Complex Spaces

In this section we introduce complex spaces, the basic objects of this book, by using the notion and elementary properties of sheaves from Appendix A.

Moreover, we introduce some basic constructions such as subspaces, image spaces and fibre products.

In order to provide, besides the formal definition, geometric understanding for the notion of a complex space we begin with analytic sets and a definition of reduced complex spaces which is modeled on the definition of a complex manifold, and which naturally leads to the concept of a (reduced)C-analytic ringed space. Then, it is only a short step to give the general definition of a complex space via structure sheaves with nilpotent elements.

From now on we are working with the fieldK=C, and we endowCⁿ with the usual Euclidean topology.

Definition 1.29. (1) LetU ⊂Cⁿbe an open subset. A complex valued func- tionf :U →Cis called(complex) analytic, orholomorphic, if it isholomor- phic atpfor allp∈U. That is, for allp= (p1, . . . , pn)∈U there is an open neighbourhoodV ⊂U and a power series

∞

|α|=0

cα(x1−p1)^α1·. . .·(xn−pn)^αn

which converges inV tof|V. In particular, thecoordinate functions x₁, . . . , x_n ofCⁿ, x_i:Cⁿ →C,p →p_i, are holomorphic.

A mapf = (f₁, . . . , f_m) :U →C^mis calledholomorphicoranalyticif the component functions fi=xi◦f are.

A holomorphic mapf :U →V,V ⊂C^mopen, is calledbiholomorphiciff is bijective, and if the inversef⁻¹:V →U is holomorphic, too. By the inverse function Theorem 1.21, we have necessarilym=n.

We call functions f1, . . . , fn:U →C (local) analytic coordinates at p, if eachfiis holomorphic atpwithfi(p) = 0, and with det_∂fi

∂x_j(p)

i,j=1...n= 0.

In other words, f1, . . . , fn are analytic coordinates at p iff f = (f1, . . . , fn)

defines a biholomorphic map between an open neighbourhood ofpinCⁿand an open neighbourhood of 0 in Cⁿ, mapping p to 0 (again, by the inverse function theorem).

Recall that a complex power series that converges in V converges uni- formly on every compact subset ofV, and that the terms can be summed up in any order. Differentiation and summation commute, that is, we can differ- entiate (respectively integrate) a power series term by term, and the radius of convergence does not change with differentiation (integration). The power series expansion of a holomorphic functionf atpis given by itsTaylor series

f(x) = ∞

|α|=0

1

α!· ∂^|α|f

∂x^α₁¹. . . ∂x^αnⁿ

(p)·(x1−p1)^α1·. . .·(xn−pn)^αn, α! :=α₁!·. . .·α_n!, which converges in some open neighbourhoodV ofp.

(2) LetU ⊂Cⁿbe an open subset. IfU =∅, we denote byO(U) theC-algebra of holomorphic functions onU,

O(U) := f :U →C holomorphic .

Moreover, we setO(∅) :={0}. The associationU → O(U) defined in this way, together with the restriction mapsO(U)→ O(V),f →f|V, forV ⊂U open, defines a presheaf OCⁿ, which is, in fact, a sheaf on Cⁿ. We identify O(U) withΓ(U,O_Cⁿ).

O_Cⁿ is called thesheaf of holomorphic functionsonCⁿ. Thesheaf of holo- morphic functions on U isOU :=O_Cⁿ|U =i⁻¹O_Cⁿ, wherei:U →Cⁿ is the inclusion map, andi⁻¹O_Cⁿ denotes the topological preimage sheaf.

We refer to the elements of the stalks OCⁿ,p, p∈Cⁿ, also as germs of holo- morphic functions at p (see A.1). That is, a germ of a holomorphic function at p is the equivalence class of a holomorphic functionf defined in an open neighbourhood of p, where two functions, defined in open neighbourhoods of p, are equivalent if they coincide in some, usually smaller, common neigh- bourhood of p. We writef_p for the class off under this relation, and call it thegerm off at p.

