Local exactness in a class of differential complexes
Sagun Chanillo, François Trèves · Journal of the American Mathematical Society · 1997
The article studies the local exactness at level q q ( 1 ≤ q ≤ n ) (1\le q\le n) in the differential complex defined by n n commuting, linearly independent real-analytic complex vector fields L 1 , … , L n L_1,\dotsc ,L_n in n + 1 n+1 independent variables. Locally the system { L 1 , … , L n } \{L_1,\dotsc ,L_n\} admits a first integral Z Z , i.e., a C ω \mathcal {C}^\omega complex function Z Z such that L 1 Z = ⋯ = L n Z = 0 L_1Z=\cdots =L_nZ=0 and d Z ≠ 0 dZ e 0 . The germs of the “level sets” of Z Z , the sets