  A complex Lie algebroid is a complex vector bundle over a smooth (real)
manifold M with a bracket on sections and an anchor to the complexified tangent
bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made
here to integrate analytic complex Lie algebroids by using analytic
continuation to a complexification of M and integration to a holomorphic
groupoid. A collection of diverse examples reveal that the holomorphic stacks
presented by these groupoids tend to coincide with known objects associated to
structures in complex geometry. This suggests that the object integrating a
complex Lie algebroid should be a holomorphic stack.

Differential Geometry