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Lie methods in deformation theory

Record Type:	Electronic resources : Monograph/item
Title/Author:	Lie methods in deformation theoryby Marco Manetti.
Author:	Manetti, Marco.
Published:	Singapore :Springer Nature Singapore :2022.
Description:	xii, 574 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Lie algebras.
Online resource:	https://doi.org/10.1007/978-981-19-1185-9
ISBN:	9789811911859$q(electronic bk.)

Lie methods in deformation theory
Manetti, Marco.

Lie methods in deformation theory[electronic resource] /by Marco Manetti. - Singapore :Springer Nature Singapore :2022. - xii, 574 p. :ill., digital ;24 cm. - Springer monographs in mathematics,2196-9922. - Springer monographs in mathematics..

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.

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.

ISBN: 9789811911859$q(electronic bk.)

Standard No.: 10.1007/978-981-19-1185-9doiSubjects--Topical Terms:

191078
Lie algebras.

LC Class. No.: QA252.3 / .M35 2022

Dewey Class. No.: 512.482

Lie methods in deformation theory
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505 0 $a 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.
520 $a 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.
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