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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.
List of 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.
About the author
Simon Felten received his PhD in 2021 from the University of Mainz. He has since been a visitor at the Institut des Hautes Études Scientifiques and a postdoctoral researcher at Columbia University and the University of Oxford. His main research interests are logarithmic algebraic geometry and its applications to singularities, infinitesimal deformation theory, and the construction of algebraic varieties
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.
