Lectures on Cycle Spaces JÓningÓlfur MagnÚsson

Lectures on Cycle Spaces JónIngólfur Magnússon

1 Introduction

These lecture notes are based on a series of lectures given by the author at Ruhr Univer- sitätBochum in February 2004. It was supposed to be a “blitz” introduction to the theory of cycle spaces for the research group in Complex Geometry of the Mathematics Depart- ment. The first half is a rough introduction to analytic geometry and it is complemented by an appendix containing algebraic supplements. I would like to thank Professor Peter Heinzner and Professor Alan Huckelberry for having invited me to give these lectures. Special thanks are also due to Christian Miebach for his invaluable help in preparing these notes. He typed the first draft based on the lectures and wrote the original version of the appendix. Equally he read carefully the final version of these notes and made several interesting and helpful remarks. Finally I would like to thank the staff of the Mathematics Department for their kind assistance during my stay in Bochum.

2 Contents

1 Complex spaces 5 1.1 Ringed spaces ...... 5 1.2 Morphisms of ringed spaces ...... 6 1.3 Local models ...... 9 1.4 Complex spaces ...... 11 1.5 Closed complex subspaces ...... 13 n n 1.6 The bijection Hol(X, C ) → OX (X) ...... 16 1.7 Geometry of complex spaces ...... 17 1.8 Maximum principle ...... 23 1.9 Analytic sets ...... 24 1.10 Normal spaces ...... 30

2 Operations on complex spaces 31 2.1 Inverse images and ﬁbers ...... 31 2.2 Images of complex spaces ...... 32 2.3 Fiber products ...... 33

3 Meromorphic functions and maps 35 3.1 Meromorphic functions ...... 35 3.2 Modiﬁcations ...... 37 3.3 Meromorphic maps ...... 39

4 Differential forms 42 4.1 Differential forms on complex spaces ...... 43 4.2 Kähler–spaces ...... 43 4.3 Integration on complex spaces ...... 46

5 Flatness 47 5.1 Flat maps ...... 47 5.2 Flat families of compact subspaces ...... 49

6 The Douady space of a complex space 50

7 The space of holomorphic maps 51 7.1 The universal holomorphic map ...... 51 7.2 The group of automorphisms ...... 52

8 Ramiﬁed coverings 53 8.1 Analytic coverings ...... 53 8.2 Ramiﬁed coverings ...... 54

9 The Barlet space of a (reduced) complex space 55 9.1 Analytic families ...... 55 9.2 The Barlet space ...... 58

3 10 The morphism Douady to Barlet 59 10.1 Fundamental cycles ...... 59 10.2 Deﬁnition of the morphism ...... 60 10.3 Some special cases ...... 61

11 Compact subsets of Barlet spaces 62 11.1 The theorem of Bishop and its consequences ...... 62 11.2 The Fujiki class ...... 64 11.3 Corresponding results for the Douady space ...... 65

A Algebraic supplements 65 A.1 Elementary theory of rings ...... 65 A.2 The full ring of fractions ...... 67 A.3 Local rings and Noetherian rings ...... 69 A.4 The ring C{z1, . . . , zn} ...... 71

4 1 Complex spaces

1.1 Ringed spaces

Deﬁnition 1.1.1 (Ringed space) A ringed space is a couple X = (|X|, SX ) where |X| is a topological space and SX is a sheaf of C–algebras having the following properties:

(i) For each x ∈ |X| the stalk SX,x is a local algebra.

(ii) For each open set U ⊂ |X| and for each section s ∈ SX (U) the map

RedU (s): U → C, x 7→ s(x) := ε(sx), is continuous, where ε denotes the evaluation map (See A.3.4).

The map RedU (s) is called the reduction of s. The sheaf SX is usually referred to as the structure sheaf of X. If no confusion is possible, the topological space |X| will also be denoted by X.

Remark. Let CX denote the sheaf of continuous functions on |X|, i.e. for every open set U of X CX (U) = {f : U → C; f is continuous}.

Then the family of maps {RedU ; U ⊂ X open}, where

RedU : S (U) → C (U), s 7→ RedU (s), is a morphism of sheaves, denoted by

Red: SX → CX .

Examples.

(1) Let X be a topological space. Then the couple (X, CX ) is a ringed space. (See A.3).

n p (2) For every open set U in R and p ∈ N0 ∪ {∞} the couple (U, CU ) is a ringed space, p where CU denotes the sheaf of complex valued p times continuously diﬀerentiable functions, i.e. for each open subset V of U

p CU (V ) := {f : V → C; f is continuously diﬀerentiable of order p}.

n (3) Let D be an open subset of C and let OD be the sheaf of holomorphic functions in D. Then the couple (D, OD) is a ringed space.

Deﬁnition 1.1.2 Let X = (|X|, SX ) be a ringed space. Let Red(SX ) ⊂ C|X| be the sheaf associated to the presheaf U → Red(SX (U)). Then

Red(X) := Xred := (|X|, Red(SX )) is a ringed space, called the reduction of X. If the morphism Red: SX → CX is injective and consequently induces an isomorphism ∼ SX = Red(SX ), then X is called a reduced ringed space and the sheaves SX and Red(SX ) will be systematically identiﬁed in that case.

5 Remarks.

(i) For a reduced ringed space X the structure sheaf SX is systematically identiﬁed with the sheaf Red(SX ) so in that case one often writes s instead of Red(s) for a local section s of SX .

(ii) The sheaf morphism Red: SX → Red(SX ) is automatically surjective and consequently induces an isomorphism ∼ Red(SX ) = SX / Ker(Red).

(iii) Let U be an open subset of X, let x ∈ U and let s ∈ SX (U). Then

s(x) = 0 is equivalent to sx ∈ mx,

where mx denotes the maximal ideal of SX,x.

Examples.

(1) The ringed spaces in examples 1, 2 and 3 are all reduced.

(2) A ringed space of the form ({x}, S ), where x is a point and S is a local C−algebra, is called a one point ringed space. It is reduced if and only if S = C.

1.2 Morphisms of ringed spaces

Let (X, SX ) be a ringed space. For every topological space Y and every continuous map f : X → Y the direct image sheaf f∗S is a sheaf of C–algebras on Y . Note however, that −1 f∗S is not necessarily a sheaf of local C–algebras because it may happen that f (V ) = ∅ for some non-empty open subset V of Y and consequently (f∗SX )y = 0 for all y ∈ V .

