Contents

Disambiguation

In middle and high schools, analytic geometry refers to elementary methods in the geometry of $n$-dimensional Euclidean space involving coordinate calculations; it is usually combined with linear algebra taught in a geometric way. Analytic there pertains to the usage of coordinates. Here we use analytic for kinds of geometry based on study of analytic functions instead.

Idea

In research mathematics, when one says analytic geometry, analytic refers to analytic functions in the sense of Taylor expansion? and by analytic geometry one usually means the study of geometry of complex (holomorphic) manifolds, their analytic subsets, Stein domain?s and related notions. Similarly to an algebraic variety, an analytic variety is locally given as a locus of a set of zeros of a finite family of holomorphic functions. A short survey can be found in a chapter of Dieudonne’s Panorama of pure mathematics.

In addition to analytic geometry over complex numbers, there is also another formalism which allows for nonarchimedean ground fields. This is the subject of rigid analytic geometry or global analytic geometry. Similarly to schemes, rigid analytic varieties are glued from Bercovich spectra of certain commutative Banach algebras, so-called affinoid?s, in a certain Grothendieck topology. (See analytic space.) There are several variants of the formalism (e.g. due Huber). The subject is closely related to formal geometry and has its main applications in arithmetic geometry and representation theory.

It is an open problem to find an appropriate analogue of rigid analytic geometry in noncommutative geometry, which is supposed to play an important role in mirror symmetry.

Local properties of analytic manifolds and spaces are studied in local analytic geometry.

Theorems

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• Ostrowski's theorem

(…)

Holomorphic functions of several complex variables

This section is about certain aspects of holomorphic functions ${ℂ}^n\to ℂ$.

Currently it concentrates on aspects of relevance in the application to AQFT, such as a version of the edge-of-the-wedge theorem.

From the viewpoint of complex manifolds this is the local theory that describes the situation in coordinate patches.

• Wikipedia

When we talk about holomorphic functions in the following and do not specify the domain, we will always assume that the domain is an open, simply connected subset of ${ℂ}^n$.

relevance to quantum field theory (QFT)

In the AQFT formulation (actually the following description is the Heisenberg picture of quantum mechanics in a nutshell) selfadjoint operators $A$ on a Hilbert space $\mathscr{H}$ are the observables of a physical system, while normed vectors $x,y\in \mathscr{H}$ represent the states the system can be in. The real number $⟨y,Ax⟩$ represents the probability? that a system starting in state $x$ will be in state $y$ after a measurement of $A$.

In AQFT we often encounter a set of operators indexed by several complex variables $z=(z_1,z_2,...)$ and try to deduce properties of the theory from the function $f(z):=⟨y,A(z)x⟩$. In this way, the theory of holomorphic functions of several variables is promoted to an irreplaceable tool in quantum field theory.

Hartogs’ theorem

One striking difference of functions of several real variables and several complex variables is described by Hartogs’ theorem on separate analyticity:

• Theorem (Hartogs): Let f be a $ℂ$-valued function defined in an open set $U\subset {ℂ}^n$. Suppose that f is analytic in each variable $z_j$ when the other coordinates $z_k$ are fixed. Then f is analytic as a function of all n coordinates.

remark: There are no other assumptions about f necessary, it needs not to be continous or even measurable. Note that in the real case the property of being partially differentiable alone encodes nearly no information about a function.

Reference:

• Paul Garrett: Hartogs theorem

relevance: When reading AQFT literature you will often encounter the claim that given functions are holomorphic, Hartogs’ theorem simplifies the task of checking these claims considerably, because you have to check the holomorphy in every single variable only.

analogues from the one-dimensional theory

Some results remain true in the multi dimensional case.

