Analytic and holomorphic structures in Lie groupoids, algebroids, and Poisson geometry

This thesis explores analytic and holomorphic structures in Lie groupoids, Lie algebroids, and Poisson geometry. The first chapter provides background material on Lie groupoids, Lie algebroids, and Poisson structures. In the second chapter, we prove that any analytic s-proper Lie groupoid is analytically linearizable. Our approach relies on constructing an analytic Haar density and an analytic 2-metric on groupoids. The third chapter introduces the notion of complexification for analytic Lie algebroids. We establish that when the anchor map is injective or surjective, the integrability of the original algebroid implies the integrability of its complexification. Moreover, if the analytic algebroid is of s-proper type, its complexification is locally integrable. In the fourth chapter, we develop a computer program for computing the holomorphic Poisson cohomology of projective spaces.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Jiang, Ning

Contributors dc:contributor

• Fernandes, Rui Loja
• Albin, Pierre
• Lerman, Eugene
• Tolman, Susan

• Poisson Geometry
• Complex Geometry
• Lie Groupoid
• Lie Algebroid

Statement dc:rights

• Copyright 2025 Ning Jiang

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129932

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129932