Deﬁnition 1.2.1 (Morphism of ringed spaces) A morphism from a ringed space X to a ringed space Y is a couple ˜ (f, f):(|X|, SX ) → (|Y |, SY ), ˜ where f : |X| → |Y | is a continuous map and f : SY → f∗SX is a morphism of sheaves of C–algebras. ˜ Deﬁnition 1.2.2 Let (f, f):(|X|, SX ) → (|Y |, SY ) and (g, g˜):(|Y |, SY ) → (|Z|, SZ ) be morphisms of ringed spaces. Then the morphism of ringed spaces ˜ (g ◦ f, g∗(f) ◦ g˜):(|X|, SX ) → (|Z|, SZ ), is called the composition of (f, f˜) and (g, g˜).

6 Notation. If there is no risk of ambiguity we usually write f instead of (f, f˜). Compo- sition of two morphisms f and g will be denoted fg or f ◦ g.

Deﬁnition 1.2.3 A morphism f : X → Y of ringed spaces is called an isomorphism if there exists a morphism g : Y → X with fg = idY and gf = idX .

Examples.

(1) Let X and Y be topological spaces and let f : X → Y be a continuous map. For each open subset V of Y the pull-back map

˜ −1 fV : CY (V ) → CY (f (V )), s 7→ s ◦ f

is a morphism of C–algebras. Hence we get a morphism of sheaves of C–algebras ˜ f : CY → f∗CX

and a morphism of ringed spaces ˜ (f, f):(X, CX ) → (Y, CY ).

n (2) Let D be an open subset of C and let i: OD → CD be the inclusion. Then

(idD, i):(D, CD) → (D, OD)

is a morphism of ringed spaces.

(3) For every ringed space X there is a canonical morphism Xred → X given by

|X| → |X|, x 7→ x, and SX → Red(SX ), s 7→ Red(s).

˜ Remark. Let (f, f):(|X|, SX ) → (|Y |, SY ) be a morphism of ringed spaces. Then, for each x ∈ X, we have an induced stalk map ˜ fx : SY,f(x) → SX,x obtained by composing the map SY,f(x) → (f∗SX )f(x) with the canonical map (f∗SX )f(x) → ˜ ˜ SX,x. The maps fx are morphisms of local C–algebras and they determine f.

The following lemma is obvious.

˜ Lemma 1.2.4 Let (f, f):(|X|, SX ) → (|Y |, SY ) be an isomorphism of ringed spaces. Then, for every x ∈ X, the map ˜ fx : SY,f(x) → SX,x is an isomorphism of local C–algebras.

Now we turn our attention to the way the morphisms of ringed spaces behave under reduction.

˜ Let X = (|X|, SX ) and Y = (|Y |, SY ) be ringed spaces and let (f, f): X → Y be a ˜ morphism. Let V ⊂ Y be an open set and let s ∈ SY (V ). Then we have f(s) ∈ −1 (f∗SX )(V ) = SX (f (V )). Deﬁne ˜ ˜ f(s)x := fx(sf(x)) ∈ SX,x.

Lemma 1.2.5 In the above setting the identity s(f(x)) = f˜(s)(x) holds for all x ∈ f −1(V ).

Proof. For each x ∈ f −1(V ) we have the unique decomposition (see remark (ii) following deﬁnition A.3.4) sf(x) = s(f(x)) +s ¯f(x) ∈ C ⊕ mY,f(x) = SY,f(x). ˜ Since fx : SY,f(x) → SX,x is a morphism of local C–algebras we get ˜ ˜ ˜ ˜ f(s)x = fx sf(x) = fx s(f(x)) +s ¯f(x) = s(f(x)) + fx s¯f(x) ∈ C ⊕ mX,x = SX,x. ˜ Hence f(s)(x) = s(f(x)). Corollary 1.2.6 Let (f, f˜): X → Y be a morphism of ringed spaces. Then there exists a ˜ unique morphism (f, fred): Xred → Yred such that the following diagram commutes:

˜ (f,fred) Xred / Yred

X / Y (f,f˜) where the vertical arrows are the canonical morphisms.

˜ Proof. Deﬁne fred : Red(SY ) → Red(SX ) by ˜ ˜ fred(Red(s)) := Red(f(s)). ˜ Then the diagram above is commutative and the uniqueness of fred follows from lemma 1.2.5. Corollary 1.2.7 Let (f, f˜): X → Y be a morphism of ringed spaces with X reduced. Then f˜ is determined by f, more precisely f˜(s) = Red(s) ◦ f.

8 1.3 Local models n n Let D be a domain in C , let f1, . . . , fk ∈ OC (D). We put

J := hf1, . . . , fki := OD · f1 + ··· + OD · fk and N := N(f1, . . . , fk) := {x ∈ D; f1(x) = ··· = fk(x) = 0}.

Lemma 1.3.1 The couple

V := V (f1, . . . , fk) := (N, (OD/J )|N) is a ringed space.

Proof. For every x in N one has

(OD/J )x = OD,x/Jx and Jx 6= OD,x, so by lemma A.3.6 each stalk of the sheaf (OD/J )|N is a local C–algebra. Moreover, if mx denotes the maximal ideal of OD,x the maximal ideal of OD,x/Jx is mx/Jx. Consider the commutative diagram

ι ε C / OD,x / OD,x/mx

id π =∼

/ ( / )/(m / ) C ¯ι / OD,x Jx ¯ε / OD,x Jx x Jx

Let U ⊂ N be an open set and let s ∈ (OD/J )(U). Then the continuity of the function ¯ε: x 7→ s(x) follows from the fact that locally the section s can be lifted to a section of (OD/J )|N.

Deﬁnition 1.3.2 (Local model) The ringed space V = V (f1 . . . , fk) is called a local model with structure sheaf OV := (OD/J )|N. The elements of OV (U), where U is an open subset of N, are called holomorphic functions on U.

Warning. A holomorphic function f ∈ OV (U) as deﬁned above need not be a function in the set theoretical sense. It has an underlying continuous function

Red f : U → C, x 7→ f(x) but diﬀerent holomorphic functions in OV (U) can have the same underlying continuous function. (See example 4 below).

n Terminology. Let V = V (f1 . . . , fk) be a local model in a domain D in C and let n z = (z1, . . . , zn) be the standard coordinates in C . Then V is often referred to as the local model deﬁned by the equations f1(z) = ··· = fn(z) = 0.

