Perverse Sheaves and Applications to Representation Theory

The book "Perverse Sheaves" is a comprehensive guide to the theory of perverse sheaves, which is a fundamental tool in geometric representation theory. It is divided into two parts, the first six chapters covering constructible and perverse sheaves on complex algebraic varieties, and the last four chapters applying this machinery to selected topics in geometric representation theory. It includes over 250 exercises and a "Quick Reference" summary of commonly used facts for computations.

Format: Paperback / softback
Length: 562 pages
Publication date: 30 August 2022
Publisher: American Mathematical Society

Since its inception around 1980, the theory of perverse sheaves has emerged as a crucial tool of fundamental importance in geometric representation theory. This comprehensive volume, written with the aim of making this theory accessible to students and researchers, is divided into two parts.

The first six chapters provide a thorough account of constructible and perverse sheaves on complex algebraic varieties, encompassing topics such as Artin's vanishing theorem, smooth descent, and the nearby cycles functor. This section of the book also includes a chapter on the equivariant derived category, as well as brief surveys of side topics including etale and $ell$-adic sheaves, $mathcal{D}$-modules, and algebraic stacks.

The last four chapters of the book demonstrate how to apply this machinery to selected topics in geometric representation theory: Kazhdan-Lusztig theory, Springer theory, the geometric Satake equivalence, and canonical bases for quantum groups. Recent developments, such as the $p$-canonical basis, are also discussed.

With over 250 exercises, many of which focus on explicit calculations with concrete examples, this book offers a hands-on approach to understanding perverse sheaves. Additionally, a 4-page "Quick Reference" section summarizes the most commonly used facts for computations, akin to a table of integrals in a calculus textbook.

This book is a valuable resource for anyone interested in exploring the world of perverse sheaves and their applications in geometric representation theory. Its clear explanations, extensive exercises, and comprehensive reference section make it an essential tool for both students and professionals in the field.

Weight: 1022g
Dimension: 184 x 255 x 32 (mm)
ISBN-13: 9781470455972