Geometric structures on surfaces and representations into Lie groups

2020 Fiscal Year Final Research Report

• Project/Area Number: 19K21023
• Project/Area Number (Other): 18H05833 (2018)
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund (2019); Single-year Grants (2018)
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Osaka University
• Principal Investigator: Shinpei Baba 大阪大学, 理学研究科, 准教授 (40822870)
• Project Period (FY): 2018-08-24 – 2021-03-31
• Keywords: 複素射影構造 / Teichmuller space / 双曲幾何学 / リーマン面 / character variety

Outline of Final Research Achievements

Complex projective structures are a type of geometric structures (locally homogeneous structures) on a surface, and it has been studies from various perspective. In general, it is important to analyze degeneration of geometric structures on a surface or a more general manifold, especially, in order to compactify it associated deformation space. Projective structures has been studied traditionally more from its analytic side, but holonomy representations of projective structures are algebraic objects, and it is fascinating to understand the relations between projective structures and their holonomy representations.
In this research project, I studied degeneration of projective structures when their holonomy converges. Under this setting, it is known that the underlying complex structure also generates. I characterized such degeneration of projective structure under the basic assumption that complex structures are pinched along a single loop.

Free Research Field

低次元幾何学および位相幾何学

Academic Significance and Societal Importance of the Research Achievements

幾何学を通して，代数的および解析的両面から結びつけている。