9 Examples.

n (1) Let D be an open set in C and let f denote the zero function on D. Then J := n hfi = 0 and N(f) = D. It follows that (OC /J )|D = OD. Thus, (D, OD) is a local model. Let z denote the usual coordinate on C. For every morphism of ringed spaces ˜ (φ, φ):(D, OD) → (C, OC) it follows from 1.2.7 that φ = z ◦ φ = φ˜(z)

so φ ∈ OD(D). Conversely, for every φ ∈ OD(D) one obtains a morphism of ringed spaces ˜ (φ, φ):(D, OD) → (C, OC) where φ˜ is deﬁned by pull-back. Hence the holomorphic functions on (D, OD) correspond exactly to the morphisms (of ringed spaces) D → C. 2 (2) Consider the holomorphic function (z, w) 7→ zw on C . Then V (zw) = (N, OV ) with

2 N = N(zw) = {(z, w) ∈ C ; z = 0 or w = 0}

and ∼ OV,(a,b) = C{z − a, w − b}/hzwi. For example, at the point (0, 1) we get, since the function w is a unit in a neighborhood of (0, 1), ∼ ∼ OV,(0,1) = C{z, w − 1}/hzwi = C{z, w − 1}/hzi = C{w − 1}.

2 2 3 2 3 (3) Let U = C , let f(z, w) = w −z and consider the local model V = V (w −z ), usually 2 3 called Neil’s parabola. In order to determine the stalk OV,(0,0) = C{z, w}/hw − z i we use the map 2 3 Φ: C{z, w} → C{t}, z 7→ t , w 7→ t .

2 3 Claim. The map Φ induces an injective morphism C{z, w}/hw −z i → C{t} whose image is   X k  akt ; ak ∈ C, a1 = 0 . k≥0 

Proof. It is clear that w2 − z3 ∈ Ker(Φ) and it is easily seen that   X k  akt ; ak ∈ C, a1 = 0 k≥0 

10 is the image of Φ, so we only have to prove that Ker(Φ) ⊂ hw2 − z3i. Let f(z, w) ∈ Ker(Φ) and write

2 3 f(z, w) = g(z, w)(w − z ) + a0(z) + a1(z)w.

This is possible because of the identity w2 = (w2 − z3) + z3. We have to show that a0(z) + a1(z)w = 0. Suppose that this is not the case, i.e. a0(z) + a1(z)w 6= 0. Then we can write

k a0(z) + a1(z)w = z (b0(z) + b1(z)w),

where (b0(0), b1(0)) 6= (0, 0). If b0(0) 6= 0 this implies

2k Φ(a0(z) + a1(z)w) = t (b0(0) + terms of order > 0) 6= 0,

and if b1(0) 6= 0 we get

2k 2 3 Φ(a0(z) + a1(z)w) = t (b0(t ) + b1(0)t + terms of order ≥ 5) 6= 0,

which in both cases contradicts the assumption f(z, w) ∈ Ker(Φ). k (4) Let k ∈ N and consider the holomorphic function z 7→ z . Then

k k V (z ) = ({0}, C{z}/hz i)

is a one point ringed space often called a k−fold point. Obviously

k k−1 OV = C{z}/hz i = C ⊕ C ε ⊕ · · · ⊕ C ε ,

where εk = 0 so V (zk) is reduced if and only if k = 1. The underlying continuous k−1 function of a holomorphic function a0 + a1 ε + ··· + ak−1 ε is {0} → C, 0 7→ a0.

Notice that any local model in C having the point 0 as an underlying topological space is of the above form.

1.4 Complex spaces

Deﬁnition 1.4.1 (Complex space) A complex space is a ringed space X = (|X|, OX ) satisfying the following conditions:

(i) |X| is Hausdorﬀ.

The sheaf OX is called the structure sheaf of X. Again, the elements of OX (U) with U ⊂ X open, are called holomorphic functions on U.

Remark. Let X be a complex space and let V be an open subset of |X|. Put OV := OX |V . Then (V, OV ) is a complex space. Such spaces are called open complex subspaces.

11 Remark. Let X be a complex space. For all x ∈ X the stalk OX,x is isomorphic to an algebra C{z1, . . . , zn}/a where a ⊂ C{z1, . . . , zn} is a ﬁnitely generated ideal. Such algebras are called analytic.

Deﬁnition 1.4.2 Let X and Y be complex spaces. Morphisms [resp. isomorphisms] X → Y are called holomorphic [resp. biholomorphic] maps. A biholomorphic map X → X will often be called an automorphism of X.

Notation. The set of holomorphic maps X → Y will be denoted by Hol(X,Y ) and the set of automorphisms of X will be denoted by Aut(X).

Remark. In proposition 1.6.1 we will see that the holomorphic functions on a complex space can be identiﬁed with the holomorphic maps X → C.

Examples.

(1) Every local model is a complex space.

n (2) Let D be an open set in C . Then every ﬁnitely generated ideal I ⊂ OD determines a complex space (X, OX ) with

X = Supp(OD/I ) and OX := (OD/I )|X.

(3) Let M be a complex manifold of dimension n and let OM be the sheaf of holomorphic functions on M. Then (M, OM ) is a ringed space which is locally isomorphic to a local n model of the form (D, OD) with D an open subset of C , so (M, OM ) is a complex space. Moreover, it is easily seen that holomorphic maps between complex manifolds in the ordinary sense are the same as the holomorphic maps in the sense of complex spaces. In other words the category of complex manifolds is a full subcategory of the category of complex spaces.

2 (4) Consider the complex spaces V1 := V (z) and V2 := V (z ). Both spaces have {0} as underlying topological space, and

OV1 = C{z}/hzi = C

and 2 OV2 = C/hz i = C + C ε, 2 where ε = 0. Therefore we see that V2 is not reduced since the reduction of the zero function and the reduction of ε are both the zero function, but ε is not the zero function. Let us now consider some sets of holomorphic maps. In the sequel f : {0} → {0} will denote the identity map; it is the unique (continuous) map {0} → {0}.

12 (a) Hol(V1,V1).

Since f∗OV1 = OV1 and since every C–algebra homomorphism maps 1 onto 1, we get

Hol(V1,V1) = {φ: C → C; φ is a C–algebra homomorphism} = {idC}.

In particular we get Hol(V1,V1) = Aut(V1,V1).

(b) Hol(V2,V2).

