Residues of Chern-Maslov classes (Q1775408)

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 \[ m_{01}= {1\over 2\pi\sqrt{-1}} \pi_* {d(\text{det\,}A(\tau))\over \text{det\,}A(\tau)} \] (\S4. Cor. 10). Let \(L'\) and \(L''\) be transversal on \(U_0= M\setminus S\). Taking the complex structure \(J_0\) on \(U_0\) as \(L''= J_0L'\), the Chern-Maslov classes are localized and by using Alexander duality, the residue \(\text{Res}_{\mu^h}(S_\alpha)\in H_{m-(4h- 3)}(S_\alpha; \mathbb{C})\) of the Chern-Maslov class on \(S_\alpha\), a connected component of \(S\), is defined (\S5, Def. 11). Then it is shown that \[ \sum_\alpha (\iota_\alpha)_* \text{Res}_{\mu^h}(S_\alpha)= \mu^h\cup [M],\text{ in }H_{m- (4h-3)}(M; \mathbb{C}) \] (\S5. Th. 12). If \(S\) is a smooth codimension 1 submanifold and \(\dim(L'|_x+ L''|_x)= 2n-1\), \(x\in S\), then the first Chern-Maslov class \(\mu^1= (0,m_1, m_{01})\) is computed taking the form \[ \begin{aligned} m_{01} &= -{\sigma\over 2\pi} \Biggl(\tan^{-1} \sqrt{{k+1/2\over k-1/2}} -\tan^{-1}\Biggl({k- 1/2\over k+1/2}\Biggr) \sqrt{{k- 1/2\over k+ 1/2}}\Biggr),\\ m^1 &= {1\over 2\pi} d\phi\phi^2+ 1,\quad k= \sqrt{{\phi+ 1\over 4}},\;\sigma= {\phi\over|\phi|},\end{aligned} \] where \(\phi\) is a function obtained from simplex flames of \(E\) on \(U\setminus S\), \(U\) a neighborhood of \(S\) (\S6.1). The case when \(S\) has singularities is also treated [\S6.2. cf. \textit{I. Vaisman}, Symplectic geometry and secondary characteristic classes (Progress in Mathematics 72, Boston: Birkhäuser) (1987; Zbl 0629.53002)]. In \S7, the cases \(\text{T}\mathbb{R}^2|_{S^1}\to S^1\) and \(M= \{(x, y)\in \mathbb{R}^2| y= x^n\}\) are treated.