Lie Methods in Deformation Theory (Springer Monographs in Mathematics)

✍ Scribed by Marco Manetti

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 576
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Contents
1 An Overview of Deformation Theory of Complex Manifolds
1.1 Proper Smooth Families of Complex Manifolds
1.2 A Short Review of Čech and Dolbeault Cohomology
1.3 Locally Trivial Families
1.4 The Kodaira–Spencer Map
1.5 Deformations over Smooth Bases
1.6 Deformations over Singular Bases
1.7 The Completeness Theorem of Kuranishi
1.8 Infinitesimal Deformations
1.9 Exercises
2 Lie Algebras
2.1 Magmatic Algebras
2.2 Lie and Pre-Lie Algebras
2.3 Differential Operators and Derivations of Pairs
2.4 Free Lie Algebras
2.5 The Baker–Campbell–Hausdorff Product
2.6 Semicosimplicial Lie Algebras
2.7 Exercises
3 Functors of Artin Rings
3.1 Artin Rings and Small Extensions
3.2 Deformation Functors
3.3 Pro-representable Functors
3.4 Automorphisms and Exponential Functors
3.5 Tangent Space and Schlessinger's Theorem
3.6 Obstruction Theory
3.7 Deformation Functors Associated to Semicosimplicial Lie Algebras
3.8 Existence of Universal Obstruction Theories
3.9 Exercises
4 Infinitesimal Deformations of Complex Manifolds and Vector Bundles
4.1 Flat Modules over Artin Local Rings
4.2 Infinitesimal Deformations of Vector Bundles
4.3 Infinitesimal Deformations of Complex Manifolds
4.4 Deformations of Pairs (Manifold, Submanifold)
4.5 Deformations of Pairs (Manifold, Vector Bundle)
4.6 Exercises
5 Differential Graded Lie Algebras
5.1 DG-Vector Spaces
5.2 Suspension and Mapping Cones
5.3 Filtered DG-Vector Spaces
5.4 Basic Homological Perturbation Theory
5.5 Differential Graded Commutative Algebras
5.6 Differential Graded Lie Algebras
5.7 Examples of DG-Lie Algebras via Derived Brackets
5.8 Examples of DG-Lie Algebras via Graded Pre-Lie Algebras
5.9 Exercises
6 Maurer–Cartan Equation and Deligne Groupoids
6.1 Factorization and Homotopy Fibres of DG-Lie Morphisms
6.2 Formal and Homotopy Abelian DG-Lie Algebras
6.3 Maurer–Cartan Equation and Gauge Action
6.4 Deformation Functors Associated to a Differential Graded Lie Algebra
6.5 Deligne Groupoids
6.6 Homotopy Invariance of Deformation Functors and Deligne Groupoids
6.7 Further Examples of Nonformal DG-Lie Algebras
6.8 Exercises
7 Totalization and Descent of Deligne Groupoids
7.1 Simplicial and Cosimplicial Objects
7.2 Kan Complexes
7.3 Differential Forms on Standard Simplices
7.4 Cochains and Totalization of Semicosimplicial DG-Vector Spaces
7.5 Semicosimplicial Groupoids
7.6 Descent of Deligne Groupoids
7.7 Homotopy Operators and Elementary Forms
7.8 Cosimplicial Totalization and Cup Product
7.9 Exercises
8 Deformations of Complex Manifolds and Holomorphic Maps
8.1 Embedded Deformations of Submanifolds
8.2 Deformations of Holomorphic Maps
8.2.1 Deformations of Holomorphic Maps Between Trivial Families
8.2.2 Unrestricted Deformations of Holomorphic Maps
8.2.3 The Stein Case
8.2.4 The General Case
8.2.5 Horikawa's Theorems
8.3 The Kodaira–Spencer Algebra