Since f∗OV2 = OV2 and since every C–algebra homomorphism φ: C⊕C ε → C⊕C ε is determined by its values on the basis 1 and ε and since φ must map 1 onto 1 and ε onto λ ε for some λ ∈ C it follows that

Hol(V2,V2) = {φλ : C ⊕ C ε → C ⊕ C ε; λ ∈ C},

where φλ(a + b ε) = a + λb ε. It is easily seen that φλ ◦ φµ = φλµ and that φλ is an automorphism if and only if λ 6= 0. Hence a group isomorphism

∼ ∗ Aut(V2) = C .

Since f∗OV1 = OV1 and since each C–algebra homomorphism φ: C⊕C ε → C must map ε to zero, the set Hol(V1,V2) consists only of the map φ(a + b ε) = a.

(d) Hol(V2,V1). As above we get

Hol(V2,V1) = {φ: C → C ⊕ C ε; φ is a C–algebra homomorphism}

which consists only of the natural inclusion of C into C ⊕ C ε. (e) From proposition 1.6.1 below we will see that ∼ Hol(Vj, C) = OVj (Vj) = OVj

for j = 1, 2.

1.5 Closed complex subspaces In this subsection we generalize the method of constructing a local model in a domain n D ⊂ C out of a ﬁnitely generated ideal I ⊂ OD.

Let (X, SX ) be a ringed space and let I ⊂ SX be an arbitrary ideal.

Deﬁnition 1.5.1 The set

N(I ) := {x ∈ X; Ix 6= SX,x} is called the zero set of I in X.

13 Remark. We have

N(I ) = {x ∈ X; Ix ⊂ mSX,x } = {x ∈ X;(SX /I )x 6= 0} = {x ∈ X; 1x 6∈ Ix}.

This implies N(I ) = Supp(SX /I ). Since the set X \ N(I ) = {x ∈ X; 1x ∈ Ix} is open in X, it follows that N(I ) is closed in X.

Deﬁnition 1.5.2 Let s1, . . . , sk ∈ SX (X). The zero set of the ideal I := hs1, . . . , ski is often called the zero set of the sections s1, . . . , sk and is usually denoted by N(s1, . . . , sk).

Remarks. (i) We obviously have

N(s1, . . . , sk) = {x ∈ X;(s1)x,..., (sk)x ∈ mSX,x } = N(s1) ∩ · · · ∩ N(sk).

(ii) Since sx ∈ mSX,x is equivalent to s(x) = 0, we get N(s) = {x ∈ X; s(x) = 0}.

Lemma 1.5.3 Every proper ideal I ⊂ SX deﬁnes a ringed space (Y, SY ), where Y := N(I ) and SY := (SX /I )|Y . Let ι: Y → X be the canonical inclusion and let ˜ι: SX → SX /I be the quotient map. Then the map (ι, ˜ι):(Y, SY ) → (X, SX ) is a morphism of ringed spaces.

Proof. One only has to notice that ι∗SY = SX /I .

Deﬁnition 1.5.4 The ringed space (Y, SY ) is called the subspace of (X, SX ) deﬁned by I .

Lemma 1.5.5 Let (X, OX ) be a complex space and let I ⊂ OX be a ﬁnitely generated ideal. Then the ringed space (Y, OY ) deﬁned by I is a complex space and the canonical inclusion (ι, ˜ι) is holomorphic.

Proof. One can assume that (X, OX ) is a local model in which case the result is obvious.

Deﬁnition 1.5.6 A complex space (Y, OY ) of the kind described in lemma 1.5.5 is called a closed complex subspace of X.

Remarks. (i) In the sequel when we refer to a complex subspace or simply a subspace of a complex space it will mean a closed complex subspace.

(ii) Let Y be a complex subspace of a complex space X. Then Y is both open and closed (as a complex subspace) if and only if Y is a union of connected components of X and OY = OX |Y .

14 n Example 1. Let D be a domain in C . Clearly, every local model V = V (f1, . . . , fk) with f1 . . . , fk ∈ O(D) is a complex subspace of D.

The following theorem is a fundamental result of analytic geometry. It is most often referred to as Oka’s coherence theorem.

n Theorem 1.5.7 (Oka) For every non-negative integer n the structure sheaf OC is a coherent sheaf of rings.

Proof. See [GrRe3].

Corollary 1.5.8 The structure sheaf of every complex space is a coherent sheaf of rings.

Proof. Let X be a complex space. Since coherence is a local property we may assume that X is a local model deﬁned by holomorphic functions f1, . . . , fm in a domain D of n some C . The structure sheaf OX is the restriction of OD/I to X where I is the ideal generated by f1, . . . , fm. By Oka’s coherence theorem the sheaf of rings, OD, is coherent and the ideal sheaf I is OD–coherent since it is a ﬁnitely generated subsheaf of OD. Consequently OX is a coherent sheaf of rings.

Remark. For any complex space X there is a one-one correspondence between the complex subspaces of X and the coherent ideals of the structure sheaf OX .

Deﬁnition 1.5.9 Let X be a complex space and let Y be a subspace of X deﬁned by a coherent ideal I . Then Y is called a hypersurface of X if Iy is a principal ideal generated by a non zero-divisor in OX,y for all y in Y .

Deﬁnition 1.5.10 Let (X, SX ) be a ringed space. The sheaf of nilpotent sections of SX is called the nilradical of SX and denoted by NS.

Remark. Since CX is without nilpotent sections, NS is contained in the kernel of the morphism Red: SX → CX .

For complex spaces we have much stronger results. These are very deep and fundamental results in analytic geometry.

Theorem 1.5.11 Let (X, OX ) be a complex space and let N be the nilradical of OX . Then

(i) (Hilbert’s Nullstellensatz) Ker(Red) = N ,

(ii) (Cartan’s Coherence Theorem) N is coherent.

Proof. See [GrRe3].

15 Remark. For a ringed space (X, SX ) the kernel of Red: SX → CX is in general bigger than the nilradical NS. If for example ({x}, S ) is the one point ringed space with S the algebra of convergent power series in one variable, then Ker(Red) is the maximal ideal of S but the nilradical of S is zero.

The following result is an obvious consequence of theorem 1.5.11.

Corollary 1.5.12 The reduction of a complex space is a complex space.

Example 2. Let X be a complex space. Then Xred is a complex subspace of X but Xred is not an open subspace of X unless X is reduced.

