nLab GAGA

Idea

GAGA is short for the title Géométrie algébrique et géométrie analytique of the article by Serre 1956, and it has come to stand for the kind of results initated in this article, establishing the close relationship between algebraic geometry over the complex numbers and complex analytic geometry, hence between algebraic spaces (algebraic varieties, schemes) over the complex numbers and complex analytic spaces.

For example, often one can compute sheaf cohomology either in the algebraic category or in the analytic category, obtaining the same result. This can be useful because one has more tools in the analytic category, such as Dolbeault resolutions. Typically GAGA-type theorems hold for proper (compact) varieties, but not for non-proper varieties.

Theorems

• analytification – Existence and fully faithfulness (this is often referred to as “the GAGA theorem”)

• Chow's theorem

• relation between algebraic and analytic vector bundles

• comparison theorem (étale cohomology)

References

The original article:

• Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Annales de l’Institut Fourier 6 (1956) 1-42 [doi:10.5802/aif.59, numdam:AIF_1956__6__1_0, ISSN:0373-0956, MR 0082175]

which compares the categories of projective integral $\mathbb{C}$-schemes of finite type with that of projective complex manifolds possibly with singularities. In

• Alexander Grothendieck, SGA I, Exposé XII

this was generalized to a comparison of proper $\mathbb{C}$-schemes locally of finite type and complex analytic spaces.

Modern textbook account:

• Amnon Neeman: Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, Cambridge University Press (2007) [ISBN:9780521709835]

Reviews and surveys include

• Wikipedia, Algebraic geometry and analytic geometry

• Yan Zhao, Géométrie algébrique et géométrie analytique, 2013 (pdf)

• Jean-Pierre Demailly, Analytic methods in algebraic geometry, lecture notes 2009 (pdf)

Discussion for real analytic spaces includes

• Johannes Huisman, section 2 of The exponential sequence in real algebraic geometry and Harnack’s Inequality for proper reduced real schemes, Communications in Algebra, Volume 30, Issue 10, 2002 (journal)

Generalization to more general analytic geometry includes

• Brian Conrad, M. Temkin: Non-Archimedean analytification of algebraic spaces, J. Algebraic Geom. 18 4 (2009) 731-788 [arXiv:0706.3441, doi:10.1090/S1056-3911-09-00497-4]

Generalization to higher geometry (stacks) includes

• Jacob Lurie, Tannaka duality for geometric stacks, (arXiv:math.AG/0412266)

• Jack Hall, Generalizing the GAGA Principle (arXiv:1101.5123)

• Anton Geraschenko, David Zureick-Brown, Formal GAGA for good moduli spaces (arXiv:1208.2882)

Discussion in the context of moduli spaces includes

• Kai-Wen Lan, Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties pdf

category: reference