The Poincaré lemma and local embeddability

Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 393-398
  • ISSN: 0392-4033

Abstract

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For pseudocomplex abstract C R manifolds, the validity of the Poincaré Lemma for 0 , 1 forms implies local embeddability in C N . The two properties are equivalent for hypersurfaces of real dimension 5 . As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for 0 , 1 forms for a large class of abstract C R manifolds of C R codimension larger than one.

How to cite

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Brinkschulte, Judith, Hill, C. Denson, and Nacinovich, Mauro. "The Poincaré lemma and local embeddability." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 393-398. <http://eudml.org/doc/195395>.

@article{Brinkschulte2003,
abstract = {For pseudocomplex abstract $CR$ manifolds, the validity of the Poincaré Lemma for $(0,1)$ forms implies local embeddability in $\mathbb\{C\}^\{N\}$. The two properties are equivalent for hypersurfaces of real dimension $\geq 5$. As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for $(0,1)$ forms for a large class of abstract $CR$ manifolds of $CR$ codimension larger than one.},
author = {Brinkschulte, Judith, Hill, C. Denson, Nacinovich, Mauro},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {393-398},
publisher = {Unione Matematica Italiana},
title = {The Poincaré lemma and local embeddability},
url = {http://eudml.org/doc/195395},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Brinkschulte, Judith
AU - Hill, C. Denson
AU - Nacinovich, Mauro
TI - The Poincaré lemma and local embeddability
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 393
EP - 398
AB - For pseudocomplex abstract $CR$ manifolds, the validity of the Poincaré Lemma for $(0,1)$ forms implies local embeddability in $\mathbb{C}^{N}$. The two properties are equivalent for hypersurfaces of real dimension $\geq 5$. As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for $(0,1)$ forms for a large class of abstract $CR$ manifolds of $CR$ codimension larger than one.
LA - eng
UR - http://eudml.org/doc/195395
ER -

References

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  2. ANDREOTTI, A.- FREDRICKS, G.- NACINOVICH, M., On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Sc. Norm. Sup. Pisa, 8 (1981), 365-404. Zbl0482.35061MR634855
  3. ANDREOTTI, A.- HILL, C. D., E.E. Levi convexity and the Hans Lewy problem I, II, Ann. Sc. norm. sup. Pisa, 26 (1972), 325-363, 747-806. Zbl0283.32013
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  5. HILL, C. D.- NACINOVICH, M., A weak pseudoconcavity condition for abstract almost C R manifolds, Invent. Math., 142 (2000), 251-283. Zbl0973.32018MR1794063
  6. HILL, C. D.- NACINOVICH, M., On the failure of the Poincaré lemma for the ¯ M - complex, Quaderni sez. Geometria Dip. Matematica Pisa, 1.260.1329 (2001), 1-10. 
  7. HÖRMANDER, L., Linear Partial Differential Operators, Springer, Berlin (1963). MR404822
  8. JACOBOWITZ, H.- TREVES, F., Aberrant C R structures, Hokkaido Math. Jour., 12 (1983), 276-292. Zbl0564.32010MR719968
  9. KURANISHI, M., Strongly pseudoconvex CR structures over small balls I-III, Ann. of Math., 115, 116 (1982), 451-500, 1-64, 249-330. Zbl0576.32033MR662117
  10. NACINOVICH, M., On the absence of Poincaré lemma for some systems of partial differential equations, Compos. Math., 44 (1981), 241-303. Zbl0487.58026MR662464
  11. NIRENBERG, L., On a problem of Hans Lewy, Uspeki Math. Naut, 292 (1974), 241-251. Zbl0305.35017MR492752
  12. ROSSI, H., LeBrun's nonrealizability theory in higher dimensions, Duke Math. J., 52 (1985), 457-474. Zbl0573.32018MR792182
  13. WEBSTER, S., On the proof of Kuranishi's embedding theorem, Ann. Inst. H. Poincaré, 9 (1989), 183-207. Zbl0679.32020MR995504

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