Deﬁnition 1.5.13 Let X and Y be complex spaces, let Z be a complex subspace of X and let ι: Z → X denote the canonical inclusion. For every holomorphic map F : X → Y one deﬁnes a holomorphic map by

F |Z := F ◦ ι: Z → Y.

It is called the restriction of the map F to the subspace Z.

n n 1.6 The bijection Hol(X, C ) → OX (X) n ˜ n Let X be a complex space, let (f, f):(X, OX ) → (C , OC ) be a holomorphic map and n n let z1, . . . , zn ∈ OC (C ) denote the coordinate functions. Then we get for every j: ˜ n −1 n f(zj) ∈ (f∗OX )(C ) = OX (f (C )) = OX (X). Hence the mapping

n n ˜ ˜ ˜ Hol(X, C ) → (OX (X)) , (f, f) 7→ (f(z1),..., f(zn)).

n n Proposition 1.6.1 The mapping Hol(X, C ) → (OX (X)) is a bijection. Moreover, if X is reduced then (f˜(z1),..., f˜(zn)) = (z1 ◦ f, . . . , zn ◦ f).

Proof. We ﬁrst show that the mapping is injective. ˜ n ˜ Let (f, f), (g, g˜) ∈ Hol(X, C ) and suppose that f(zj) =g ˜(zj) for all 1 ≤ j ≤ n. Then lemma 1.2.5 implies

zj(f(x)) = f˜(zj)(x) =g ˜(zj)(x) = zj(g(x)) for all x ∈ X. and consequently f = g. n In order to prove f˜ =g ˜ we ﬁx a point x ∈ X and set a := f(x) = g(x) ∈ C . Since the n n ˜ analytic algebras OC ,a and OC ,0 are isomorphic we can assume a = 0. Then fx andg ˜x n are C–algebra morphisms OC ,0 → OX,x. Now, OX,x is Noetherian, z1, . . . , zn generate n ˜ the maximal ideal of OC ,0 and fx andg ˜x coincide on these generators so corollary A.3.11 tells us that f˜ =g ˜. Hence the injectivity of the mapping.

16 In order to prove the surjectivity of the mapping choose f1, . . . , fn ∈ OX (X). Suppose m ﬁrst that X is a local model in a domain D in C and that f1, . . . , fn are induced by F1,...,Fn ∈ OD(D) = O(D), more precisely

fj := Fj ◦ ι, for all 1 ≤ j ≤ n, where ι: X → D is the inclusion. Consider the holomorphic map

n F : D → C , y 7→ (F1(y),...,Fn(y))

n and let f : D → C be its restriction to X, i. e. f := F ◦ ι.

For every 1 ≤ j ≤ n we have F˜(zj) = zj ◦ F = Fj and consequently

f˜(zj) = ˜ι ◦ F˜(zj) = ˜ι(Fj) = Fj ◦ ι = fj.

Now consider an arbitrary complex space X. First we choose an open covering {Uα}α∈A with each Uα biholomorphic to a local model as above and then we apply the construction to f1|Uα, . . . , fn|Uα for every α ∈ A. Since we have already shown that the map

n Hol(Uα ∩ Uβ, C ) → OX (Uα ∩ Uβ)

n n is injective, two maps fα : Uα → C and fβ : Uβ → C , which are obtained by this con- ˜ struction, coincide on Uα ∩ Uβ. Hence, the mappings(fα, fα) glue together to deﬁne a map n (f, f˜) ∈ Hol(X, C ), such that f˜(zj) = fj for all 1 ≤ j ≤ n.

1.7 Geometry of complex spaces A germ of a complex space is a pair (X, x) where X is a complex space and x is a point in X. Germs of complex spaces are often referred to as space germs. Let (X, x) and (Y, y) be two space germs. The germs of holomorphic maps at x from open neighborhoods of x into Y which map x to y are called morphisms of space germs. Let U be an open neighborhood of x in X and f : U → Y be a holomorphic map with f(x) = y. Then its germ at x will be denoted by fx and one writes fx :(X, x) → (Y, y). It is easy to see that fx is an isomorphism if and only if there exists an open neighborhood V of x and an open neighborhood W of y such that f induces a biholomorphic map between V and W . With each space germ (X, x) there is associated the analytic algebra OX,x and with each morphism fx :(X, x) → (Y, y) there is associated the algebra morphism ˜ fx : OY,y → OX,x.

Hence a contravariant functor from the category of space germs to the category of analytic algebras.

Theorem 1.7.1 The above contravariant functor is an antiequivalence.

17 Proof. The proof consists of verifying the three following statements.

• For every analytic algebra A there exists a space germ (X, x) such that OX,x is isomorphic to A.

• For every algebra morphism φ: OY,y → OX,x there exists a morphism fx :(X, x) → (Y, y) such that φ = f˜x.

• If fx and gx are two morphisms from (X, x) to (Y, y) such that f˜x =g ˜x then fx = gx.

For the details see [Gro1]. Compare also [Fis] and [Re].

The above theorem enables us to translate local geometric properties into algebraic properties and vice versa.

Deﬁnition 1.7.2 Let X = (|X|, OX ) be a complex space. A point x ∈ X is called

(i) smooth or non-singular, if OX,x is regular, i. e. if it is isomorphic to C{z1, . . . , zn} for some n ∈ N. Otherwise x is called singular.

(ii) irreducible, if OX,x is an integral domain.

(iii) reduced, if OX,x is reduced.

(iv) normal, if OX,x is reduced and integrally closed in its full ring of fractions.

Remark. A complex space X is reduced if and only if every point of X is reduced.

Examples. (1) Let M be a complex manifold of dimension n and let x ∈ M. Then by choosing a

∼ n chart centered at x we get OM,x = OC ,0. This shows that every point in M is smooth. (2) Let V = V (zw). Clearly the analytic algebra

OV,(0,0) = C{z, w}/hzwi is not an integral domain so the point (0, 0) is not irreducible. For the same reason the stalk OV,(0,0) cannot be isomorphic to C{z1, . . . , zn} for any n ∈ N0. Hence, the point (0, 0) is singular.

(3) Let V = V (w2 − z3). Every point in V is a smooth point of V except (0, 0) and from example 3 in subsection 1.3 we see that (0, 0) is an irreducible point of V .

n (4) The complex space (D, OD) where D denotes a domain in C is everywhere reduced. Thus, a complex manifold is everywhere reduced, too.

