Cosimplicial meromorphic functions cohomology on complex manifolds

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.