The traces of holomorphic functions on real submanifolds

Author:
Gary Alvin Harris

Journal:
Trans. Amer. Math. Soc. **242** (1978), 205-223

MSC:
Primary 32C05

DOI:
https://doi.org/10.1090/S0002-9947-1978-0477120-1

MathSciNet review:
0477120

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Abstract: Suppose *M* is a real-analytic submanifold of complex Euclidean *n* = space and consider the following question: Given a real-analytic function *f* defined on *M*, is *f* the restriction to *M* of an ambient holomorphic function? If *M* is a C.R. submanifold the question has been answered completely. Namely, *f* is the trace of a holomorphic function if and only if *f* is a C.R. function. The more general situation in which *M* need not be a C.R. submanifold is discussed in this paper.

A complete answer is obtained in case the dimension of *M* is larger than or equal to *n* and *M* is generic in some neighborhood of each point off its C.R. singularities. The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function *f* and a holomorphic mapping , when does there exist a holomorphic mapping *F* such that ?

**[E-H]**Paul M. Eakin and Gary A. Harris,*When 𝐹(𝑓) convergent implies 𝑓 is convergent*, Math. Ann.**229**(1977), no. 3, 201–210. MR**0444651**, https://doi.org/10.1007/BF01391465**[G-R]**H. Grauert and R. Remmert,*Analytische Stellenalgebren*, Springer-Verlag, Berlin-New York, 1971 (German). Unter Mitarbeit von O. Riemenschneider; Die Grundlehren der mathematischen Wissenschaften, Band 176. MR**0316742****[T]**Giuseppe Tomassini,*Tracce delle funzioni olomorfe sulle sottovarietà analitiche reali d’una varietà complessa*, Ann. Scuola Norm. Sup. Pisa (3)**20**(1966), 31–43 (Italian). MR**0206992**

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DOI:
https://doi.org/10.1090/S0002-9947-1978-0477120-1

Keywords:
Holomorphic trace,
real-analytic submanifold of

Article copyright:
© Copyright 1978
American Mathematical Society