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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