Overview
- Combines a rigorous introduction to LCK geometry with an extensive survey of the most recent research results
- Emphasizes connections to algebraic geometry, topology, and complex analysis
- Discusses over 100 open problems in the area to inspire future research
Part of the book series: Progress in Mathematics (PM, volume 354)
Buy print copy
Tax calculation will be finalised at checkout
Keywords
- Locally conformally Kähler manifold
- Kähler geometry
- Kähler metric
- Stein space
- Vaisman manifold
- Gauduchon metric
- Ample bundle
About this book
This monograph introduces readers to locally conformally Kähler (LCK) geometry and provides an extensive overview of the most current results. A rapidly developing area in complex geometry dealing with non-Kähler manifolds, LCK geometry has strong links to many other areas of mathematics, including algebraic geometry, topology, and complex analysis. The authors emphasize these connections to create a unified and rigorous treatment of the subject suitable for both students and researchers.
Part I builds the necessary foundations for those approaching LCK geometry for the first time with full, mostly self-contained proofs and also covers material often omitted from textbooks, such as contact and Sasakian geometry, orbifolds, Ehresmann connections, and foliation theory. More advanced topics are then treated in Part II, including non-Kähler elliptic surfaces, cohomology of holomorphic vector bundles on Hopf manifolds, Kuranishi and Teichmüller spaces for LCK manifolds with potential, and harmonic forms on Sasakian and Vaisman manifolds. Each chapter in Parts I and II begins with motivation and historic context for the topics explored and includes numerous exercises for further exploration of important topics.
Part III surveys the current research on LCK geometry, describing advances on topics such as automorphism groups on LCK manifolds, twisted Hamiltonian actions and LCK reduction, Einstein-Weyl manifolds and the Futaki invariant, and LCK geometry on nilmanifolds and on solvmanifolds. New proofs of many results are given using the methods developed earlier in the text. The text then concludes with a chapter that gathers over 100 open problems, with context and remarks provided where possible, to inspire future research.
Authors and Affiliations
Bibliographic Information
Book Title: Principles of Locally Conformally Kähler Geometry
Authors: Liviu Ornea, Misha Verbitsky
Series Title: Progress in Mathematics
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
Hardcover ISBN: 978-3-031-58119-9Due: 07 June 2024
Softcover ISBN: 978-3-031-58122-9Due: 07 June 2024
eBook ISBN: 978-3-031-58120-5Due: 07 June 2024
Series ISSN: 0743-1643
Series E-ISSN: 2296-505X
Edition Number: 1
Number of Pages: XXI, 736