Algebraic Cycles and Motives by Jan Nagel – A Cambridge University Press Volume on Algebraic Geometry, Chow Groups, and

Introduction

Algebraic geometry stands as one of the most profound and active branches of modern mathematics, bridging the worlds of geometry, number theory, and abstract algebra. For Indian students and researchers delving into this intricate field, Algebraic Cycles and Motives by Jan Nagel offers a rigorous yet accessible gateway into the study of algebraic cycles from a motivic perspective. Published by Cambridge University Press, this hardbound volume is an essential resource for anyone seeking a deeper understanding of the conjectures and techniques that shape contemporary algebraic geometry.

Book Overview

This 2007 volume is part of a two-book set that provides a comprehensive and self-contained account of algebraic cycles and motives. The book brings together twenty-two contributions from leading mathematicians, each chapter presenting cutting-edge research and surveying key strands in the field. The central theme revolves around Alexander Grothendieck’s visionary idea that algebraic cycles should be studied through the lens of motives—a framework that unifies cohomological and geometric methods. Whether you are a postgraduate student or an established researcher, this book serves as both a reference and a springboard for new investigations.

Key Highlights

• Expert Contributions: Features chapters authored by prominent figures in complex algebraic geometry and arithmetic geometry, offering diverse perspectives and novel results.
• Self-Contained Approach: Designed to be accessible to readers with a basic grounding in algebraic geometry, with careful exposition of foundational concepts.
• Focus on Modern Conjectures: Delves into the Bloch-Beilinson conjectures, Murre’s conjectures on filtrations of Chow groups, and Bloch’s conjecture, providing a state-of-the-art overview.
• Motivic Framework: Explores Voevodsky’s triangulated category of mixed motives and the theory of finite-dimensional motives, equipping readers with powerful tools for research.

Inside the Book

The volume is structured to guide readers from classical ideas to advanced contemporary research. Early chapters introduce the study of algebraic cycles using Abel-Jacobi maps, regulator maps, and normal functions. As the narrative progresses, the focus shifts to motivic cohomology, the construction of triangulated categories, and the structure of Chow groups. Each chapter is self-contained, making it easy to dip into specific topics without losing context. The book also includes detailed proofs and illustrative examples that clarify abstract concepts.

Key Topics

• Algebraic cycles and their equivalence relations
• Abel-Jacobi maps and regulator maps
• Normal functions and their applications
• Voevodsky’s triangulated category of mixed motives
• Finite-dimensional motives and their properties
• Bloch-Beilinson and Murre conjectures on filtrations
• Bloch’s conjecture on zero-cycles
• Chow groups and their structure

Reader Benefits

This book is more than a collection of articles—it is a carefully curated resource that helps readers navigate the complexities of motivic theory. By studying this volume, you will gain a solid grasp of the conjectures driving current research, learn to apply motivic techniques to concrete geometric problems, and understand how algebraic cycles interact with cohomology theories. The contributions are written with clarity, making advanced material accessible without sacrificing depth.

Learning Outcomes

• Understand the role of algebraic cycles in modern algebraic geometry
• Master the use of Abel-Jacobi and regulator maps in studying cycle classes
• Gain familiarity with Voevodsky’s motivic categories and their applications
• Analyze filtrations on Chow groups and their conjectural properties
• Evaluate the impact of Grothendieck’s motivic philosophy on contemporary research

Who Should Read

This book is ideal for graduate students and researchers in pure mathematics, particularly those specializing in algebraic geometry, arithmetic geometry, or number theory. It is also well-suited for mathematicians seeking to broaden their understanding of motives and cycles. Indian readers preparing for advanced study or teaching courses in algebraic geometry will find this volume an invaluable addition to their library.

About the Author

Jan Nagel is a respected mathematician known for his contributions to algebraic geometry and the theory of motives. With a career spanning research institutions and universities, Nagel has edited and contributed to several seminal volumes in the field. His deep expertise ensures that the content is both authoritative and pedagogically sound.

About the Publisher

Cambridge University Press is one of the world’s oldest and most prestigious academic publishers. Known for its rigorous editorial standards and commitment to scholarly excellence, Cambridge University Press has been a trusted source for mathematical literature for over four centuries. This volume upholds that tradition, offering a high-quality hardcover edition that will withstand years of use in any library or study.

Conclusion

Algebraic Cycles and Motives is an indispensable resource for anyone serious about understanding the deep connections between algebraic cycles, motives, and modern geometry. With its blend of survey articles, original research, and clear exposition, this book stands as a milestone in the literature. Whether you are beginning your journey into motivic theory or looking to deepen your expertise, this volume deserves a prominent place on your bookshelf.