complex geometry

# Contents

## Idea

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 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.

## References

The original article is

• Jean-Pierre Serre, Géométrie algébrique et géométrie analytique Université de Grenoble. Annales de l’Institut Fourier 6: 1–42, (1956) doi:10.5802/aif.59, 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

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

A good modern textbook account is in

• Amnon Neeman, Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, 2007 (publisher)

Reviews and surveys include

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 (pdf)

Generalization to more general analytic geometry includes

• Brian Conrad, M. Temkin, Non-Archimedean analytification of algebraic spaces, J. Algebraic Geom. 18 (2009), no. 4, 731–788 (arXiv:0706.3441)

Generalization to higher geometry (stacks) includes

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

catgeory: reference

Last revised on May 9, 2018 at 04:37:22. See the history of this page for a list of all contributions to it.