Holomorphic Jacobi Manifolds and Holomorphic Contact Groupoids

TL;DR

This paper proves that holomorphic Jacobi manifolds can be integrated into complex contact groupoids using a homogenization scheme that links them to homogeneous holomorphic Poisson and symplectic groupoids.

Contribution

It introduces a homogenization scheme that connects holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and proves their integration into complex contact groupoids.

Findings

Holomorphic Jacobi manifolds integrate to complex contact groupoids.

The homogenization scheme links Jacobi and Poisson structures.

Holomorphic contact groupoids correspond to homogeneous complex symplectic groupoids.

Abstract

This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to complex contact groupoids. A crucial tool in our proof is what we call the "homogenization scheme", which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids.