Cosimplicial meromorphic functions cohomology on complex manifolds

TL;DR

This paper introduces a new cosimplicial cohomology framework for meromorphic functions on complex manifolds, linking it to Lie algebra formal series and applications in physics and geometry.

Contribution

It develops a canonical cosimplicial cohomology for meromorphic functions, relating it to sheaf cohomology and singular cohomology, with applications in multiple mathematical and physical theories.

Findings

Computed graded differential cohomology of sheaves of Lie algebras.

Established relations between cosimplicial cohomology and singular cohomology.

Proposed applications in conformal field theory, deformation theory, and foliations.

Abstract

Developing ideas of \cite{Fei}, 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 $M$. Graded differential cohomology of a sheaf of Lie algebras $\mathcal G$ via the cosimplicial cohomology of $\mathcal G$-formal series for any covering by Stein spaces on $M$ is computed. A relation between cosimplicial cohomology (on a special set of open domains of $M$) of formal series of an infinite-dimensional Lie algebra $\mathcal G$ and singular cohomology of auxiliary manifold associated to a $\mathcal G$-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.