Integration of Holomorphic Lie Algebroids

TL;DR

This paper establishes that the integrability of holomorphic Lie algebroids is equivalent to that of their underlying real Lie algebroids, extending existing criteria to the holomorphic setting and providing new proofs for related theorems.

Contribution

It demonstrates the equivalence of integrability conditions for holomorphic and real Lie algebroids, applying known criteria without modifications and offering alternative proofs for holomorphic Poisson manifold integrability.

Findings

Holomorphic Lie algebroid integrability iff real Lie algebroid is integrable

Crainic-Fernandes criteria apply in holomorphic context

Holomorphic Poisson manifold integrability characterized by real/imaginary parts

Abstract

We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic Poisson manifold is integrable if, and only if, its real (or imaginary) part is integrable as a real Poisson manifold.