Note that, forp= (p1, . . . , pn)∈Cⁿ, the Taylor series expansion of holo- morphic functions atpprovides an isomorphism

OCⁿ,p ∼= C{x1−p1, . . . , xn−pn} ∼= C{x1, . . . , xn}. In particular,OCⁿ,pis an analyticC-algebra.

Definition 1.30.Let D⊂Cⁿ be an open subset. Then a subset A⊂D is called analytic at p∈D if there are an open neighbourhoodU ⊂D ofpand holomorphic functionsf1, . . . , fk∈ O(U) such that

U ∩A=V(f1, . . . , fk) := a∈Uf1(a) =. . .=fk(a) = 0 .

Ais called ananalytic subset ofD if it is analytic at everyp∈D. It is called a locally analyticsubset if it is analytic at every p∈A.

An analytic subset ofD is closed inD, and a locally analytic subset islocally closed, that is, it is the intersection of an open and a closed subset ofD.

In particular, Cⁿ =V(0) and∅=V(1) are analytic sets in Cⁿ. IfA⊂D is analytic andB ⊂Ais open, thenB is locally analytic inD, hence analytic in some open subset ofD.

A mapf :A→Bbetween analytic sets is calledholomorphic(oranalytic, or amorphism) if it is locally the restriction of a holomorphic map between open subsets of someCⁿ. IfA⊂Cⁿ,B ⊂C^mare locally analytic, this means that eachp∈Ahas an open neighbourhoodU ⊂Cⁿsuch thatf|U∩A=f|U∩A

for some holomorphic mapf= f1, . . . ,fm

:U →C^m,fi∈ O(U). Note that holomorphic maps between analytic sets inCⁿ are automatically continuous, and that the composition of holomorphic maps is again holomorphic.

f is calledbiholomorphic(or an isomorphism) if it is bijective, and if the inverse f⁻¹:B→A is also holomorphic. f is called biholomorphic at p, if there exist open neighbourhoodsU ⊂Cⁿ ofpandV ⊂C^moff(p) such that f|U∩A:U∩A→V ∩B is biholomorphic.

Remark 1.30.1.Letf :A→B be a holomorphic map between analytic sets A⊂Cⁿ, B⊂C^m which is biholomorphic at p∈A. If m=n, then f can be lifted to a biholomorphic map f:U →V for open neighbourhoods U of p, and V of q=f(p) (by the lifting Lemma 1.23). If m=n, then f can, of course, not be lifted to a biholomorphic map U →V. But, by the em- bedding Lemma 1.24, there exists some e≤min{m, n} and analytic coor- dinatesu₁, . . . , u_n at p∈Cⁿ, and v₁, . . . , v_m at q∈C^m, such that the pro- jections π₁: (u₁, . . . , u_n)→(u₁, . . . , u_e), andπ₂: (v₁, . . . , v_m)→(v₁, . . . , v_e), map U, V to open neighbourhoods U of π₁(p) and V of π₂(q) in C^e, and such that the restrictions of π₁, π₂, π₁:A∩U →π₁(A∩U) =:A and π₂:B∩V →π1(B∩V) =:B are both biholomorphic. Now, the biholomor- phic map π₂◦f◦(π₁)⁻¹:A →B can be lifted locally to a biholomorphic map between open neighbourhoods inC^e.

In the following, we present three possible definitions of a reduced complex space. The first definition is modeled on that of a complex manifold, while the other two will be sheaf theoretic definitions. The advantage of the first definition is that it provides an easier access to reduced complex spaces and their geometry. However, the sheaf theoretic description is needed later for the more general notion of a complex space. That both definitions are equivalent is a consequence of Cartan’s coherence Theorem 1.75.

