Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces

TL;DR

This paper extends the Relative Riemann-Hilbert and Newlander-Nirenberg theorems to torsion-free sheaves on complex analytic spaces, establishing correspondences with local systems and representations.

Contribution

It generalizes classical theorems to torsion-free sheaves on maximal and homogeneous spaces, connecting flat connections with local systems and representations.

Findings

Relative Riemann-Hilbert theorem holds up to torsion for certain morphisms.

Generalized $ar{ abla}$-operators correspond to relative complex analytic connections.

On maximal and homogeneous spaces, flat connections correspond to local systems and fundamental group representations.

Abstract

In this paper it is shown that for locally trivial complex analytic morphisms between some reduced spaces the Relative Riemann-Hilbert Theorem still holds up to torsion, i.e. tame flat relative connections on torsion-free sheaves are in 1-to-1 correspondence with torsion-free relative local systems. Subsequently, it is shown that generalised $\bar{\partial}$-operators on real analytic sheaves over complex analytic spaces can be viewed as relative complex analytic connections on the complexification of the underlying real analytic space with respect to a canonical morphism. By means of complexification, the Relative Riemann-Hilbert Theorem then yields a Newlander-Nirenberg type theorem for $\bar{\partial}$-operators on torsion-free real analytic sheaves over some complex analytic varieties. In the non-relative case, this result shows that on all maximal and homogeneous analytic spaces tame flat analytic connections are in 1-to-1 correspondence with local systems, which in turn are in 1-to-1 correspondence with linear representations of the fundamental group assuming connectedness.