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Complex codimension one foliations leaving a compact submanifold invariant. (English) Zbl 0647.57017
Dynamical systems and bifurcation theory, Proc. Meet., Rio de Janeiro/Braz. 1985, Pitman Res. Notes Math. Ser. 160, 295-317 (1987).
[For the entire collection see Zbl 0621.00019.]
\({\mathcal F}\) is a (complex) codimension one singular foliation on a complex \((n+1)\)-manifold V, \(n\geq 1\), with singular set) S(\({\mathcal F})=\cup_{\alpha \in I}S_{\alpha}\), \(S_{\alpha}\) an analytic subvariety of V, of codimension at least 2. There is a global codimension 1 foliation of \(V-S({\mathcal F})\), with “regular leaves of \({\mathcal F}''\). \({\mathcal F}\) is assumed to leave invariant a compact connected codim. 1 submanifold \(M\subset V\). \(M-S({\mathcal F})\) is a regular leaf of \({\mathcal F}\), and let \[ S({\mathcal F})\cap M=S_ 1\cup S_ 2\cup...\cup S_{\beta}. \] Let L be the normal line bundle of M in V, \(L^ k=L\oplus L\oplus...\oplus L\) (k times), and \(c_ k=kth\) Chern form of \(L^ k\), i.e. \(c_ k=(i/2\pi)^ k\Theta^ k\) for \(\Theta=\) curvature form of a connection in L. The author proves: for \(\phi\) a \(C^{\infty}\) closed \((2n-2)\)-form on M, \[ \int_{M}c_ 1\wedge \phi =\sum^{\beta}_{i=1}i(S_ j,{\mathcal F},M)\int_{S}\phi, \] in particular \[ \int_{M}c_ n=\sum^{\beta}_{j=1}i(S_ j,{\mathcal F},M)\int_{S_ j}c_{n-1} \] and \[ [c_ 1]=[\sum^{\beta}_{j=1}i(S_ j,{\mathcal F},M)\Omega_ j] \] where \([\mu]=de\) Rham class of \(\mu\) (for \(\mu\) a closed form), \([\Omega_ j]=\) Poincaré dual of \(S_ j\), \(1\leq j\leq \beta\), \(i(S_ j,{\mathcal F},M)\) is the “normal index of \(S_ j\) in M with respect to \({\mathcal F}''\) which can be expressed by a residue. A second theorem deals with the existence of V with a singular foliation \({\mathcal F}\) if one starts with a compact connected complex n-manifold M and a subvariety \(S=\cup^{\beta}_{j=1}S_ j\subset M\) with irreducible components \(S_ j\). The crucial condition that is to be satisfied is \[ [\sum^{\beta}_{j=1}\lambda_ j\Omega_ j]\in H^ 2(M;{\mathbb{Z}}) \] for some nonzero complex numbers \(\lambda_ 1,\lambda_ 2,...,\lambda_{\beta}\). In the Kähler case, a singular foliation of the considered type exists on a line bundle L over M if \[ [\sum^{\beta}_{j=1}\lambda_ j\omega_ j]=[c_ 1(L)] \] for some nonzero complex numbers \(\lambda_ 1,\lambda_ 2,...,\lambda_{\beta}\) [cf. also the author’s paper “Construction of singular holomorphic vector fields and foliations in dimension two” (to appear in J. Differ. Geom.)].
Reviewer: A.Aeppli

MSC:
57R30 Foliations in differential topology; geometric theory
32L99 Holomorphic fiber spaces
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
57R20 Characteristic classes and numbers in differential topology