Local exactness in a class of differential complexes

Chanillo, Sagun¹ ; Treves, François¹

¹ Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Journal of the American Mathematical Society, Tome 10 (1997) no. 2, pp. 393-426

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-97-00231-2

Bibliographie

[1] Baouendi, M. S., Trã¨Ves, F. A property of the functions and distributions annihilated by a locally integrable system of complex vector fields Ann. of Math. (2) 1981 387 421

[2] Chanillo, Sagun The first eigenvalue of analytic level surfaces on spheres Math. Res. Lett. 1994 159 166

[3] Cordaro, Paulo, Hounie, Jorge On local solvability of underdetermined systems of vector fields Amer. J. Math. 1990 243 270

[4] Cordaro, Paulo, Trã¨Ves, Franã§Ois Homology and cohomology in hypo-analytic structures of the hypersurface type J. Geom. Anal. 1991 39 70

[5] Cordaro, Paulo D., Trã¨Ves, Franã§Ois Hyperfunctions on hypo-analytic manifolds 1994

[6] Cordaro, Paulo D., Trã¨Ves, Franã§Ois Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields Invent. Math. 1995 339 360

[7] Hardt, Robert M. Slicing and intersection theory for chains associated with real analytic varieties Acta Math. 1972 75 136

[8] Hardt, Robert M. Triangulation of subanalytic sets and proper light subanalytic maps Invent. Math. 1976/77 207 217

[9] Hã¶Rmander, Lars Propagation of singularities and semiglobal existence theorems for (pseudo)differential operators of principal type Ann. of Math. (2) 1978 569 609

[10] Hã¶Rmander, Lars Pseudodifferential operators of principal type 1981 69 96

[11] ŁOjasiewicz, S. Sur le problème de la division Studia Math. 1959 87 136

[12] Mendoza, G. A., Trã¨Ves, F. Local solvability in a class of overdetermined systems of linear PDE Duke Math. J. 1991 355 377

[13] Nirenberg, L., Treves, F. Solvability of a first order linear partial differential equation Comm. Pure Appl. Math. 1963 331 351

[14] Séminaire Schwartz. 4e année, 1959/60. Unicité du problème de Cauchy. Division des distributions 1960

[15] Sussmann, H. J. Real analytic desingularization and subanalytic sets: an elementary approach Trans. Amer. Math. Soc. 1990 417 461

[16] Treves, Franã§Ois Study of a model in the theory of complexes of pseudodifferential operators Ann. of Math. (2) 1976 269 324

[17] Trã¨Ves, Franã§Ois On the local solvability and the local integrability of systems of vector fields Acta Math. 1983 1 38

[18] Trã¨Ves, Franã§Ois Hypo-analytic structures 1992