nLab analytification

under construction

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.

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).

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.

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

Properties

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 $\mathbb{A}^1$-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

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

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

Related concepts

• GAGA

References

Complex analytification

Original articles include

• Michael Artin, Algebraization of formal moduli: II. Existence of modifications, Annals of Math., 91 no. 1 (1970), pp. 88–135.

• Alexander Grothendieck, SGA I, Exposé XII

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

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

• Jacob Lurie, Tannaka duality for geometric stacks, (arXiv:math.AG/0412266)

• Jacob Lurie, Structured Spaces, 2008

• Jack Hall, Generalizing the GAGA Principle (arXiv:1101.5123)

• Anton Geraschenko, David Zureick-Brown, Formal GAGA for good moduli spaces (arXiv:1208.2882)

See also

• Walter Gubler. Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros) (arXiv:1303.7364)

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

• Daniel Isaksen, Étale realization of the $\mathbb{A}^1$-homotopy theory of schemes, 2001 (K-theory archive)

• Daniel Dugger and Daniel Isaksen, Hypercovers in topology, 2005 (pdf, K-Theory archive)

Non-archimedean analytification

Discussion in more general rigid analytic geometry is in

• Brian Conrad, Michael Temkin, Non-archimedean analytification of algebraic spaces (arXiv:0706.3441)