Cosimplicial meromorphic functions cohomology on complex manifolds

Abstract

Developing ideas of [B. L. Feigin, Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, in Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vols. 1 and 2 (Mathematical Society of Japan, Tokyo, 1991), pp. 71–85], we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold [Formula: see text]. Graded differential cohomology of a sheaf of Lie algebras [Formula: see text] via the cosimplicial cohomology of [Formula: see text]-formal series for any covering by Stein spaces on [Formula: see text] is computed. A relation between cosimplicial cohomology (on a special set of open domains of [Formula: see text]) of formal series of an infinite-dimensional Lie algebra [Formula: see text] and singular cohomology of auxiliary manifold associated to a [Formula: see text]-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.