Analytic and holomorphic structures in Lie groupoids, algebroids, and Poisson geometry
Jiang, Ning
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https://hdl.handle.net/2142/129932
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
Language
eng
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.
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