Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank

Giuseppe Zampieri

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 291-296
  • ISSN: 0066-2216

Abstract

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Let M be a real-analytic submanifold of ℂ n whose “microlocal” Levi form has constant rank s M + + s M - in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees s M - , s M + (and 0).  This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. s M - + s M + = n - c o d i m M ). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively. The idea of our proof, which relies on the microlocal theory of sheaves of [3], is new.

How to cite

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Giuseppe Zampieri. "Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank." Annales Polonici Mathematici 74.1 (2000): 291-296. <http://eudml.org/doc/208373>.

@article{GiuseppeZampieri2000,
abstract = {Let M be a real-analytic submanifold of $ℂ^n$ whose “microlocal” Levi form has constant rank $s^\{+\}_\{M\} + s^\{-\}_\{M\}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s^\{-\}_\{M\}$, $s^\{+\}_\{M\}$ (and 0).  This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. $s^\{-\}_\{M\} + s^\{+\}_\{M\} = n - codim M$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively. The idea of our proof, which relies on the microlocal theory of sheaves of [3], is new.},
author = {Giuseppe Zampieri},
journal = {Annales Polonici Mathematici},
keywords = {CR manifolds; ∂̅ and $∂̅^b$ problems; tangential CR complex; tangential -system; Levi form},
language = {eng},
number = {1},
pages = {291-296},
title = {Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank},
url = {http://eudml.org/doc/208373},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Giuseppe Zampieri
TI - Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 291
EP - 296
AB - Let M be a real-analytic submanifold of $ℂ^n$ whose “microlocal” Levi form has constant rank $s^{+}_{M} + s^{-}_{M}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s^{-}_{M}$, $s^{+}_{M}$ (and 0).  This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. $s^{-}_{M} + s^{+}_{M} = n - codim M$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively. The idea of our proof, which relies on the microlocal theory of sheaves of [3], is new.
LA - eng
KW - CR manifolds; ∂̅ and $∂̅^b$ problems; tangential CR complex; tangential -system; Levi form
UR - http://eudml.org/doc/208373
ER -

References

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  1. [1] A. Andreotti, G. Fredricks and M. Nacinovich, On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Scuola Norm. Sup. Pisa 8 (1981), 365-404. Zbl0482.35061
  2. [2] L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudodifferential operators, Comm. Pure Appl. Math. 27 (1974), 585-639. Zbl0294.35020
  3. [3] M. Kashiwara and P. Schapira, Microlocal theory of sheaves, Astérisque 128 (1985). 
  4. [4] C. Rea, Levi-flat submanifolds and holomorphic extension of foliations, Ann. Scuola Norm. Sup. Pisa 26 (1972), 664-681. Zbl0272.57013
  5. [5] M. Sato, M. Kashiwara and T. Kawai, Hyperfunctions and Pseudodifferential Operators, Lecture Notes in Math. 287, Springer, 1973, 265-529. 
  6. [6] G. Zampieri, Microlocal complex foliation of ℝ-Lagrangian CR submanifolds, Tsukuba J. Math. 21 (1997), 361-366. Zbl0893.32008

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