Global logarithmic deformation theory / Simon Felten.

Format:

Book

Author/Creator:

Felten, Simon, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 v. 2373.

Lecture notes in mathematics, 0075-8434 ; volume 2373

Language:

English

Subjects (All):

Geometry, Algebraic.

Calabi-Yau manifolds.

Physical Description:

xlviii, 627 pages : color illustrations ; 24 cm.

Place of Publication:

Cham : Springer, [2025]

Summary:

This monograph provides the first systematic treatment of the logarithmic Bogomolov-Tian-Todorov theorem. Providing a new perspective on classical results, this theorem guarantees that logarithmic Calabi-Yau spaces have unobstructed deformations. Part I develops the deformation theory of curved Batalin-Vilkovisky calculi and the abstract unobstructedness theorems which hold in quasi-perfect curved Batalin-Vilkovisky calculi. Part II presents background material on logarithmic geometry, families of singular log schemes, and toroidal crossing spaces. Part III establishes the connection between the geometric deformation theory of log schemes and the purely algebraic deformation theory of curved Batalin-Vilkovisky calculi. The last Part IV explores applications to the Gross-Siebert program, to deformation problems of log smooth and log toroidal log Calabi-Yau spaces, as well as to deformations of line bundles and deformations of log Fano spaces. Along the way, a comprehensive introduction to the logarithmic geometry used in the Gross-Siebert program is given. This monograph will be useful for graduate students and researchers working in algebraic and complex geometry, in particular in the study of deformation theory, degenerations, moduli spaces, and mirror symmetry.

Contents:

Chapter 1. Introduction

Chapter 2. Related Works

Part I. Abstract Unobstructedness Theorems

Chapter 3. Algebraic Structures

Chapter 4. Gauge Transforms

Chapter 5. The Extended Maurer-Cartan Equations

Chapter 6. The Two Abstract Unobstructedness Theorems

Part II. Logarithmic Geometry

Chapter 7. Logarithmic Geometry

Chapter 8. Families of Singular Log Schemes

Chapter 9. Toroidal Crossing Spaces

Part III. Global Deformation Theory

Chapter 10. Generically Log Smooth Deformations

Chapter 11. Deformations with a Vector Bundle

Chapter 12. Geometric Families of P-Algebras

Chapter 13. The Characteristic Algebra

Part IV. Applications

Chapter 14. Log Toroidal Families of Gross-Siebert Type

Chapter 15. The Gerstenhaber Calculus of Log Toroidal Families

Chapter 16. Deformations of Line Bundles

Chapter 17. Algebraic Deformations

Chapter 18. Modifications of the Log Structure.

Notes:

Includes bibliographical references (pages 609-621) and index.

ISBN:

9783031987502

3031987500

OCLC:

1522716226

Publisher Number:

CIPO000287449