Lie Methods in Deformation Theory

ISBN-13: 9789811911873 / Angielski / Miękka / 2023

Marco Manetti

Opis

This book furnishes a comprehensive treatment of differential graded Lie algebras, L-infinity algebras, and their use in deformation theory. We believe it is the first textbook devoted to this subject, although the first chapters are also covered in other sources with a different perspective.

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. 

Recenzje

"The book is very clearly written and nicely structured with abstract, algebraic parts, followed by concrete applications that make it very suitable as a base for courses either in the general theory of deformation of complex manifolds or to various special aspects." (Andrei D. Halanay, Mathematical Reviews, September, 2023)

"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)

Spis treści

1. An Overview of Deformation Theory of Complex Manifolds.- 2. Lie Algebras.- 3. Functors of Artin Rings.- 4. Infinitesimal Deformations of Complex Manifolds and Vector Bundles.- 5. Differential Graded Lie Algebras.- 6. Maurer–Cartan Equation and Deligne Groupoids.- 7. Totalization and Descent of Deligne Groupoids.- 8. Deformations of Complex Manifolds and Holomorphic Maps.- 9. Poisson, Gerstenhaber and Batalin–Vilkovisky Algebras.- 10. L1-algebras.- 11. Coalgebras and Coderivations.- 12. L1-morphisms.- 13. Formal Kuranishi Families and Period Maps.- References.

O autorze

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).