Some Books and References in Learning Algebraic Geometry
In this blog, we will give a long list of books and notes about many areas in algebraic geometry.
1. Dictionary-type reference
1.1. The Stacks project. An open source textbook and reference work on algebraic geometry: The Stacks project. Including so many preliminaries, such as category theory, commutative algebra, homological algebra, sheaf and site theory and so on. This is nearly the most detailed references in the world about scheme theory, topics of scheme theory, algebraic space/stack-theory and moduli stack theory.
2. Classical Algebraic Geometry
2.1. Mircea Mustata’s Lecture notes. See Lecture notes for Math 631/632: Introduction to algebraic geometry. This is a very comprehensive notes using the language of classical variety to introduce many mordern results in basic algebraic geometry, such as cohomology of sheaves, the theorem on formal functions and Serre duality.
2.2. GTM 133. Algebraic Geometry, a First Course. By Joe Harris. Including many examples in geometry.
2.3. Fulton’s ALGEBRAIC CURVES, An Introduction to Algebraic Geometry. Some very basic and classical things about varieties and the theorems over curves.
2.4. Introduction to Algebraic Geometry by Justin R. Smith. Basic introduction of the classical language of varieties and some introductions of scheme theory. In main text this book haven’t use the cohomology (but in appendix, these things were introduced).
3. The Basic Algebraic Geometry using Scheme Theory
3.1. GTM 52. Algebraic Geometry by R. Hartshorne.
3.2. THE RISING SEA, Foundations of Algebraic Geometry by R. Vakil. See THE RISING SEA.
3.3. Algebraic Geometry and Arithmetic Curves by Qing Liu.
3.4. Algebraic Geometry I: Schemes by Görtz/Wedhorn. See Görtz/Wedhorn-I.
3.5. Algebraic Geometry and Commutative Algebra by Siegfried Bosch. See AGCA.
3.6. Introduction to Schemes by Geir Ellingsrud/John Christian Ottem. Baby but more detailed version of Hartshorne’s GTM 52, see Introduction to Schemes.
3.7. Algebraic Geometry I/II by Peter Scholze. See AG-I and AG-II.
3.8. GTM 76. Algebraic Geometry by S. Itaka.
3.9. Topics in Algebraic Geometry by Luc Illusie. See Illusie.
4. Cohomology Theory of Algebraic Geometry
4.1. Etale Cohomology Theory by Lei Fu.
4.2. Notes on Etale cohomology by B. Conrad. See Etconrad.
4.3. Lecture Notes for Étale Cohomology I by Jens Franke. See Franke.
4.4. LECTURE NOTES ON ÉTALE COHOMOLOGY by JOHANNES ANSCHÜTZ. See ANSCHÜTZ.
4.5. Lectures on Etale Cohomology by Milne. See MilneEtNotes.
4.6. Etale Cohomology by Milne. The only book compute the cohomology of surfaces.
4.7. Introduction to etale cohomology by Lombardo/Maffei. See LM.
5. Hilbert scheme, Quot scheme and Picard scheme
5.1. FGA by A. Grothendieck and so on.
5.2. FGA-explained.
6. Duality of Schemes
6.1. Residues and Duality by Robin Hartshorne. Lecture Notes of a seminar on the original work of A. Grothendieck, given at Harvard 1963/64. There are many mistakes in the book.
6.2. Grothendieck Duality and Base Change by Brian Conrad. A detailed and corrected version of the Hartshorne’s book.
6.3. Foundations of Grothendieck Duality for Diagrams of Schemes by Lipman/Hashimoto. The second method of duality.
6.4. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability by Amnon Neeman. Published in J. Amer. Math. Soc. 9 (1996), 205-236. See GDJAMS. This is the modern method.
7. Intersection Theory
7.1. Intersection Theory by W. Fulton. The Bible of intersection Theory.
7.2. 3264 AND ALL THAT, A Second Course in Algebraic Geometry by Eisenbud/Harris. A new book.
7.3. Virtual Classes for the Working Mathematician. See VWM.
8. Theory of Algebraic Curves
8.1. Geometry of Algebraic Curves, Volume I. By E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris. Introducing the theory of complex algebraic curves.
