Complex symplectic geometry and solvability of systems of analytic vector fields

Abstract

This note reviews briefly the classes of involutive systems $${\varvec{L}} =\left( L_{1},\ldots ,L_{\nu }\right)$$ of analytic vector fields for which necessary and sufficient conditions for the local solvability, or local exactness in a given degree of the associated differential complex, are known. We point out that these conditions, at the microlocal level, can be reformulated in terms of a locally exact $$\left( 1,0\right)$$ -form on the $$complex$$ characteristic bundle of $${\varvec{L}}$$ and of the primitive of its imaginary part. The results in the known cases raise natural questions about their generalization.