form \(c^{2h-1}(\nabla',\nabla'')\) and its de Rham class, the \(h\)-th Chern-Maslov class \(\mu^h(E, L',L'')\in H^{4h-3}_{DR}(M;\mathbb{C})\) are defined. Here, \((E\to M,\omega)\) is a symplectic vector bundle of real rank \(2n\) with symplectic form \(\omega\), \(L'\), \(L''\) are two Lagrangian subbundles, and \(\nabla'\), \(\nabla''\) are \(L'\) and \(L''\)-orthogonal unitary connections respectively (\S3, Def. 2). The Chern-Maslov forms are computed in terms of a fixed complex structure on \(E\) but they do not depend on the choice of complex structure. Hence we have \(c^{2h-1}(\nabla_1', \nabla^{\prime\prime}_1)- c^{2h-1}(\nabla_0',\nabla^{\prime\prime}_0)= dm_{01}\). The explicit form of \(m_{01}\) is computed in \S4 [Prop. 8 and Th. 9. cf. \textit{T. Suwa}, Indices of vector fields and residues of singular holomorphic foliations (Actualités Mathématiques, Paris: Hermann) (1998; Zbl 0910.32035)]. If \(h= 1\) and \(A\) is a change of the \(\widetilde g\)-orthogonal frame of \(L'\times I\) and \(L''\times I\), \(\widetilde g(x,y)= \omega(x,\widetilde J y)\), \(x,y\in E\times I\), then it is shown that \[ …
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 2164234
zbMATH Keywords: Maslov class
zbMATH Keywords: Maslov index
zbMATH Keywords: characteristic class
zbMATH Keywords: residue class
zbMATH Keywords: secondary class
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1160745811
OpenAlex ID: W2038006337
DOI: 10.2969/JMSJ/1160745811
description: scientific article; zbMATH DE number 2164234