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Residues of holomorphic foliations relative to a general submanifold. (English) Zbl 1088.32018
Since the celebrated result of P. Baum and R. Bott [J. Differ. Geom. 7, 279–342 (1972; Zbl 0268.57011)], residue theorems have become an essential analytical tool in the study of complex (possibly singular) foliations. They are used, for example, in the study and classification of foliations of complex compact surfaces, as well as trajectories of a holomorphic vector field around an isolated singular point, whose blow-up produces an invariant exceptional divisor (the so-called non-dicritical situation). Usually, residues are given relative to a complex analytic subset that is tangent to the foliation (i.e., a union of leaves and singular points of the foliation). This is the case for the aforementioned examples.
The present paper extends to residue theorems relative to general subvarieties that are not necessarily tangent to the foliation. Examples given are residues relative to the base for foliations generically transverse to a holomorphic vector bundle, and residues relative to a locally complete intersection. The latter case includes the example of blow-up of dicritical type.

MSC:
32S65 Singularities of holomorphic vector fields and foliations
53C12 Foliations (differential geometric aspects)
57R20 Characteristic classes and numbers in differential topology
55N15 Topological \(K\)-theory
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