Algebraic geometry and analytic geometry

There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century. Some of the more important advances are listed here in chronological order.

This principle permits the carrying over of some results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.

Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.