Residues, currents, and their relation to ideals of holomorphic functions. (English) Zbl 0633.32005

Passare, Mikael

Math. Scand. 62, No. 1, 75-152 (1988).

This thesis deals with the problem of deciding whether a holomorphic function of several variables belongs to a given ideal, and also, in case it does, how the function can be expressed as a combination of generators for the ideal. Under the assumption that generators can be found which form a complete intersection, several differential criteria for the solvability of the above division problem are obtained.
Some of the conditions are of a global, algebraic nature in demanding that certain cohomology classes be in the kernel for a residue homomorphism. Others are stated in purely local, analytic terms such as the vanishing of residue currents. In the latter context representation formulas are obtained, which occur as limits of certain integral formulas and which explicitly give coefficients serving to express the function by means of the generators. The expressions for these coefficients involve generalized residue currents, which depend on the generators and whose existence is proved even without the hypothesis of complete intersections.
Examples are included to elucidate the properties of residue currents.

MSC:

32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A27	Residues for several complex variables
32C30	Integration on analytic sets and spaces, currents
14M10	Complete intersections
14F25	Classical real and complex (co)homology in algebraic geometry
32B20	Semi-analytic sets, subanalytic sets, and generalizations
32T99	Pseudoconvex domains
32A38	Algebras of holomorphic functions of several complex variables

Keywords:

holomorphic function; ideal; complete intersection; residue; residue currents; Dolbeault cohomology; integral representation formula; semimeromorphic form; pseudoconvex domain