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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Local exactness in a class of differential complexes
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by Sagun Chanillo and François Treves PDF
J. Amer. Math. Soc. 10 (1997), 393-426 Request permission

Abstract:

The article studies the local exactness at level $q$ $(1\le q\le n)$ in the differential complex defined by $n$ commuting, linearly independent real-analytic complex vector fields $L_1,\dotsc ,L_n$ in $n+1$ independent variables. Locally the system $\{L_1,\dotsc ,L_n\}$ admits a first integral $Z$, i.e., a $\mathcal {C}^\omega$ complex function $Z$ such that $L_1Z=\cdots =L_nZ=0$ and $dZ\ne 0$. The germs of the “level sets” of $Z$, the sets $Z=z_0\in \mathbb {C}$, are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension $q-1$, of these level sets is sufficient for local exactness at the level $q$. The condition was already known to be necessary.
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Additional Information
  • Sagun Chanillo
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
  • MR Author ID: 47385
  • Email: chanillo@math.rutgers.edu
  • François Treves
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
  • Email: treves@math.rutgers.edu
  • Received by editor(s): May 17, 1996
  • Received by editor(s) in revised form: November 11, 1996
  • Additional Notes: The first author was partially supported by NSF Grant DMS-9401782, and the second author by NSF Grant DMS-9201980
  • © Copyright 1997 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 10 (1997), 393-426
  • MSC (1991): Primary 35A07, 35F05
  • DOI: https://doi.org/10.1090/S0894-0347-97-00231-2
  • MathSciNet review: 1423030