(5) In the space V (zk) the point 0 is not reduced if k ≥ 1.

Remark. Every non–irreducible point is singular.

Deﬁnition 1.7.3 Sing(X) := {x ∈ X; x singular} is called the singular locus of X.

18 Example. For X = V (zw) we have Sing(X) = {(0, 0)}.

Lemma 1.7.4 Let X be a complex space. Then X \ Sing(X) is an open subspace of X. Moreover, X \ Sing(X) is a complex manifold.

Proof. Let x ∈ X be a smooth point. Then there exists an isomorphism φx : OX,x → n OC ,0 for some n ∈ N0 and by 1.7.1 there exists an open neighborhood of x in X which n is biholomorphic to an open neighborhood of 0 in C . Consequently every point in that neighborhood of x is non-singular.

Deﬁnition 1.7.5 The Zariski–tangent space or simply the tangent space of X at x is the vector space 2 ∗ TxX := (mx/mx) . The vector space ∗ 2 Tx X := mx/mx is called the Zariski–cotangent space or the cotangent space of X at x.

Examples.

n (1) Let D be a domain in C with 0 ∈ D. Then we have OD,0 = C{z1, . . . , zn}. Moreover, the maximal ideal of C{z1, . . . , zn} is given by

m = hz1, . . . , zni.

It follows that 2 m/m = C ε1 ⊕ · · · ⊕ C εn, 2 where εj is the image of zj in m/m for 1 ≤ j ≤ n. It then follows that the image of an element f of m is equal to ∂f ∂f (0) ε1 + ··· + (0) εn . ∂z1 ∂zn

2 n If we identify m/m with C via the linear isomorphism determined by

εj 7→ (0,..., 1,..., 0)

n for 1 ≤ j ≤ n, the projection map m → C is given by

 ∂f (0) ∂z1  .  f 7→  .  . ∂f (0) ∂zn

Now suppose that f1, . . . , fm generate the maximal ideal m. Then from the above considerations it follows that ∂f rk i (0) = n. ∂zj 1≤i≤m 1≤j≤n

19 k (2) Let X = V (z ) and let m denote the maximal ideal of OX,0. Then we have

k k−1 OX,0 = C{z}/hz i = C ⊕ C ε ⊕ · · · ⊕ C ε with εk = 0 and k k−1 m = hzi/hz i = C ε ⊕ · · · ⊕ C ε . k 2 2 k−1 Moreover, we see from ε = 0 that m = C ε ⊕ · · · ⊕ C ε . (This is understood to be zero if k = 1.) so 0 , if k = 1 m/m2 = . C ε , if k ≥ 2 (3) Let M be a complex manifold of dimension n, let x ∈ M and let m denote the

∼ n maximal ideal of OM,x. By choosing a chart centered at x we get OM,x = OC ,0 and consequently 2 dimC(m/m ) = dimC M. n (4) Let D ⊂ C be a domain and let V = V (f1, . . . , fk) be the local model given by f1, . . . , fk ∈ O(D). For every x = (x1, . . . , xn) in N(f1, . . . , fk) we have OV,x = OD,x/Jx where J is the ideal generated by the functions f1, . . . , fk and

mx = h(z1 − x1),..., (zn − xn)i/Jx

is the maximal ideal of OV,x. Consequently the images of the elements z1 −x1, . . . , zn − 2 xn in the cotangent space mx/mx generate it but they need not be linearly independent. Hence 2 mx/mx = C ε1 + ··· + C εn, 2 where εj denotes the image of zj −xj in mx/mx and in particular we get the inequality 2 dimC(mx/mx) ≤ n. (∗)

Deﬁnition 1.7.6 Let X and Y be complex spaces. (i) A holomorphic map f : X → Y is called an embedding, if there exist a complex subspace Z of Y and a biholomorphic map g : X → Z, such that the diagram

f X / Y AA O AA ι g AA A Z is commutative, where ι denotes the canonical inclusion

(ii) We say that X can be embedded into Y at x ∈ X, if there exist an open neighborhood U of x in X, an open subset V of Y and an embedding f : U → V . As an abbreviation we say that f :(X, x) → (Y, f(x)) is a local embedding. In this case we also say that the space germ (X, x) can be embedded into the space germ (Y, f(x)).

(iii) The embedding dimension of X at x is the minimal integer k such that X can be k embedded into C at x. It will be denoted by embx X.

20 Proposition 1.7.7 A holomorphic map (f, f˜): X → Y is an embedding if and only if the following two properties are satisﬁed (i) f(X) is a closed subset of Y and the induced map X → f(X) is a homeomorphism. ˜ ˜ (ii) The sheaf morphism f : OY → f∗OX is surjective and the ideal Ker f ⊂ OY is ﬁnitely generated.

Proof. A proof can be found in [GrRe3].

n Proposition 1.7.8 Let X = V (f1 . . . , fk) be a local model in a domain D ⊂ C and let x ∈ X. Then we have ∂fi embx V + rk (x) = n. ∂zj

Proof. Let x ∈ X and put ∂fi e := embx X and r := rk (x) . ∂zj First we prove e ≤ n − r. Without loss of generality we may assume that ∂f det i (x) 6= 0. ∂zj 1≤i,j≤r From the implicit function theorem it follows that there exist an open neighborhood U = n−r r V × W ⊂ D, with V ⊂ C and W ⊂ C , and functions g1, . . . , gr ∈ O(V ) such that the set N := {y ∈ U; f1(y) = ··· = fr(y) = 0} is the graph of the holomorphic map

r V → C , (y1, . . . , yn−r) 7→ (g1(y1, . . . , yn−r), . . . , gr(y1, . . . , yn−r)). If we equip N with the reduced complex structure, it follows that N is biholomorphically n−r equivalent to V ⊂ C . Since X ∩ (V × W ) is a complex subspace of N, this implies e ≤ n − r.

e Now we show r ≥ n − e. By deﬁnition of e there exists an embedding (X, x) → (C , 0) and the corresponding algebra morphism C{z1, . . . , ze} → OX,x is surjective. Thus for 0 0 any choice of generators (g1)x,..., (ge)x ∈ OX,x for the maximal ideal mx of OX,x we can choose holomorphic functions g1, . . . , ge in an open neighborhood of x in D such that

0 (gj|X)x = (gj)x for all 1 ≤ j ≤ e. Then we get

0 0 n OC ,x/h(f1)x,..., (fk)x, (g1)x,..., (ge)xi = OX,x/h(g1)x,..., (ge)xi = C and it follows that the Jacobi matrix of f1, . . . , fk, g1, . . . , ge has rank n. Therefore, the Jacobian of f1, . . . , fk has rank at least n − e and that proves the assertion.

