×

Boundary continuity of proper holomorphic correspondences. (English) Zbl 0596.32027

Sémin. analyse P. Lelong - P. Dolbeault - H. Skoda, Années 1983/84, Lect. Notes Math. 1198, 47-64 (1986).
[For the entire collection see Zbl 0583.00011.]
If \(\Omega\) and D denote bounded domains in \({\mathbb{C}}^ n\), then a subvariety \(V\subset \Omega \times D\) is a holomorphic correspondence. It yields a set value function \(f(z)=\pi_ D(\pi_{\Omega}^{-1}(z))\). Here, \(\pi_ D\) and \(\pi_{\Omega}\) are the projections of V to D and \(\Omega\) respectively. Details may be found in K. Stein, Rocky Mountain J. Math. 2, 443-463 (1972; Zbl 0272.32001).
In the paper under review, the authors discuss boundary properties of proper holomorphic correspondences. They first show that an irreducible proper correspondence \(f: \Omega\)-\(\circ D\) yields a proper holomorphic map \(\hat f:\) \(\Omega\) \(\to D^ p_{sym}\), where \(D^ p_{sym}\) denotes the p-fold symmetric product of D. The main result then is that if \(\Omega\) has \(C^ 2\) boundary and D either has \(C^ 2\) boundary and is strictly pseudoconvex, or has real-analytic boundary and is weakly pseudoconvex, then \(f: \Omega\)-\(\circ D\) extends to a continuous holomorphic mapping \(\hat f:\) \({\bar \Omega}\to \bar D^ p_{sym}.\)
This main result is combined with some results of S. I. Pinchuk [Sib. Math. J. 15(1974), 644-649 (1975); translation from Sib. Mat. Zh. 15, 909-917 (1974; Zbl 0289.32011); Math. USSR, Sb. 27(1975), 375-392 (1977); translation from Mat. Sb., n. Ser. 98(140), 416-435 (1975; Zbl 0366.32010); Math. USSR, Sb. 34, 503-519 (1978); translation from Mat. Sb., n. Ser. 105(147), 574-593 (1978; Zbl 0389.32008)] to deduce results in the case where \(\Omega\) and D are simply connected, strongly pseudoconvex with real-analytic boundaries. In this case \(\hat f\) will actually split into a union of biholomorphic mappings. Moreover, V ”extends” to a neighborhood of \({\bar \Omega}\times \bar D\), \(\pi_{\Omega}: V\to \Omega\) and \(\pi_ D: V\to D\) are unbranched covering maps, and V is nonsingular. When moreover \(\Omega =D\), then f is an automorphism of \(\Omega\). If \(f: \Omega\) \(\to D\), D a complex space, is proper, then there is a group \(\Gamma\) \(\subset Aut (\Omega)\) such that \(f\circ g=f\) for all \(g\in \Gamma\) and \(f^{-1}(f(z))=\cup_{g\in \Gamma}g(z)\) for all \(z\in \Omega\).
Reviewer: E.Straube

MSC:

32E35 Global boundary behavior of holomorphic functions of several complex variables
32H99 Holomorphic mappings and correspondences
32H35 Proper holomorphic mappings, finiteness theorems