Integration of holomorphic Lie algebroids

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 (Theorem 4.1 in Crainic, Fernandes in Ann Math 2:157, 2003) do also apply in the holomorphic context without any modification. As a consequence we prove that a holomorphic Poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real Poisson manifold.