Definition 1.31 (Reduced complex spaces I).LetXbe a Hausdorff topo- logical space. Then a set of pairs (Ui, ϕi)i∈I

is called ananalytic atlas (or aholomorphic atlas) forX if{Ui|i∈I}is an open covering ofX, and if, for eachi∈I,ϕi:Ui→Aiis a homeomorphism onto a locally closed analytic setAi⊂Cⁿⁱ such that, for all (i, j)∈I×I with Ui∩Uj=∅, the transition functions

ϕij :=ϕj◦ϕ⁻_i¹:ϕi(Ui∩Uj)−→ϕj(Ui∩Uj)

are morphisms of analytic sets (hence, isomorphisms withϕ⁻_ij¹=ϕji).

Each element (Ui, ϕi) of an analytic atlas is called an analytic (or holo- morphic) chart, and two analytic atlases are called equivalent if their union defines an analytic atlas for X, too.

The topological space X together with an equivalence class of analytic atlases is called a reduced complex space.

A reduced complex spaceX is acomplex manifold of dimensionniff there exists an analytic atlas {(Ui, ϕi)|i∈I} such that each ϕi is a homeomor- phism onto an open subset Di⊂Cⁿ.

Example 1.31.1.Each local analytic subset A⊂Cⁿ with the (class of the) standard atlas, that is, the atlas consisting of the global chart (A,id_A), is a reduced complex space.

In particular, we always considerCⁿas a reduced complex space, equipped with the standard atlas{(Cⁿ,id_Cn)}.

Definition 1.32.A morphism of reduced complex spaces X, Y with analytic atlases (U_i, ϕ_i)i∈I

, (V_j, ψ_j)j∈J

, is a continuous map f :X→Y such that for all (i, j)∈I×J with f⁻¹(V_j)∩U_i=∅the composition

ϕi

f⁻¹(Vj)∩Ui

ϕ⁻¹_i

−→f⁻¹(Vj)∩Ui

−→f Vj ψj

−→ψj(Vj) is a morphism of analytic sets.

Such an f is called an isomorphism of reduced complex spaces if it is a bijection, and if the inverse f⁻¹ is a morphism of reduced complex spaces, too.

A morphismf :X →Cis called ananalytic(orholomorphic)functionon X. We denote byO(X) the set of analytic functions onX, which is obviously a C-algebra.

IfX is a reduced complex space, and ifU ⊂X is an open subset, thenU is a complex space, too, with atlas (U∩U_i, ϕ_i|U∩Ui)

. Such aU is called anopen subspace ofX. ForV ⊂U open inX, the restriction mapO(U)→ O(V) is a morphism of C-algebras. Thus, we get a presheaf OX on X, which is in fact a sheaf, called the sheaf of analytic (or holomorphic) functions on X. Note that, by definition, each analytic functionU →Cis continuous. Thus,OX is a subsheaf of the sheaf CX of continuous complex valued functions onX.

Similar to the above, we refer to the elements of the stalksOX,p, p∈X, as germs atpof holomorphic functions onX. Each such germ is represented by a holomorphic functionf ∈ OX(U), defined on an open neighbourhoodU of p. Conversely, each f ∈ OX(U) defines a unique germ at p∈U, which is denoted by fp.

If (U, ϕ) is an analytic chart with p∈U, and with ϕ a homeomorphism from U onto a locally closed analytic set A⊂Cⁿ, then f →f◦ϕ⁻¹ defines an isomorphism of sheavesOA∼=ϕ_∗(OX|U). In particular, we get

OX,p∼=OA,ϕ(p).

Next, we show thatOA,ϕ(p) (hence OX,p) is an analytic C-algebra. Without restriction, assume that ϕ(p) =0, and let A be an analytic subset of some open neighbourhoodV ⊂Cⁿ of0. Then each germ at0of an analytic func- tion onA is the restriction to A of a germ in OV,0=OCⁿ,0. Moreover, two germsf0, g0∈ OCⁿ,0induce the same element ofOA,0iff they have small holo- morphic representatives f, g:W →C satisfying f|A∩W −g|A∩W = 0. Thus, OA,0=OCⁿ,0/I(A), where I(A) denotes the ideal

I(A) := f0∈ O_Cⁿ,0∃f ∈ O_Cⁿ(W) representing f0and f|A∩W = 0 . SinceOCⁿ,0 is Noetherian,I(A) is finitely generated, thusOA,0is an analytic C-algebra.

