Cosimplicial cohomology of restricted meromorphic functions on foliated manifolds

TL;DR

This paper develops a cosimplicial cohomology framework for restricted meromorphic functions on foliated manifolds, establishing invariants that generalize classical foliation invariants like the Godbillon–Vay invariant.

Contribution

It introduces a novel cosimplicial cohomology theory for restricted meromorphic functions on foliations, including the construction of invariants that are independent of choices and coordinate changes.

Findings

Constructed double chain-cochain complexes and coboundary operators.

Proved invariants are non-vanishing and independent of transversal basis.

Generalized the Godbillon–Vay invariant for higher codimension foliations.

Abstract

Starting from the axiomatic description of meromorphic functions with prescribed analytic properties, we introduce the cosimplicial cohomology of restricted meromorphic functions defined on foliations of smooth complex manifolds. Spaces for double chain-cochain complexes and coboundary operators are constructed. Multiplications of several restricted meromorphic functions with non-commutative parameters, as well as for elements of double complex spaces are introduced and their properties are discussed. In particular, we prove that the construction of invariants of cosimplicial cohomology of restricted meromorphic functions is non-vanishing, independent of the choice of the transversal basis for a foliation, and invariant with respect to changes of coordinates on a smooth manifold and on transversal sections. As an application, we provide an example of general cohomological invariants, in particular, generalizing the Godbillon--Vay invariant for codimension one foliations.