Hyperbolic Complex Spaces

Let X be a set. A pseudo-distance d on X is a function on X × X with values in the non-negative real numbers satisfying the following axioms:

In this section we prove Ahlfors’ generalization of the classical Schwarz lemma in function theory of one complex variable and its variants. For the general theory of intrinsic distances, we need only the classical Schwarz-Pick lemma in the form (2.1.7).

Throughout this section we denote the unit disc by D, its Poincaré metric by ds ², and the Poincaré distance by ρ, (see Chapter 2, Section 1).

Let X be a complex space. We denote its Carathéodory pseudo-distance by c _x , (see (3.1.1)), and the induced inner pseudo-distance by c ⁱ _x , (see (1.1.2)). While the Kobayashi pseudo-distance d _x is always inner (see (3.1.15)), the Carathéodory pseudo-distance c _x need not be (see Examples (3.1.25), (3.1.26), (3.1.27) and (3.1.28).

Let X and Y be complex spaces. Let C(X, Y) denote the family of continuous maps from X into Y with compact-open topology. Let D(X, Y) be the subfamily of distance-decreasing maps from X into Y with respect to their intrinsic pseudo-distances d _X and d _Y . Then D(X, Y) is closed in C(X, Y). The family Hol(X, Y) of holomorphic maps from X into Y is a closed subset of D(X, Y).

The classical little Picard theorem states that every entire function f missing two values must be constant. In (3.10.2) we stated E. Borel’s generalization to a system of entire functions. One of its geometric consequences is that given n + p hyperplanes H ₁,..., H _{n +p} in P _n C in general position, every \(f \in Hol(C,{P_n}C - \cup _{i = 1}^{n + p}{H_i})\) has its image in a linear subspace of dimension ≤ [n/p], see (3.10.7). For p = n + 1, this means that \(f \in Hol(C,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i})\) must be constant, (see (3.10.8)). This has been further strengthened to the statement that \({P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\) is complete hyperbolic and is hyperbolically imbedded in P _n C, (see (3.10.9)), which is a statement on Hol(\(D,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\)) rather than on Hol(\(C,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\)) since the hyperbolicity and hyperbolic imbeddedness of \(X = {P_2}C - \cup _{i = 1}^{2n + 1}{H_i}\) is defined in terms of d _X which, in turn, is constructed by Hol(\(D,{P_n}C - \cup _{i = 1}^{2n + 1}{H_i}\)). Thus, (3.10.9) is a statement en termes finis in the sense of Bloch [1].

In general, given a topological space X with a pseudo-distance d and a non-negative real number k, the k-dimensional Hausdorff measure m _k is defined as follows. For a subset E ⊂ X, we set , where δ(E _i ) denotes the diameter of E _i . If X is a complex space, then the pseudo-distances c _x and d _x induce Hausdorff measures on X. Since every holomorphic map is distance-decreasing with respect to these intrinsic pseudo-distances, it is also measure-decreasing with respect to the Hausdorff measures they define. There are other intrinsic measures on complex spaces. For a systematic study of intrinsic measures on complex manifolds, see Eisenman [1]. In Section 2 we shall discuss the intrinsic mesaures which may be considered as direct generalizations of c _x and d _x In this section we discuss their infinitesimal forms.

We fix n, and for each k, 0 ≤ k ≤ n, consider ∧ ^{k +1} C ^{n +1}. Set so that ∧ ^{k +1} C ^{n +1} ≃ C ^{n (k)+1} Let G (n, k) be the Grassmannian of k-planes in P _n C, i.e., the Grassmannian of (k + 1)-dimensional subspaces in C ^{n +1} Then dim G (n, k) = (n − k)(k + 1). To a (k + 1)-dimensional subspace spanned by a ₀, …, a _k ∈ C ^{n +1}, we assign a decomposable (k + 1)-vector A = a ₀ ∧ … ∧ a _k ∈ ∧ ^{k +1} C ^{n +1} which is determined, up to a constant factor, by the subspace. Conversely, each decomposable (k + 1)-vector A determines a k-plane in P _n C, i.e., a (k + 1)-dimensional vector subspace of C ^{n +1} both of which will be denoted by the same symbol [A]. This correspondence defines the Plücker imbedding