Overview
- Introduces differential graded Lie algebras, L-infinity algebras, and their homotopy classification in detail
- Applies the differential graded Lie approach to deformation theory of complex manifolds—the first book to do so
- Includes practice exercises at the end of every chapter
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (14 chapters)
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Front Matter
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Back Matter
Keywords
About this book
Deformation theory is an important subject in algebra and algebraic geometry, with an origin that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and Grothendieck. In the last 30 years, a new approach, based on ideas from rational homotopy theory, has made it possible not only to solve long-standing open problems, but also to clarify the general theory and to relate apparently different features. This approach works over a field of characteristic 0, and the central role is played by the notions of differential graded Lie algebra, L-infinity algebra, and Maurer–Cartan equations.
The book is written keeping in mind graduate students with a basic knowledge of homological algebra and complex algebraic geometry as utilized, for instance, in the book by K. Kodaira, Complex Manifolds and Deformation of Complex Structures. Although the main applications in this book concern deformation theory of complex manifolds, vector bundles, and holomorphic maps, the underlying algebraic theory also applies to a wider class of deformation problems, and it is a prerequisite for anyone interested in derived deformation theory.
Researchers in algebra, algebraic geometry, algebraic topology, deformation theory, and noncommutative geometry are the major targets for the book.
Reviews
“This book provides an accessible and self-contained approach to the field through the particular lens of Lie theoretical techniques. ... The book is introductory in its nature. ... The book is wonderfully well-written and it is always balanced. ... the book is also full of fun and relevant exercises, and the proofs are clear and concise when possible ... . The more abstract chapters are balanced out too by a wealth of examples to read along … .” (Camilo Andres Angulo Santacruz, zbMATH 1509.14001, 2023) “The book contains many aspects of deformation theory in large including Deligne groupoids, Batalin-Vilkovisky and L1-algebras. … I think that the book of Marco Manetti can be recommended to any researcher who works in these fields.” (Victor Palamodov, Tel Aviv University)
Authors and Affiliations
About the author
Professor Marco Manetti was born in 1966. He is full professor of geometry at the Sapienza University of Roma, Italy (since 2001). His research interests involve algebraic geometry, deformation theory, homotopical algebra and higher operations in geometry. He is the author of the book “Topologia”, Springer UTX (2008).
Bibliographic Information
Book Title: Lie Methods in Deformation Theory
Authors: Marco Manetti
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-981-19-1185-9
Publisher: Springer Singapore
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 Singapore Pte Ltd. 2022
Hardcover ISBN: 978-981-19-1184-2Published: 02 August 2022
Softcover ISBN: 978-981-19-1187-3Published: 03 August 2023
eBook ISBN: 978-981-19-1185-9Published: 01 August 2022
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XII, 574
Number of Illustrations: 23 b/w illustrations
Topics: Category Theory, Homological Algebra, Commutative Rings and Algebras, Differential Geometry