Proposition 1.7.9 A holomorphic map f :(X, x) → (Y, f(x)) is a local embedding if and ˜ only if fx : OY,f(x) → OX,x is surjective.

21 Proof. For a proof see [Fis].

˜ Remark. Note that fx : OY,f(x) → OX,x is surjective if and only if the induced linear map TxX → Tf(x)Y is injective.

Example. Consider the complex space V = V (z2 − w). From proposition 1.7.8 we see that the embedding dimension of V at the origin is 1. Since the maximal ideal m0 of OV,0 is generated by z + hz2 − wi and w + hz2 − wi, we get the following two generators of 2 m0/m0: 2 2 2 2 (z + hz − wi) + m0 and (w + hz − wi) + m0. Because of 2 2 2 2 w + hz − wi = z + hz − wi ∈ m0 2 2 we have (w + hz − wi) + m0 = 0. Notice that the map V → C,(z, w) 7→ w, is not injective near 0 while the map (z, w) 7→ z deﬁnes even a global embedding V → C. At every point a = (a1, a2) ∈ V \{(0, 0)} the maps (z, w) 7→ w and (z, w) 7→ z deﬁne local embeddings. This is due to the fact that both 2 2 2 2 ((z − a1) + hz − wi) + ma and ((w − a2) + hz − wi) + ma 2 are non-zero elements in ma/ma.

The following proposition will make this correspondence clearer.

Proposition 1.7.10 Let X be a complex space, let x be a point in X and let m denote 2 the maximal ideal of OX,x. Then embx X = dimC(m/m ).

n Proof. Put n := embx X. Then there exists an embedding ι:(X, x) → (C , 0) so we n can assume that X is a local model in C given by equations f1 = ··· = fk = 0 in a n neighborhood of the origin. Let I be the ideal generated f1, . . . , fk in OC ,0, let n be the n n maximal ideal of OC ,0 and let m be the maximal ideal of OX,0 = OC ,0/I. The algebra morphism

˜ n fx : OC ,0 → OX,0 is surjective and it is easy to see that it induces a surjective morphism from n onto m and hence a surjective C−linear mapping

n/n2 → m/m2.

2 The kernel of this C−linear mapping is obviously the image of I in n/n .

From proposition 1.7.8 we see that the Jacobian matrix of (f1, . . . , fk) is zero at the 2 origin which means that the ideal I is contained in n . Consequently the the above C– linear mapping is an isomorphism and in particular the two vector spaces are of the same dimension. Since we have already shown that

2 n = dimC n/n

22 we ﬁnally get 2 n = dimC m/m .

1.8 Maximum principle Deﬁnition 1.8.1 A holomorphic map f : X → Y is said to be open at a point x in X if for every neighborhood U of x in X the image f(U) is a neighbourhood of f(x) in Y .

Theorem 1.8.2 (Open mapping theorem) Let X be a complex space and let x be a point in X. If f is a holomorphic function on X that is not constant in any neighborhood of x then the mapping f : X → C is open at x.

Proof. See [GrRe3]. Corollary 1.8.3 Let f be a holomorphic function on a complex space X which is not constant on any non empty open subset of X. Then the mapping f : X → C is open.

Notation. For a complex valued function f on a set S we put

|f|S := sup{|f(s)| : s ∈ S}.

Deﬁnition 1.8.4 Let f be a holomorphic function on a complex space X. (i) We say that f has a local maximum at a point x in X if there exists a neighbourhood U of x in X such that |f(x)| = |f|U . (ii) We say that f has an absolute maximum in X if there exists a point x in X such that |f(x)| = |f|X . Theorem 1.8.5 (Maximum principle) Let f be a holomorphic function on a complex space X. (i) If f has a local maximum at a point x in X then f is constant in a suﬃciently small neighbourhood of x. (ii) If f has an absolute maximum in X and X is connected then f is constant on X.

Proof. (i) Suppose f is not constant in any neighborhood of x and choose a neighborhood U of x such that |f(x)| = |f|U . Then f(U) is not a neighborhood of f(x) in contradiction to the open mapping theorem.

−1 (ii) Let x be a point in X such that |f(x)| = |f|X and put F := f (f(x)). By (i) the function f is constant in a neighborhood of every point in F so F is open. Since F is also closed and non-empty we can conclude that F = X.

Corollary 1.8.6 Every holomorphic function on a compact connected complex space is constant.

23 Proof. Obvious.

1.9 Analytic sets Deﬁnition 1.9.1 Let X denote a complex space. A subset A of X is called an analytic subset of X, if there exists a complex subspace Z of X such that

A = |Z|.

Remark. A subset A of a complex space X is an analytic subset of X if and only if it satisﬁes the following equivalent conditions:

• A is a closed subset of X and for every a in A there exists an open neighborhood U of a in X and a ﬁnite number of holomorphic functions f1, . . . , fk in U such that

A ∩ U = {x ∈ X ; f1(x) = ··· = fk(x) = 0}.

• For every x in X there exists an open neighborhood U of x in X and a ﬁnite number of holomorphic functions f1, . . . , fk in U such that

A ∩ U = {x ∈ X ; f1(x) = ··· = fk(x) = 0}.

n Lemma 1.9.2 Let D ⊂ C be a domain and let A ( D be an analytic subset of D. Then A is nowhere dense in D, or equivalently

A˚ = ∅.

Proof. Assume that A˚ 6= ∅. Since A is closed in D we have A˚ ⊂ A = A. Thus there exist for each x ∈ A˚ an open connected neighborhood U of x in D and holomorphic functions f1, . . . , fk ∈ O(U) such that

A ∩ U = N(f1, . . . , fk).

By our assumption we have U ∩ A˚ 6= ∅ so that

f1|(U ∩ A˚) = ··· = fk|(U ∩ A˚) = 0.

The identity theorem then implies that f1|U = ··· = fk|U = 0, which means that we have U ∩ A = U so x is an interior point of A. Thus we have shown that

A˚ = A,˚ in other words we have shown that A˚ is closed in D. Now D is connected so this implies A = D, in contradiction to the hypotheses of the lemma.