These considerations show that each reduced complex space in the sense of Definition 1.31 is, in a natural way, a reduced complex space in the sense of the following definition:

Definition 1.33 (Reduced complex spaces II). Areduced complex space is a C-analytic ringed space (X,OX), where X is a Hausdorff topological space, and whereOX is a subsheaf ofCX satisfying

each pointp∈X has an open neighbourhoodU ⊂X such that (U,OX|U) is isomorphic to (A,OA) asC-analytic ringed space, where Ais a locally closed analytic subset in someCⁿ and where OAis the sheaf of holomorphic functions on A.

(1.3.1)

Amorphism f, f

: (X,OX)→(Y,OY) of reduced complex spaces is just a morphism ofC-analytic ringed spaces, that is,f :X →Y is continuous, and f:OY →f_∗OX is a morphism of sheaves of localC-algebras (A.6).

The equivalence between Definitions 1.31, 1.32 and Definition 1.33 is specified by the following proposition:

Proposition 1.34.Associating to a complex spaceX (in the sense of Defini- tion 1.31) theC-analytic ringed space(X,OX), withOX the sheaf of holomor- phic functions on X, and associating to a morphism f :X →Y of complex spaces the morphism(f, f) : (X,OX)−→(Y,OY), with

f:OY −→f_∗OX, g −→g◦f forg∈ OY(V), V ⊂Y open, defines a functor from the category of reduced complex spaces to the full⁵ subcategory ofC-analytic ringed spaces satisfying the conditions of Definition 1.33. This functor is an equivalence of categories. In particular, two reduced complex spacesX, Y are isomorphic iff(X,OX)and(Y,OY) are isomorphic asC-analytic ringed spaces.

5 LetC be a category. Then a subcategoryB of C is called afull subcategory if Hom_B(A, B) = Hom_C(A, B) for any two objectsA, B ofB.

Proof. Letf :X→Y be a morphism of reduced complex spaces (in the sense of Definition 1.32). Then, using local charts, it follows that the induced maps f_p:OY,f(p)→ OX,pare morphisms ofC-analytic algebras. Hence, (f, f) is a morphism ofC-analytic ringed spaces. The functor properties are obvious.

It remains to see that this functor defines an equivalence of categories.

The key point is that for a morphism (f, f) : (A,OA)→(B,OB), withA, B analytic subsets of some open sets V ⊂Cⁿ, W ⊂C^m, the continuous map f :A→Buniquely determinesf. Indeed, as for eachp∈A, the induced map of stalks f_p is a morphism ofC-algebras, we have the commutative diagram

OB,f(p)

f_p

OA,p

OB,f(p)/m_B,f(p) C ^id C OA,p/mA,p.

(1.3.2)

Iff = (f1, . . . , fn), and ifxi∈ OB(B),i= 1, . . . , m, are induced by the coor- dinate functions onW ⊂C^m, we read from this diagram that

f_k(p) =

(x_k)_f(p) modmB,f(p)

=

f_p(x_k)_p modmA,p

=f(x_k)(p). Since each (continuous) map A→B is uniquely determined by the values at all points p∈A, we get f_k =f(x_k) and it follows from Remark 1.1.1 (5) and Lemma 1.14 that f is uniquely determined by the images f(xk), k= 1, . . . , m.

Now, we can define the inverse functor: let (f, f) : (X,OX)→(Y,OY) be a morphism of C-analytic ringed spaces satisfying the requirements of Defi- nition 1.33. Then the property (1.3.1) implies that there is an open covering {Ui|i∈I} of X and isomorphisms (ϕi, ϕ_i) : (Ui,OX|Ui)→(Ai,OAi) of C- analytic ringed spaces with A_i⊂Cⁿⁱ locally analytic.