8.4 Deformations of Products and Examples of Obstructed Manifolds
8.4.1 Two Examples of Obstructed Manifolds
8.5 Holomorphic Cartan Homotopy Formulas
8.5.1 Inner Products
8.5.2 Holomorphic Cartan Homotopy Formulas
8.6 Ambient Cohomology Annihilates Obstructions
8.7 Semi-Regularity Maps
8.8 Cartan Homotopies and the Abstract BTT Theorem
8.9 Exercises
9 Poisson, Gerstenhaber and Batalin–Vilkovisky Algebras
9.1 Graded Poisson and Gerstenhaber Algebras
9.2 The Koszul–Tian–Todorov Lemma
9.3 Koszul Braces and Differential Operators
9.4 Batalin–Vilkovisky Algebras
9.5 Poisson and Symplectic Manifolds
9.6 Holomorphic Poisson and Symplectic Manifolds
9.7 Exercises
10 Linfty-Algebras
10.1 Symmetric Powers and Koszul Sign
10.2 Formal Neighbourhoods of Graded Vector Spaces
10.3 A Simple Model of Infinity Structure
10.4 Linfty-Algebras
10.5 Maurer–Cartan and Deformation Functors
10.6 Décalage Isomorphisms and Linfty[1]-Algebras
10.7 Derived Brackets
10.8 Exercises
11 Coalgebras and Coderivations
11.1 Graded Coalgebras
11.2 Comodules and Coderivations
11.3 The Reduced Tensor Coalgebra
11.4 Symmetrization
11.5 The Reduced Symmetric Coalgebra
11.6 Scalar Extension and Restitution
11.7 Symmetric Coalgebras and Their Coderivations
11.8 Exercises
12 Linfty-Morphisms
12.1 Formal Pointed DG-Manifolds
12.2 Linfty-Morphisms of Linfty[1]-Algebras
12.3 Linfty-Morphisms of Linfty and DG-Lie Algebras
12.4 Transferring Linfty[1] Structures
12.5 Homotopy Classification of Linfty and Linfty[1]-Algebras
12.6 Homotopy Classification of DG-Lie Algebras
12.7 Homotopy Abelian DG-Lie and Linfty-Algebras
12.8 Exercises
13 Formal Kuranishi Families and Period Maps
13.1 Linfty-Morphisms and Deformation Functors
13.2 Formal Kuranishi Families
13.3 Cartan Homotopies and Linfty-Morphisms
13.4 Formal Pointed Grassmann Functors
13.5 Formal Period Maps
13.6 Period Data of Differential Graded BV-Algebras
13.7 Toward an Algebraic Proof of the Kodaira Principle
13.8 Exercises
14 Tree Summation Formulas
14.1 Rooted Trees and Forests
14.2 A Tree Summation Formula for the BCH Product
14.3 Automorphisms of barTc(V), barSc(V) and Inversion Formulas
14.4 Tree Summation Formula for Homotopy Transfer
14.5 An Example: Linfty[1] Structures on Mapping cones
14.6 Exercises
Appendix A Topics in the Theory of Analytic Algebras
A.1 Analytic Algebras
A.2 Weierstrass Polynomials
A.3 Smooth Algebras
A.4 The Holomorphic Curve Selection Lemma
A.5 Dimension Bounds
A.6 Artin's Theorem on the Solution of Analytic Equations
A.7 Exercises
Appendix B Special Obstructions and T1-Lifting
B.1 The Approximation Theorem
B.2 Curvilinear Obstructions
B.3 Primary and Semitrivial Obstructions
B.4 The Ran–Kawamata T1-Lifting Theorem
B.5 Exercises
Appendix C Kähler Manifolds
C.1 Alternating Forms on Hermitian Vector Spaces
C.2 The Lefschetz Decomposition
C.3 Kähler Identities and Harmonic Forms
C.4 Hodge Decomposition in Compact Hermitian Manifolds
C.5 Compact Kähler Manifolds
C.6 Degeneration of Spectral Sequences
C.7 Exercises
Appendix References
Index