9. Theory of Algebraic Surfaces
9.1. Algebraic Surfaces by Lucian Badescu. A pure-alge-geometry approach of the theory of classifying the surfaces in any charactristics.
9.2. Complex Algebraic Surfaces by Arnaud Beauville. A classical and important book in this area.
9.3. Compact Complex Surfaces. By Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, Antonius Van de Ven. A very important book about complex surfaces. Everyone need to read this book.
9.4. Lectures on K3 Surfaces by Daniel Huybrechts. A good book about K3 surface.
9.5. K3 Surfaces by Shigeyuki Kondo.
10. Birational Geometry and Higher-Dimensional Algebraic Geometry
10.1. Birational Geometry of Algebraic Varieties by Kollar/Mori. A classical book in this area.
10.2. Positivity in Algebraic Geometry I/II by R. Lazarfeld.
10.3. Higher-Dimensional Algebraic Geometry by Olivier Debarre.
10.4. Rational Curves on Algebraic Varieties by J. Kollar.
10.5. Singularities of the Minimal Model Program by J. Kollar.
10.6. Introduction to the Mori Program by Kenji Matsuki.
10.7. Classification of Higher Dimensional Algebraic Varieties by Hacon/Kovács. Aiming to introduce the BCHM.
10.8. The Minimal Model Program for Varieties of Log General Type by Hacon. See LMMP.
10.9. Foundations of the Minimal Model Program by Fujino.
11. Theory of Moduli Spaces
11.I. Algebraic stacks and spaces
11.1. Algebraic Spaces and Stacks by Martin Olsson.
11.2. Stacks and Moduli by Jarod Alper. See SM.
11.3. Champs algébriques by Gérard Laumon/Laurent Moret-Bailly.
11.4. Algebraic Spaces by Donald Knutson.
11.II. Moduli of Varieties
11.5. Geometry of Algebraic Curves, Vol II by Arbarello/Cornalba/Griffiths. Very detailed book.
11.6. Moduli of Curves by Harris/Morrison. I don’t like this book.
11.7. Families of varieties of general type by János Kollár. An important book aiming to introducing the recent works about the moduli space of general type varieties.
11.8. Moduli of Varieties by C. Birkar. A series of papers about moduli of more general varieties. See ModuliVar.
11.III. Moduli of Bundles and Sheaves
11.9. The Geometry of Moduli Spaces of Sheaves by Huybrechts/Lehn.
11.10. Lectures on Vector Bundles by J. Le Potier.
11.11. Algebraic Stacks and Moduli of Vector Bundles by Neumann/Leicester. See ASMVB.
12. Hodge Theory
12.1. Théorie de Hodge: I/II by Deligne. Classical papers.
12.2. Hodge Theory and Complex Algebraic Geometry I/II by Claire Voisin.
12.3. Introduction to Hodge Theory by Bertin/Demailly/Illusie/Peters.
12.4. Topics in Trascendental Geometry by Griffith.
12.5. Mixed Hodge Structures by Peters/Steenbrink.
12.6. Introduction to Nonabelian Hodge Theory by Raboso/Rayan. See NAHT.
13. Algebraic Groups and Abelian Varieties
13.1. Algebraic Groups by J.S. Milne.
13.2. Linear Algebraic Groups I/II by B. Conrad. See I, II.
13.3. Abelian varieties by B. Bhatt. See I and II.
13.4. Abelian varieties by Edixhoven/van der Geer/Moonen. SeeA-V.
13.5. Abelian Varieties by David Mumford.
14. Derived Category of Coherent Sheaves and Fourier-Mukai Theory
14.1. Fourier–Mukai transforms in algebraic geometry by Huybrechts.
14.2. Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics by Bartocci/Bruzzo/Ruipérez.
14.3. Bridgeland Stability Notes by Debray. See M392C.
15. Motives
15.1. Lectures on the Theory of Pure Motives. By Jacob P. Murre, Jan Nagel and Chris A. M. Peters. Give a introduction of pure motives (chow motives).
15.2. Mixed Motives by Marc Levine.
16. Others
16.1. Principles in Algebraic Geometry by Griffith/Harris.
16.2. Calabi-Yau Manifolds and Related Geometries by Gross/Huybrecht.