By the above, the components of the transition functionsϕ_ij :=ϕ_j◦ϕ⁻_i ¹ are given by ϕ_ij(xk) =

(ϕ⁻_i¹)◦ϕ_j

(xk),k= 1, . . . , ni. Thus, they are holo- morphic functions. It follows that{(Ui, ϕi)|i∈I}is an analytic atlas forX, and the equivalence class of this atlas is independent of the chosen covering and isomorphisms. EquippingX andY in this way with analytic atlases, it is clear that f :X →Y is a morphism of reduced complex spaces in the sense

of Definition 1.32.

Next, we come to the definition of a general complex space. The definition is similar to Definition 1.33, except thatOX may have nilpotent elements and, hence, cannot be a subsheaf of the sheaf of continuous functions onX. Nilpo- tent elements appear naturally when we consider fibres of holomorphic maps.

Indeed, the behaviour of the fibres of a morphism can only be understood if we take nilpotent elements into account.

Definition 1.35 (Complex Spaces). LetD⊂Cⁿ be an open subset.

(1) An ideal sheafJ ⊂ OD is called of finite type if, for every pointp∈D, there exists an open neighbourhoodU ofpin D, and holomorphic functions f1, . . . , fk ∈ O(U) generatingJ overU, that is, such that

J |U =f1OU +. . .+fkOU. ThenOD/J is a sheaf of rings onD, and we define

V(J) := p∈DJp=OD,p

= p∈D(OD/J)p= 0 .

to be theanalytic set inD defined byJ. This is also thesupportofOD/J: V(J) = supp (OD/J).

For eachp∈Dwe haveJp=OD,p ifff(p) = 0 for allf ∈ Jpand, hence, for a neighbourhoodU as above

V(J)∩U = V(f₁, . . . , f_k) = p∈Uf₁(p) =. . .=f_k(p) = 0 . In particular, forJ of finite type,V(J) is an analytic subset ofD.

(2) ForJ ⊂ OD an ideal sheaf of finite type andX :=V(J), we set OX := (OD/J)

X. Then (X,OX) =

V(J),(OD/J)|X

is a C-analytic ringed space, called a complex model space or thecomplex model space defined byJ.

(3) Acomplex space, orcomplex analytic space, is a C-analytic ringed space (X,OX) such thatX is Hausdorff and, for everyp∈X, there exists a neigh- bourhoodU ofpsuch that (U,OX|U) is isomorphic to a complex model space asC-analytic ringed space.

We usually writeX instead of (X,OX).OXis called thestructure sheaf of X, and, for U ⊂X open, each sectionf ∈Γ(U,OX) is called aholomorphic function onU.

Amorphism f, f

: (X,OX)→(Y,OY)of complex spaces is just a mor- phism ofC-analytic ringed spaces. Such a morphism is also called aholomor- phic map. We write Mor(X, Y) for the set of morphisms (X,OX)→(Y,OY).

An isomorphism of complex spaces is also called abiholomorphic map.

Let (X,OX) be a complex model space, andp∈X. Then OX,p ∼= OCⁿ,0/J0 ∼= C{x1, . . . , xn}/f1, . . . , fk

for somef1, . . . , fk ∈C{x}=C{x1, . . . , xn}. We say thatx1, . . . , xn arelocal (analytic) coordinates and thatf1, . . . , fk arelocal equations forX atp.

On the other hand, given convergent power seriesf1, . . . , fk∈C{x}, there is an open neighbourhoodU ⊂Cⁿof0such that eachfidefines a holomorphic

function fi:U →C. Setting J :=f1OU +. . .+fkOU, the complex (model) space

(X,OX) :=

V(J),(OU/J)|V(J)

satisfiesOX,0∼=C{x}/f₁, . . . , f_k. Thus, each analyticC-algebra appears as the stalk of the structure sheaf of a complex space.

Definition 1.36 (Reduced Complex Spaces III). A complex space (X,OX) is calledreduced if, for eachp∈X, the stalkOX,pis a reduced ring, that is, has no nilpotent elements.

