## Local exactness in a class of differential complexes

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- by Sagun Chanillo and François Treves
- J. Amer. Math. Soc.
**10**(1997), 393-426 - DOI: https://doi.org/10.1090/S0894-0347-97-00231-2
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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.## References

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## Bibliographic 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