On q -Runge pairs

Mihnea Colţoiu

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 2, page 231-235
  • ISSN: 0391-173X

Abstract

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We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a 4 -complete open subset D ⊂ ℂ 6 (which is an analytic complement in the unit ball) such that the restriction map H 3 ( ℂ 6 , ℱ ) → H 3 ( D , ℱ ) has a dense image for every ℱ ∈ C o h ( ℂ 6 ) but the pair ( D , ℂ 6 ) is not a 4 -Runge pair.

How to cite

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Colţoiu, Mihnea. "On $q$-Runge pairs." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.2 (2003): 231-235. <http://eudml.org/doc/84501>.

@article{Colţoiu2003,
abstract = {We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a $4$-complete open subset $D \subset \mathbb \{C\}^6$ (which is an analytic complement in the unit ball) such that the restriction map $H^3(\mathbb \{C\}^6,\mathcal \{F\}) \rightarrow H^3(D, \mathcal \{F\})$ has a dense image for every $\mathcal \{F\} \in Coh(\mathbb \{C\}^6)$ but the pair $(D, \mathbb \{C\}^6)$ is not a $4$-Runge pair.},
author = {Colţoiu, Mihnea},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {231-235},
publisher = {Scuola normale superiore},
title = {On $q$-Runge pairs},
url = {http://eudml.org/doc/84501},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Colţoiu, Mihnea
TI - On $q$-Runge pairs
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 2
SP - 231
EP - 235
AB - We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a $4$-complete open subset $D \subset \mathbb {C}^6$ (which is an analytic complement in the unit ball) such that the restriction map $H^3(\mathbb {C}^6,\mathcal {F}) \rightarrow H^3(D, \mathcal {F})$ has a dense image for every $\mathcal {F} \in Coh(\mathbb {C}^6)$ but the pair $(D, \mathbb {C}^6)$ is not a $4$-Runge pair.
LA - eng
UR - http://eudml.org/doc/84501
ER -

References

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  1. [A-G] A. Andreotti – H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259. Zbl0106.05501MR150342
  2. [Ba] W. Barth, Lokale Cohomologie bei isolierten Singulatitäten analytischer Mengen, Schriftenreihe Math. Inst. Univ. Münster (2) 5 (1971), 59 pp. Zbl0231.32005MR302936
  3. [Col] M. Colţoiu, q -convexity. A survey, In: “Complex analysis and geometry XII”, Pitman Research Notes in Math. 366 (1987), 83-93. Zbl0883.32016MR1477441
  4. [Col1] M. Colţoiu, On the relative homology of q -Runge pairs, Ark. Math. 38 (2000), 45-52. Zbl1039.32016MR1749357
  5. [Co-Sil] M. Colţoiu – A. Silva, Behnke-Stein theorem on complex spaces with singularities, Nagoya Math. J. 137 (1995), 183-194. Zbl0820.32007MR1324548
  6. [Fo] G. Forni, Homology and cohomology with compact supports for q -convex spaces, Ann. Mat. Pura Appl. (4) 159 (1991), 229-254. Zbl0758.32008MR1145099
  7. [Gra] H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460-472. Zbl0108.07804MR98847
  8. [Gre] M. Greenberg, “Lectures on algebraic topology”, W.A. Benjamin, New-York, 1967. Zbl0169.54403MR215295
  9. [Pr] D. Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375-386. Zbl0179.12301MR210944
  10. [So] G. Sorani, Homologie des q -paires de Runge, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17 (1963), 319-332. Zbl0126.09701MR165143
  11. [Va] V. Vajaitu, Approximations theorems and homology of q -Runge domains, J. Reine Angew. Math. 449 (1994), 179-199. Zbl0795.32004MR1268585

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