• Identity theorem: If a holomorphic function is zero in a neighbourhood of a point, it is the zero function.

remark: As usual the domain is supposed to be an open, simply connected (not necessarily proper) subset of ${ℂ}^n$, which implies that the point of the precondition of the theorem is an interior point of the domain.

domains of holomorphy

One of the most notably difference of the theory of one complex variable and of several complex variables is that the riemann mapping theorem fails in several complex variables, which is in a certain sense the reason why in several complex variables there are domains which can be enlarged such that all holomorphic functions extend to the larger domain.

handwaving why this is not possible in one dimension: According to the riemann mapping theorem every domain (open, simply connected proper subset of $ℂ$) is biholomorph equivalent to the open disk $E:=\{z:\mid z\mid <1\}$, which means that the rings of holomorphic functions are isomorph, too. But the ring of holomorphic functions on E has to every point in the boundary of E a function that has a pole in this point, so that E cannot be enlarged in a way that all holomorphic functions are extentable. Therefore this applies to every domain.

Some domains in ${ℂ}^n$ do have the property that they cannot be enlarged, and since this is an interesting property, the name domain of holomorphy was coined for these, and the question how they could be described was promoted to an interesting research topic in the beginning of the 20th century.

• Wikipedia

edge of the wedge theorems

These theorems desribe situations where holomorphic functions defined on specific domains (the wedges) can be continued to holomorphic functions of larger domains.

• Wikipedia

They are a valuable tool in AQFT (and were in fact discovered by one of the fathers of the theory, Nikolay Bogolyubov).

We state here one version that will be of use to the nLab:

• theorem (edge of the wedge): Let $K:=\{z\in {ℂ}^n:\mid z\mid \le r\}$ be a ball in ${ℂ}^n$ and let $𝒞\subset {ℝ}^n$ be an open convex cone such that $𝒞\cap (-𝒞)\ne \varnothing$. We put $z=x+\mathrm{iy}$ with $x,y\in {ℝ}^n$ and define an open domain G by $G:=\{z=x+\mathrm{iy}:z\in K,y\in 𝒞\}$. Let $f$ be a holomorphic function in G and assume that

  $$\underset{y\to 0,y\in 𝒞}{\lim }f(x+\mathrm{iy})$$\lim_{y \to 0, y \in \mathcal{C}} f(x + iy)

  exists for all $x\in B_r\subset {ℝ}^n$, where $B_r$ is an open ball with radius r. (The limit may not depend on the specific sequence chosen). Then $f$ is holomorph extendable into an open region $G\cup G_0$ with

  $$G_0:=\bigcup _{x\in B_r}\{z:\mid z-x\mid \le \theta \cdot \mathrm{dist}(x,\partial U)\}$$G_0 := \bigcup_{x \in B_r} \{ z: |z-x| \le \theta \cdot dist(x, \partial U) \}

  with $0<\theta <1$ a constant that is independent from $x,B_r$, and $f$.

proof: V.S.Vladimirov, “theory of functions of several complex variables” (ZMATH entry)

Related concepts

• several complex variables, analytic spectrum, analytic space, Stein space?

• Oka principle, B1-homotopy theory

References

Course notes on analytic geometry are in

• Frédéric Paugam, Global analytic geometry and the functional equation (2010) (pdf)

and for rigid analytic geometry in

• Kiran Sridhara Kedlaya, Introduction to Rigid Analytic Geometry (web)

• Brian Conrad, Several approaches to non-archimedean geometry (pdf)

A gentle and modern introduction to complex manifolds that starts with an extensive exposition of the local theory is this:

• Daniel Huybrechts, Complex geometry. An introduction. (ZMATH entry)

• Hans Grauert, Reinhold Remmert, Theory of Stein spaces, Grundlehren der Math. Wissenschaften 236, Springer 1979, xxi+249 pp.; Coherent analytic sheaves, Grundlehren der Math. Wissenschaften 265, Springer 1984. xviii+249 pp.; Komplexe Räume, Math. Ann. 136, 1958, 245–318, DOI

For more see the references at rigid analytic geometry and at analytic space.