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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Local exactness
in a class of differential complexes


Authors: Sagun Chanillo and François Treves
Journal: J. Amer. Math. Soc. 10 (1997), 393-426
MSC (1991): Primary 35A07, 35F05
MathSciNet review: 1423030
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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
Email: chanillo@math.rutgers.edu

François Treves
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Email: treves@math.rutgers.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-97-00231-2
PII: S 0894-0347(97)00231-2
Keywords: Differential complex, local solvability, singular homology, subanalytic sets
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
Article copyright: © Copyright 1997 American Mathematical Society