Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles

TL;DR

This paper extends the non-abelian Hodge correspondence to quasi-regular Sasakian manifolds, relating flat bundles and Higgs orbibundles, and characterizes numerically flat bundles in this setting.

Contribution

It introduces notions of quasi-regularity for vector bundles on Sasakian manifolds and establishes a correspondence between flat bundles and Higgs orbibundles in the quasi-regular case.

Findings

Established a canonical correspondence between semi-simple orbifold representations and Higgs orbibundles.

Extended the non-abelian Hodge correspondence to flat bundles and basic Higgs bundles under quasi-regularity.

Proved a Sasakian analogue of the Demailly-Peternell-Schneider characterization of numerically flat bundles.

Abstract

In this continuation of \cite{BK} we investigate the non-abelian Hodge correspondence on compact Sasakian manifolds with emphasis on the quasi-regular case. On quasi-regular Sasakian manifolds, we introduce the notions of quasi-regularity and regularity of basic vector bundles. These notions are useful in relating the vector bundles over a quasi-regular Sasakian manifold with the orbibundles over the orbifold defined by the orbits of the Reeb foliation of the Sasakian manifold. We note that the non-abelian Hodge correspondence on quasi-regular Sasakian manifolds gives a canonical correspondence between the semi-simple representations of the orbifold fundamental groups and the Higgs orbibundles on locally cyclic complex orbifolds admitting Hodge metrics. Under the quasi-regularity of Sasakian manifolds and vector bundles, we extend this correspondence to one between the flat bundles and the basic Higgs bundles. We also prove a Sasakian analogue of the characterization of numerically flat bundles given by Demailly, Peternell and Schneider.