Remark 1.36.1.The three definitions of a reduced complex space coincide.

The equivalence of Definitions 1.31 and 1.33 was already shown (Proposition 1.34). Definition 1.33 implies 1.36 by the fact that OA∼=OD/J(A) where J(A)⊂ ODis the full ideal sheaf ofA⊂D(see Definition 1.37 below), which is of finite type by Cartan’s Theorem 1.75. On the other hand, ifXis a reduced complex space in the sense of Definition 1.36 then, locally,OX is isomorphic to (OD/J)|X with X =V(J)⊂D. Then J is contained in the full ideal sheaf J(X) and, since Jp is a radical ideal for p∈X, the Hilbert-R¨uckert Nullstellensatz (Theorem 1.72) implies thatJ =J(X).

Let D⊂Cⁿ be any open set such that A is an analytic subset of D. Then each holomorphic function on Alocally lifts to a holomorphic function on an open set U ⊂D. Moreover, two holomorphic functions f, g onU induce the same holomorphic function on A∩U iff (f−g)(p) = 0 for allp∈A. Thus,

OA∼=

OD/J(A)

A, where J(A)⊂ OD is the full ideal sheaf ofA⊂D:

Definition 1.37.Let (X,OX) be a complex space andM ⊂X any subset.

Then thefull ideal sheaf or thevanishing ideal sheaf J(M) ofM is the sheaf defined by

J(M)(U) = f ∈ OX(U)M ∩U ⊂V(f) , forU ⊂X open.

Remark 1.37.1. (1) It is easy to see thatJ(M) is a radical sheaf (A.5), and that for an analytic set A⊂D we haveA=V(J(A)).

Cartan’s Theorem 1.75 says that the full ideal sheafJ(A) is coherent and the Hilbert-R¨uckert Nullstellensatz (Theorem 1.72) says thatJ(A) =√

J for each ideal sheaf J such that A=V(J).

(2) The full idealJ(X) coincides with the nilradicalNil(OX) =

0ofOX. Definition 1.38. (1) Aclosed complex subspaceof a complex space (X,OX) is a C-analytic ringed space (Y,OY), given by an ideal sheaf of finite type JY ⊂ OX such thatY =V

JY

:= supp OX

JY

andOY = OX

JY

|Y.

In particular, we define the reduction of a complex space (X,OX) to be the (reduced) closed complex subspace

Xred:= (X,OX/J(X))

defined by the full ideal sheafJ(X) ofX. We also say thatOX/J(X) is the reduced structure (sheaf )onX.

Anopen complex subspace (U,OU) of (X,OX) is given by an open subset U ⊂X andOU =OX|U.

(2) A morphismf :X→Y of complex spaces is called anopen(resp.closed) embedding if there exists an open (resp. closed) subspaceZ⊂Y and an iso- morphismg:X −→^∼= Z such that f =i◦g, wherei:Z →Y is the inclusion map.

Remark 1.38.1. (1) If (Y,OY) is a closed complex subspace of a complex space (X,OX), thenY is closed inX and (Y,OY) is a complex space.

Indeed, by the coherence theorem of Oka (Theorem 1.63, below), the struc- ture sheaf of any complex space is coherent. As the ideal sheafJY ⊂ OX is supposed to be of finite type, it is also coherent, and the same holds for the quotient OX/JY (A.7, Fact 3). Hence, Y is the support of a coherent OX- sheaf, thus closed inX (A.7, Fact 1).

To see that (Y,OY) is a complex space, we may assume that (X,OX) is a complex model space, defined by an ideal sheafI ⊂ OD of finite type (with D⊂Cⁿ an open subset). Let p∈Y. Since JY is of finite type, there is an open neighbourhoodU ⊂D ofpand functionsf₁, . . . , f_k ∈ OD(U) such that the corresponding residue classesf₁, . . . , f_k ∈ OX(X∩U) generateJY|X∩U. Then (Y ∩U,OY|Y∩U) is the complex model space inU defined by the finitely generated idealI|U+f₁OU+. . .+f_kOU ⊂ OU.

