geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
GAGA is short for the title Géométrie algébrique et géométrie analytique of the article (Serre 56), and more generally 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 computes 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.
analytification – Existence and fully faithfulness (this is often referred to as “the GAGA theorem”)
The original article is
which compares the categories of projective integral $\mathbb{C}$-schemes of finite type with that of projective complex manifolds possibly with singularities. In
this was generalized to acomparison of proper $\mathbb{C}$-schemes locally of finite type and complex analytic spaces.
A good modern textbook account is in
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
Generalization to more general analytic geometry includes
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
catgeory: reference