Let F be a codimension one holomorphic foliation whose singular set Σ is contained in a compact leaf S of F.
When F is of dimension one, Σ is a set of isolated points {q1, ..., qr}, C. Camacho and P. Sad define the index of F at each point qk and prove that the sum of these indices equals the Euler class c1(E) of the fibre bundle E normal to S.
Generally, whenever Σ is of any dimension m, we can define a such index iα along the maximal dimension strates {Σα} of a suitable stratification of the complex variety Σ. Let σα be the fundamental cycle of Σα, σ the 2m-cycle of S defined by σ = Σiα.σα and σ* the 2-cocycle dual to σ by Poincaré isomorphism H2(S) → H2m(S), we prove that the cohomology class [σ*] equals the Euler class c1(E).
@article{urn:eudml:doc:41154,
title = {Sur les feuilletages holomorphes singuliers de codimension 1.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {229-240},
zbl = {0760.32019},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41154}
}
Gmira, Bouchra. Sur les feuilletages holomorphes singuliers de codimension 1.. Publicacions Matemàtiques, Tome 36 (1992) pp. 229-240. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41154/