Holomorphic Functions and Integral Representations in Several Complex Variables

In §1 and §2 of this chapter we present the standard local properties of holomorphic functions and maps which are obtained by combining basic one complex variable theory with the calculus of several (real) variables. The reader should go through this material rapidly, with the goal of familiarizing himself with the results, notation, and terminology, and return to the appropriate sections later on, as needed. The inclusion at this stage of holomorphic maps and of complex submanifolds, i.e., the level sets of nonsingular holomorphic maps, is quite natural in several variables. In particular, it allows us to present elementary proofs of two results which distinguish complex analysis from real analysis, namely: (i) the only compact complex submanifolds of ℂ ⁿ are finite sets, and (ii) the Jacobian determinant of an injective holomorphic map from an open set in ℂ ⁿ into ℂ ⁿ is nowhere zero. Section 3, which gives an introduction to analytic sets, may be omitted without loss of continuity. We have included it mainly to familiarize the reader with a topic which is fundamental for many aspects of the general theory of several complex variables, and in order to show, by means of the Weierstrass Preparation Theorem, how algebraic methods become indispensable for a thorough understanding of the deeper local properties of holomorphic functions and their zero sets.

In 1906 F. Hartogs discovered the first example exhibiting the remarkable extension properties of holomorphic functions in more than one variable. It is this phenomenon, more than anything else, which distinguishes function theory in several variables from the classical one-variable theory. Hartogs’ discovery marks the beginning of a genuine several-variable theory, in which fundamental new concepts like domains of holomorphy and the various notions of convexity used to characterize them have become indispensable. In particular, the property now generally referred to as “pseudoconvexity” originates with Hartogs, and even today it still is one of the richest sources of intriguing phenomena and deep questions in complex analysis. (See, for example, the remarks at the end of §2.8.) We will say more about this in Chapter VII.

In this chapter we collect the technical tools from the calculus of differential forms and from complex differential geometry which will be needed in the following chapters. Section 1 deals with differentiable manifolds; the principal goal here is a thorough understanding of Stokes’ Theorem in the language of differential forms. In §2 we discuss the additional structures which arise when the manifold under consideration is complex. The main topics here are the natural intrinsic complex structure on the (real) tangent space of a complex manifold M, the direct sum decomposition of the algebra of complex valued differential forms into forms of type (p, q), 0 ≤ p, q ≤ dim_ℂ M, and the Cauchy-Riemann complex with its associated ∂̅-cohomology groups. In §3 we discuss the elementary aspects of Riemannian geometry in ℂ ⁿ in complex form. Of major importance for our purposes are the inner product of differential forms defined by integration over regions in ℂ ⁿ , the Hodge *- operator, which allows us to freely go back and forth between the geometric inner product and the algebraic wedge product of forms, the various formulas for integration by parts, and the natural differential operators associated to the Cauchy-Riemann operator, i.e., the (formal) adjoint ϑof ∂̅ and the complex Laplacian \( \square = \vartheta \overline \partial + \overline \partial \vartheta . \). In this paragraph, which is more computational than the preceding ones, we consider only the case of ℂ ⁿ rather than general Hermitian manifolds; not only does this simplify matters quite a bit, but it allows us to state certain basic formulas in exact form without having to introduce the numerous error terms which occur in the general setting.

In this chapter we develop the basic machinery of integral representations of functions and differential forms in ℂ ⁿ as it relates to the Cauchy-Riemann operator. These representations have their roots in potential theory, the link being the relationship between the complex Laplacian □ and the ordinary Laplacian Δ established in Chapter III, §3.6.

In the preceding chapter we developed a general integral representation formula for differential forms, and we saw, in the case of convex domains, some of its major applications whenever there is a generating form which is globally holomorphic in the parameter z. In this chapter we apply these techniques to a strictly pseudoconvex domain D. Here the geometric information is only local, and there is no simple way to find a globally holomorphic generating form.

In §1 of this chapter we first extend the fundamental vanishing theorem \( H_{\overline \partial }^q \left( K \right) = 0\,for\,q \geqslant 1 \) for q ≥ 1 on a Stein compactum K proved in Corollary V.2.6 to arbitrary open Stein domains (Theorem 1.4). The proof involves an approximation theorem for holomorphic functions on compact analytic polyhedra which is of independent interest, and which generalizes the classical Runge Approximation Theorem in the complex plane. We also discuss several variations of this approximation theorem. In particular we consider the Runge property for the exhaustion of a pseudoconvex domain D by strictly pseudoconvex domains which arises from the existence of a strictly plurisubharmonic exhaustion function on D. Together with the results of Chapter V this yields the solution of the Levi problem for arbitrary pseudoconvex domains. In §2 we apply these methods to solve the Cauchy-Riemann equations directly on a pseudoconvex domain D, i.e., we show that \( H_{\overline \partial }^q \left( D \right) = 0\,for\,q \geqslant 1 \) for q ≥ 1, and we prove that this property characterizes Stein domains. §3 deals with some topological properties of Stein domains D, for example, we show that if D ⊂ ℂ ⁿ , then H ^r (D, ℂ) = 0 for r > n. This section may be skipped without loss of continuity. Finally, in §4 and §5, the vanishing of \( H_{\overline \partial }^1 \) for Stein domains D is used to generalize to several variables the classical theorems of Mittag-Leffler and Weierstrass on the existence of global meromorphic functions with prescribed poles and zero sets on regions in the complex plane. §5 includes a detailed discussion of the new—strictly higher dimensional—phenomenon of a topological obstruction in the analog of the Weierstrass theorem.

Every domain in the complex plane with C ² boundary is strictly (Levi) pseudoconvex. In this chapter we generalize several classical function theoretic results from planar domains to strictly pseudoconvex domains in ℂ ⁿ . In contrast to the results on arbitrary domains of holomorphy discussed in Chapter VI, the emphasis here will be on the behavior of holomorphic functions and other analytic objects up to the boundary of the domain. In somewhat more detail, we will present the construction and basic properties of two analogues of the Cauchy kernel for a strictly pseudoconvex domain D, and of a solution operator for ∂̅ on D with L ^P estimates for 1 ≤ p ≤ ∞. Moreover, we will discuss applications of these results to uniform and L ^p approximation by holomorphic functions and to ideals in the algebra A(D) of holomorphic functions with continuous boundary values. The highlight will be a regularity theorem for the Bergman projection based on a rather explicit representation of the abstract Bergman kernel, and its application to the study of boundary regularity of biholomorphic maps.