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Erschienen in:
2015 | OriginalPaper | Buchkapitel
3. Index Theorems
verfasst von : Marco Brunella
Erschienen in: Birational Geometry of Foliations
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Abstract
In this chapter we prove two important results of Baum–Bott [3] and Camacho–Sad [10, 11], the first one concerning the computation of
\(N_{\mathcal{F}}\cdot N_{\mathcal{F}}\)
for a foliation
\(\mathcal{F}\)
on a compact surface, the second one concerning the computation of
\(C \cdot C\)
for a compact curve C invariant by a foliation. These two results, which are in fact two manifestations of the same “vanishing principle”, have a quite different nature than the easier formulae of the previous chapter: here the integrability of
\(\mathcal{F}\)
, that is the existence of leaves, plays a fundamental role. We shall also give several applications of these formulae. A comprehensive reference for these index theorems, and much beyond, is [45], especially Chapter V; see also [6, 7].