# nLab analytification

Contents

under construction

### Context

#### Analytic geometry

analytic geometry (complex, rigid, global)

## Basic concepts

analytic function

analytification

GAGA

complex geometry

# Contents

## Idea

Analytification is the process of universally turning an algebraic space into an analytic space.

## Definition

Let $X \to Spec(\mathbb{C})$ be a scheme of locally finite type over the complex numbers. Its set $X(\mathbb{C})$ of “complex points” is the set of maximal ideals, since $\mathbb{C}$ is an algebraically closed field, e.g. Neeman 07, prop. 4.2.4).

This set $X(\mathbb{C})$ canonically carries the complex analytic topology. As such it is a topological space written $X^{an}$. Equipped with the canonical structure sheaf $\mathcal{O}_{X^{an}}$ this is a complex analytic space. This $(X^{an}, \mathcal{O}_{X^{an}})$ is called the analytification of $X$.

This construction extends to a functor from the category of schemes over $\mathbb{C}$ to that of complex analytic spaces.

Generalization to structured (infinity,1)-toposes is in (Lurie 08, remark 4.4.13).

## Examples

The analytification of the projective space $\mathbb{P}^1$ is the complex projective space $(\mathbb{P}^1)^{an} \simeq \mathbb{C}\mathbb{P}^1$, hence the Riemann sphere.

The analytification of an elliptic curve is the complex torus.

## Properties

### Existence and fully faithfulness (GAGA)

The analytification of an algebraic space over the complex numbers which is

Moreover, under suitable conditions analytification is a fully faithful functor.

This is a classical result due to (Artin 70, theorem 7.3). A textbook account of the proof is in (Neeman 07, section 10). Discussion in more general analytic geometry is in (Conrad-Temkin 09, section 2.2).

Generalization to algebraic stacks/Deligne-Mumford stacks/geometric stacks is in (Lurie 04, Hall 11, Geraschenko & Zureick-Brown 12).

### As geometric realization in $\mathbb{A}^1$-homotopy theory

For $k \hookrightarrow \mathbb{C}$ a field, then the functor that takes a smooth complex scheme to the the homotopy type underlying its analytification induces geometric realization

$Sh_\infty(Sch^{sm}_k) \to Sh_\infty(Sch^{sm}_k)^{\mathbb{A}^1} \to \infty Grpd$

## References

### Complex analytification

Original articles include

A review of that is in

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

Textbook accounts include

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

• Vladimir Danilov, chapter 3 of Cohomology of algebraic varieties, in I. Shafarevich (ed.), Algebraic Geometry II, volume 35 of Encyclopedia of mathematical sciences, Springer 1991 (GoogleBooks))

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)

Generalizations to higher geometry are in