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

Jiang, Ning

Permalink

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

Description

Title

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

Author(s)

Jiang, Ning

Issue Date

2025-07-16

Director of Research (if dissertation) or Advisor (if thesis)

Fernandes, Rui Loja

Doctoral Committee Chair(s)

Albin, Pierre

Committee Member(s)

• Lerman, Eugene
• Tolman, Susan

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2025-08

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

Copyright 2025 Ning Jiang

Owning Collections