CONTENTS Introduction...................................................................................................................................................................5 1. Notation and some definitions...................................................................................................................................6 (a) Analytic families.....................................................................................................................................................6 (b) Coordinate neighbourhoods and the normal bundle of a submanifold..................................................................7 (c) A characteristic system for an analytic family.........................................................................................................9 (d) Mutual positions of submanifolds.........................................................................................................................10 2. Sufficient conditions for triviality of the normal bundle.............................................................................................11 (a) Formulations of the conditions.............................................................................................................................11 (b) Connection of condition $(**)_{[U_i]}$ with the normal bundle.............................................................................12 (c) Construction and some properties of functions $f_i$...........................................................................................14 (d) Proof of triviality of the normal bundle.................................................................................................................16 (e) Formulation of the results of Section 2................................................................................................................19 3. A maximum principle for 0-cochains of holomorphic functions on a connected compact complex manifold.............20 4. Construction of an analytic family............................................................................................................................21 (a) Formulation of the main theorem.........................................................................................................................21 (b) Construction of first coefficients of formal power series.......................................................................................23 (c) Some lemmas......................................................................................................................................................25 (d) The induction step - construction of the coefficients $φ_{i|ν+1}$........................................................................28 (e) Proof of the convergence of the formal power series..........................................................................................31 (0 Proof of the main theorem....................................................................................................................................33 Appendix....... .............................................................................................................................................................36 (a) Formulation of sufficient conditions for the convergence of formal power series.................................................36 (b) Some estimates...................................................................................................................................................38 (c) Proof of Proposition A.1.......................................................................................................................................42 References.................................................................................................................................................................44
The purpose of this paper is to take a closer look at uniform semi-global (i.e. on compact subsets) holomorphic approximation of CR functions on tubular submanifolds in ℂ².
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