Fourier-Mukai Transforms in Algebraic Geometry by D. Huybrechts – A Detailed Study of Derived Categories of Coherent She

Introduction

Fourier-Mukai transforms have become an indispensable tool in modern algebraic geometry, bridging deep connections between derived categories and classical geometric invariants. This rigorous yet accessible text by D. Huybrechts offers a comprehensive introduction to the subject, tailored for postgraduate students and researchers who wish to master the core ideas and techniques. Published by OUP Oxford, this hardcover edition is an essential addition to any serious mathematics library in India.

Book Overview

Based on a course delivered at the Institut de Mathématiques de Jussieu, this book systematically develops the theory of Fourier-Mukai transforms, starting from the foundations of derived categories of coherent sheaves on smooth projective varieties. The author expertly guides readers through the interplay between algebraic geometry and homological algebra, illustrating how these transforms unify diverse topics such as abelian varieties, K3 surfaces, and Hodge theory. Every major result is accompanied by complete proofs, making the book self-contained for students with a basic grounding in algebraic geometry.

Key Highlights

• Comprehensive treatment: Covers derived categories, Fourier-Mukai functors, and their applications to moduli spaces and birational geometry.
• Full proofs: All theorems are proved in detail, allowing readers to follow the logical flow without gaps.
• Exercises throughout: Each chapter includes carefully chosen problems that reinforce understanding and extend the theory.
• Interdisciplinary connections: Integrates singular cohomology, Hodge structures, and the geometry of abelian varieties and K3 surfaces.
• Authoritative authorship: Written by a leading researcher known for clear exposition and deep insights.

Inside the Book

The text begins with a review of derived categories and triangulated structures, then moves to the construction of Fourier-Mukai transforms and their fundamental properties. Subsequent chapters explore equivalences between derived categories, the role of the canonical bundle, and applications to stability conditions. Special attention is given to examples from abelian varieties and K3 surfaces, where Fourier-Mukai transforms reveal hidden symmetries. The final chapters discuss open problems and recent developments, providing a springboard for independent research.

Key Topics

• Derived categories of coherent sheaves
• Triangulated categories and exact functors
• Fourier-Mukai functors and kernels
• Equivalences and reconstruction theorems
• Hodge theory and singular cohomology
• Abelian varieties and their Fourier-Mukai partners
• K3 surfaces and derived autoequivalences
• Moduli spaces and stability conditions

Reader Benefits

By working through this book, readers will gain a firm command of one of the most active research areas in algebraic geometry. The clear, step-by-step exposition reduces the steep learning curve often associated with derived categories. The inclusion of exercises allows for self-assessment and deeper engagement with the material. Moreover, the book's focus on concrete examples—especially from abelian varieties and K3 surfaces—helps bridge abstract theory with geometric intuition, making it an ideal companion for seminars and self-study.

Learning Outcomes

• Understand the construction and properties of derived categories of coherent sheaves.
• Master the definition and basic properties of Fourier-Mukai transforms.
• Prove equivalences between derived categories using kernel functors.
• Apply Fourier-Mukai techniques to study abelian varieties and K3 surfaces.
• Connect derived categories with Hodge theory and singular cohomology.
• Develop the ability to read and contribute to current research literature.

Who Should Read

This book is designed for postgraduate students in mathematics who have completed a first course in algebraic geometry—covering schemes, sheaves, and cohomology. It is equally valuable for researchers in algebraic geometry, representation theory, and mathematical physics who wish to incorporate derived categories into their work. Advanced undergraduates with strong backgrounds in algebra and geometry will also find the book accessible with additional effort. Indian students preparing for competitive exams or pursuing PhDs in pure mathematics will benefit greatly from the systematic approach.

About the Author

Daniel Huybrechts is a professor of mathematics at the University of Bonn, renowned for his contributions to algebraic geometry, particularly in the study of derived categories, moduli spaces, and K3 surfaces. He has authored several influential monographs and is celebrated for his ability to present complex ideas with clarity and precision. His research has shaped modern understanding of Fourier-Mukai transforms and their applications.

About the Publisher

Oxford University Press (OUP) is a world-leading academic publisher with a distinguished history of disseminating high-quality research in mathematics and the sciences. This hardcover edition is produced to the highest standards of durability and readability, ensuring it remains a reliable reference for years to come.

Conclusion

Fourier-Mukai Transforms in Algebraic Geometry is an indispensable resource for anyone seeking a deep and rigorous understanding of this transformative subject. With its blend of thorough proofs, illuminating examples, and practical exercises, it stands as a definitive guide for students and researchers alike. Order your copy today from Bookshops.in and embark on a journey into the heart of modern algebraic geometry.