Purely complex local Lie algebras (Q750590)

Let M be a complex manifold, L be a complex line bundle over M, and \(\Gamma\) (L) be the vector space of its smooth sections. A Lie algebra structure on \(\Gamma\) (L) is called local if supp([f,g])\(\subset \sup p(f)\cap \sup p(g)\), for any two sections f,g of L. To such a structure one can associate an integrable distribution \(P\subset TM\otimes C\). The structure is called purely complex if P(x) has constant dimension, \(P\subset T''M\), and \(P+\bar P\) is integrable; if \(P=T''M\), the local Lie algebra structure on \(\Gamma\) (L) is called transitive. The aim of this note is to announce some results concerning the classification of transitive local Lie algebra structures on \(\Gamma\) (L). The real analog of this problem was studied earlier by \textit{A. A. Kirillov} [Usp. Mat. Nauk 31, No.4(190), 57-76 (1976; Zbl 0352.58014)] and by \textit{F. Guedira} and \textit{A. Lichnerowicz} [J. Math. Pures Appl., IX. Sér. 63, 407-484 (1984; Zbl 0562.53029)].