ABSTRACT
In this chapter we explain the classical Leray theory of residues and its main applications. The context is the following. Let X be a compact complex manifold, D ⊂ X a smooth divisor, i.e. a smooth complex hypersurface. We are interested in the differential forms ω on the open set X \ D having logarithmic poles along D; they are called logarithmic forms. This means that in local coordinates around any x ∈ X we can write
ω = α ∧ dz z
+ β
where z = 0 is the local equation of D. The Leray residue of ω is the restriction
Resω = α|D which is, in fact, a global form on D. The differential dω of a logarithmic form is logarithmic: locally we can write dω = dα ∧ dzz + dβ, so that
Res dω = dα|D = dResω
hence the residue commutes to the differential. The above definition has a local nature: we can define a logarithmic form on Y \D for any open subset Y ⊂ X . It follows that the sheaf EkX〈logD〉 of the logarithmic k-forms is well defined, and E·X〈logD〉 is a complex of fine sheaves on X (the reader should take note: on X , not only on X \D). The logarithmic forms are particular differential forms on X \D, hence we have an inclusion
E·X〈logD〉 ⊂ ρ∗E·X\D where ρ : X \D ↪→ X is the natural inclusion map. A theorem due to Leray states that the above inclusion of complexes induces an isomorphism in cohomology. This means exactly the following local statements: let U be an open neighborhood in X of a point x ∈ D, isomorphic to a ball in CN ; then
to a form;
• every logarithmic differential form on U \D which is exact, is the differential of a logarithmic form.
