We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a -complete open subset (which is an analytic complement in the unit ball) such that the restriction map has a dense image for every but the pair is not a -Runge pair.
@article{ASNSP_2003_5_2_2_231_0, author = {Col\c{t}oiu, Mihnea}, title = {On $q${-Runge} pairs}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {231--235}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {2}, year = {2003}, mrnumber = {2004963}, zbl = {1170.32308}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_2_231_0/} }
Colţoiu, Mihnea. On $q$-Runge pairs. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 2, pp. 231-235. http://www.numdam.org/item/ASNSP_2003_5_2_2_231_0/
[A-G] Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259. | Numdam | MR | Zbl
- ,[Ba] Lokale Cohomologie bei isolierten Singulatitäten analytischer Mengen, Schriftenreihe Math. Inst. Univ. Münster (2) 5 (1971), 59 pp. | MR | Zbl
,[Col] -convexity. A survey, In: “Complex analysis and geometry XII”, Pitman Research Notes in Math. 366 (1987), 83-93. | MR | Zbl
,[Col1] On the relative homology of -Runge pairs, Ark. Math. 38 (2000), 45-52. | MR | Zbl
,[Co-Sil] Behnke-Stein theorem on complex spaces with singularities, Nagoya Math. J. 137 (1995), 183-194. | MR | Zbl
- ,[Fo] Homology and cohomology with compact supports for -convex spaces, Ann. Mat. Pura Appl. (4) 159 (1991), 229-254. | MR | Zbl
,[Gra] On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460-472. | MR | Zbl
,[Gre] “Lectures on algebraic topology”, W.A. Benjamin, New-York, 1967. | MR | Zbl
,[Pr] Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375-386. | MR | Zbl
,[So] Homologie des -paires de Runge, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17 (1963), 319-332. | Numdam | MR | Zbl
,[Va] Approximations theorems and homology of -Runge domains, J. Reine Angew. Math. 449 (1994), 179-199. | MR | Zbl
,