(2) A complex space (X,OX) is reduced iffXred= (X,OX).

Example 1.38.2. (1) An important example of non-reduced complex spaces are fat points (or Artinian complex space germs). As such we denote non- reduced complex spaces X satisfying Xred={pt}. That is, the underlying topological space of a fat point consists only of one point, and the struc- ture sheaf of X is uniquely determined by the stalk at this point. When defining a fat point, we usually specify only this stalk. For instance, we call Tε:=

{pt},C[ε]

thefat point of length two, since the definingC-algebra is non-reduced (asε²= 0) and a two-dimensional complex vector space. More generally, each analyticC-algebraAwith 1<dim_CA <∞defines a fat point.

Note thatTεmay be embedded as a closed complex subspace in each fat point.

(2) Let (X,OX) =

V(y),O_C²/xy, y²

⊂

C²,O_C²

then the reduction of X is Xred=

V(y),OC²/y∼= (C,OC). Here, X is the union of the x-axis and a fat point with support {0}. Indeed, the primary decomposition of I=xy, y²=y ∩ x, y²yieldsX =V(y)∪V(x, y²) withV(x, y²)∼=T_ε.

Definition 1.39.LetXbe a complex space,p∈Xandmpthe maximal ideal ofOX,p. Then we define

dimpX := Krull dimension ofOX,p, thedimension ofXatp, dimX := sup

p∈X

dim(X, p), thedimension ofX,

edimpX := dim_Cmp/m²_p, theembedding dimension of Xatp.

Note that dim_pX = dim_pX_red (see B.2), while the embedding dimension of Xred atpmay be strictly smaller than the embedding dimension of X atp.

We refer to a reduced complex space X as a curve (respectively as a surface) if dimpX = 1 (respectively dimpX = 2) for allp∈X.

Remark 1.39.1.Locally at a pointp∈X, we can identify each complex space (X,OX) with a complex model space

V(J),OD/J

, where D is an open set in Cⁿ, and whereJ =f1OD+. . .+fkOD⊂ OD. WhileOD/J is part of the structure, the embedding X ⊂D⊂Cⁿ and, hence,J is not part of the structure. Indeed, we may embed (X,OX) in different ways as a subspace of C^mfor variousm. By the embedding Lemma 1.24, the minimal possiblemis edim_pX, which is the reason for calling edim_pX the embedding dimension of X atp(Exercise 1.3.3).

Definition 1.40.A complex spaceX is calledregular at p∈X, if dimpX = edimpX ,

that is, if OX,p is a regular local ring. Thenpis also called aregular point of X. A point ofX is calledsingular if it is not a regular point ofX.

By Proposition 1.48 below, a complex spaceX is a complex manifold iffX is regular at eachp∈X.

Definition 1.41.A morphism f, f

:X→Y of complex spaces is called regular at p∈X if the induced ring map f_p:OY,f(p)→ OX,p is a regular morphism of analyticK-algebras.

Here, a morphism ϕ:A=Kx/I →B of analytic K-algebras is called regular, orBis called aregularA-algebra, ifBis isomorphic (asA-algebra) to a free power series algebra overA, that is, ifB ∼=Ay:=Kx,y/IKx,y, where x= (x1, . . . , xn),y= (y1, . . . , ym) are disjoint sets of variables.

Instead of “regular”, the notionssmooth ornon-singular are used as well.

Remark 1.41.1.Recall from the proof of Proposition 1.34 that a morphism (f, f) : (X,OX)→(Y,OY) ofreduced complex spaces is uniquely determined byf. If (X,OX) is not reduced, this is no longer true. As a concrete example, consider the fat point T_ε=

{pt},C[ε]

. We may supplement the continuous mapTεpt →0∈Cto a morphism

f, f

:Tε→Cby settingf(x) :=aε, where a∈Cis arbitraily chosen.

No documento Introduction to (páginas 46-66)