nLab analytification


under construction



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


Let XSpec()X \to Spec(\mathbb{C}) be a scheme of locally finite type over the complex numbers. Its set X()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()X(\mathbb{C}) canonically carries the complex analytic topology. As such it is a topological space written X anX^{an}. Equipped with the canonical structure sheaf 𝒪 X an\mathcal{O}_{X^{an}} this is a complex analytic space. This (X an,𝒪 X an)(X^{an}, \mathcal{O}_{X^{an}}) is called the analytification of XX.

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

See e.g. (Neeman 07, section 4, p.71, Danilov 91, chapter 3, paragraph 1, section 1.1 (p.61)

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


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

The analytification of an elliptic curve is the complex torus.

see e.g. (Danilov 91, example in chapter 3, paragraph 1, section 1.1. (p. 61))


Existence and fully faithfulness (GAGA)

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

  1. locally of finite type

  2. locally separated

is a complex analytic space.

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 𝔸 1\mathbb{A}^1-homotopy theory

For kk \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 (Sch k sm)Sh (Sch k sm) 𝔸 1Grpd Sh_\infty(Sch^{sm}_k) \to Sh_\infty(Sch^{sm}_k)^{\mathbb{A}^1} \to \infty Grpd

(Isaksen 01, Dugger-Isaksen 05, theorem 5.2)


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

See also

Discussion in the context of hypercovers and A1-homotopy theory is in

Non-archimedean analytification

Discussion in more general rigid analytic geometry is in

Last revised on November 28, 2018 at 22:05:45. See the history of this page for a list of all contributions to it.