ABSTRACT
In this chapter, we shall explain the theory of residues for a smooth hypersurface. The theory has a long history. It starts from the Cauchy formula in classical holomorphic function theory. In the 19th century, it is used to construct the theory of abelian integrals of the 1st, 2nd and 3rd kinds on a compact Riemann surface. The idea is to realize the compact Riemann surface as an algebraic curve in P2, possibly with “standard” singularities and to obtain the various differentials of degree 1 as residues. From that construction, one could deduce all the classical theory of algebraic curves. Picard and Poincare´ defined the notion of residue of rational forms on algebraic surfaces, in particular studied the periods of rational differentials and proved that they are periods of abelian integrals on the polar curve. The modern theory was defined by Leray to study the Cauchy problem of holomorphic partial differential equations. Actually, Leray constructed a residue theory for a hypersurface which is the union of divisors with normal crossing. But the main result, namely that the cohomology of the complementary of a hypersurface can be realized with forms with poles of order at most 1 was proved by Leray only for a smooth hypersurface.
