A remark on the relative Lie algebroid connections and their moduli spaces

Abstract

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space. © 2026 World Scientific Publishing Company.