7 New Algebraic Geometry Books Shaping 2025

What if everything you knew about algebraic geometry was wrong? Renaud Gauthier challenges traditional approaches by weaving together schemes, simplicial sets, and higher categories into a cohesive framework for derived algebraic geometry. You’ll find a detailed exploration of advanced topics like Goodwillie Calculus and Supersymmetry, alongside new material on Derived Motivic Spectra that reflects the latest mathematical advances. This book suits mathematicians and researchers keen on deepening their understanding of both classical and spectral algebraic geometry through a modern, elemental lens. If you’re looking to bridge physical mathematics with cutting-edge categorical methods, this text offers precisely that.

Unlike most algebraic geometry texts that dive straight into complex abstractions, David Anderson and William Fulton offer a clear pathway through equivariant cohomology's foundational concepts and applications, making this advanced subject accessible to early graduate students and researchers alike. You’ll explore core ideas like finite-dimensional approximations and the localization theorem, then see these principles in action with concrete examples such as toric varieties and Grassmannians. The book’s strength lies in its balance between theoretical depth and practical computations, supported by exercises and fresh material not previously compiled in textbook form. If you’re delving into algebraic geometry or combinatorics and want a solid grasp of equivariant methods without overwhelming prerequisites, this book serves as a reliable guide.

This personalized book explores the latest breakthroughs and emerging concepts in algebraic geometry as of 2025, tailored specifically to your background and goals. It examines how recent discoveries are reshaping classical notions and introduces you to cutting-edge developments such as derived methods, spectral techniques, and novel geometric structures. By focusing on your interests, this tailored guide helps you navigate complex topics like moduli spaces, cohomology theories, and equivariant phenomena with clarity and depth. The content matches your expertise level and addresses the precise aspects you want to master, providing a uniquely customized learning experience that keeps you at the forefront of this evolving field.

After decades immersed in algebraic geometry, Herbert Lange crafted this text to clarify the intricate world of abelian varieties through complex geometric constructions. You’ll explore foundational concepts starting with complex tori and theta functions, then advance to detailed studies of moduli spaces and key examples like Jacobian and Prym varieties. The book dives into sophisticated tools such as the Fourier–Mukai transform and its applications to sheaves and Chow groups, offering insights that bridge algebraic geometry with number theory and mathematical physics. If you’re pursuing a deeper understanding of these structures or aiming to apply them across related fields, this course-style book lays out the concepts with precision and rigor.

What started as an effort to advance the foundational understanding of algebraic geometry, Ulrich Görtz and Torsten Wedhorn deliver a deep dive into the cohomology of schemes, a central topic in modern algebraic geometry. You’ll explore detailed treatments of smooth, unramified, and étale morphisms, alongside the étale fundamental group, before moving into the powerful formalism of derived categories for cohomology of quasi-coherent sheaves. The book then unfolds advanced themes like numerical intersection theory, Grothendieck-Riemann-Roch, and Grothendieck duality, culminating with concrete applications to curves and abelian schemes. If you’re seeking to master the theoretical backbone and techniques that underpin current research, this book provides rigorous explanations and exercises to deepen your insight.

Masayoshi Miyanishi's extensive experience in algebraic geometry drives this rigorous exploration into affine varieties, a fundamental yet challenging area of the field. You’ll gain a nuanced understanding of polynomial rings and affine spaces, starting from comprehensive introductions to commutative algebra and advancing through the intricacies of affine lines and algebraic surfaces. The book delves into key topics such as the Abhyankar–Moh–Suzuki Theorem and the Jacobian conjecture, providing detailed explanations and problem sets with hints to deepen your grasp. If you’re engaged in advanced mathematical research or graduate studies focusing on the algebraic structure of affine varieties, this work offers a thorough and demanding treatment that's hard to find elsewhere.

This tailored book explores the forefront of algebraic geometry as it evolves through 2025, focusing on emerging theories and discoveries that will shape the field’s future. It presents a personalized journey, aligning complex concepts with your existing knowledge and interest areas to deepen understanding of topics such as derived schemes, spectral geometry, and moduli spaces. By concentrating on the latest developments and anticipated challenges, it reveals how fresh insights integrate with classical foundations, offering a uniquely tailored exploration that keeps you ahead in this dynamic discipline. With its customized approach, this book matches your background and goals, providing a focused lens on cutting-edge research. It encourages active engagement with new mathematical tools and perspectives, turning your study into an individualized roadmap through algebraic geometry’s unfolding landscape.

When Kelli Francis-Staite introduced the concept of C^∞-schemes with corners, she bridged a gap between algebraic and differential geometry that had long puzzled researchers. This book explores singular spaces that incorporate boundaries and corners, expanding tools for studying complex geometric objects like derived manifolds and orbifolds with corners. By focusing on applications to symplectic geometry and moduli spaces of J-holomorphic curves, you gain insights into cutting-edge mathematical frameworks that can enrich your understanding of both fields. If you work with singular spaces or derived geometry, this text offers a precise, technical foundation rather than broad overviews.

Drawing from Emil Artin's profound influence on mathematics, this text delves into the foundational and modern aspects of algebraic geometry with clarity and precision. You gain insights into fundamental algebraic equations alongside contemporary methods that have shaped the field, making it a distinctive blend of theory and evolving practice. The book is especially suited for those deeply invested in understanding the structural underpinnings of algebraic geometry, offering a thoughtful progression through complex ideas that challenge and expand your mathematical perspective. Chapters carefully build up the necessary algebraic tools, ensuring you develop a solid grasp of key principles without unnecessary abstraction.