24 Example. Let D be an open subset of C and let A ( D be an analytic subset. From lemma 1.9.2 it follows that A is locally the common zero set of holomorphic functions f1, . . . , fk all of which are not identically zero. Hence A is discrete.

Theorem 1.9.3 (Riemann removable singularity theorem) Let D be a domain in n C and let A ( D be an analytic subset. Let f ∈ O(D \ A) be a holomorphic function which is bounded near every point x ∈ A. Then we can extend f holomorphically to D, in other words there exists a function F ∈ O(D) such that

F |(D \ A) = f holds.

Proof. See [KK].

n Corollary 1.9.4 Let D denote a domain in C and let A ( D be analytic. Then D \ A is connected.

Proof. If D \ A were not connected there would be two disjoint open sets U and V of D \ A such that U ∪ V = D \ A.

The function f : D \ A → C which takes the constant value 0 on U and the constant value 1 on V is holomorphic. This function is clearly not holomorphically extendable to D in contradiction to theorem 1.9.3.

n Corollary 1.9.5 Let n ≥ 2, let D be a domain in C and let x ∈ D. Then every holomorphic function f ∈ O(D \{x}) extends holomorphically to D.

Proof. Choose an ε > 0 such that

n U := {y ∈ C ; ky − xk < ε} ⊂ D.

Since n ≥ 2 we can ﬁnd for every y ∈ U a complex line L through y such that x is not contained in L. From the maximum principle it follows that

|f(y)| ≤ sup|f(bd(L ∩ U))| ≤ sup|f(bd U)|.

Hence, by theorem 1.9.3 we can extend f holomorphically to D.

n Corollary 1.9.6 Let n ≥ 2, let D be a domain in C and consider f ∈ O(D). Then f possesses no isolated zeros.

25 1 Proof. If f had an isolated zero x ∈ D then the function f would extend holomorphically to x, which is absurd.

Deﬁnition 1.9.7 Let X be a complex space. An analytic subset A of X is called

(i) thin, if it is nowhere dense, i. e. if it possesses no interior points,

(ii) analytically rare, if for every open U ⊂ |X| the restriction map

U ρU\A : OX (U) → OX (U \ A)

is injective.

Remark. From lemma 1.9.2 it follows that each analytic subset A ( D of a domain D n in C is thin.

Lemma 1.9.8 Let X be a reduced complex space and let A be an analytic subset of X. Then A is thin if and only if it is analytically rare.

Proof. Let A be an analytic subset of X. Consider the commutative diagram

U ρU\A OX (U) / OX (U \ A)

Red Red

(U) (U \ A) C|X| U / C|X| ρU\A

Since X is reduced, the morphism Red is injective. Hence the injectivity of

U ρU\A : OX (U) → OX (U \ A) is equivalent to the injectivity of

U ρU\A : C|X|(U) → C|X|(U \ A).

U If A is thin, it is clear that ρU\A : C|X|(U) → C|X|(U \ A) is injective, which implies that A is analytically rare. U On the other hand, if A is not thin, then the map ρU\A : C|X|(U) → C|X|(U \ A) is clearly not injective, hence A is not analytically rare. Thus, our assertion is proved.

Remark. Note that in general each analytically rare set is thin while the converse need not be true, as can be seen from the example here below.

26 2 2 Example. Let X be the subspace of C given by the equations xy = 0 and y = 0 and put A := {(0, 0)}. Obviously A is a thin subset of X. Consider the function f(x, y) = y on X. It is identically zero outside of the origin but its germ in OX,(0,0) is not zero. Hence the function f is not the zero function but it belongs to the kernel of the restriction map

U ρU\A : OX (U) → OX (U \ A).

Consequently A is not an analytically rare subset of X.

n Lemma 1.9.9 Let A be a compact analytic subset of an open set U in C . Then A is ﬁnite.

Proof. For each j the natural projection

pj : U → C;(z1, . . . , zn) 7→ zj is a holomorphic function on the compact space A. It then follows from the maximum principle that each of the natural projections is locally constant on A. Hence A is both discrete and compact and consequently ﬁnite.

Deﬁnition 1.9.10 A holomorphic map between complex spaces is called

(i) proper, if its preimage of every compact set is compact,

(ii) finite, if it is proper and if each fiber contains only finitely many elements.

n m Lemma 1.9.11 Let U ⊂ C and V ⊂ C be open sets, let f : U → V be a holomorphic map. Then f is proper if and only if it is ﬁnite.

Proof. Suppose that f : U → V is finite. Then f is by definition proper, so there is nothing to prove. On the other hand, assume that f is proper. Since every point x ∈ V is compact, it follows that each fiber f −1(x) is a compact analytic set. Now Lemma 1.9.9 implies that f −1(x) is finite, which was to be proved.

Deﬁnition 1.9.12 Let X be a complex space, let x ∈ X. The dimension of X at x is the smallest integer k, such that there exist an open neighborhood U of x in X and functions f1, . . . , fk ∈ O(U) with {x} = N(f1, . . . , fk).

The dimension of X at x will be denoted by dimx X.

Remark. Clearly, we have dimx X ≤ embx X for all x ∈ X.

Deﬁnition 1.9.13 The global dimension of X is deﬁned as

dim X := sup dimx X ∈ N0 ∪ {∞}. x∈X

Lemma 1.9.14 Let X and Y be complex spaces.

27 (i) If X is a complex manifold with dim X = k, we have dimx X = k for all x ∈ X.

(ii) A point x ∈ X is isolated if and only if dimx X = 0. n (iii) If X is a local model in C , then dimx X ≤ n.

(iv) If f : X → Y is ﬁnite, then dimx X ≤ dimf(x) Y .

(v) For all x ∈ X we have dimx Xred = dimx X.

Proof. (i) By taking local coordinates near x ∈ X we can assume that X is an open k k set in C and that x = 0. Thus it is suﬃcient to prove that dim0 C = k. Obviously k dim0 C ≤ k (just consider the k coordinate functions). For the proof of the inequality k dim0 C ≥ k see [GrRe3].

(ii) This is more or less deﬁnition since the zero set of no holomorphic function on an open set U is the whole set U and since x is an isolated point of X if and only if {x} is an open subset of X.

n (iii) Let X be a local model in C and let x ∈ X. If we denote by z1, . . . , zn the usual n coordinates on C we see that

{x} = N(z1 − x1, . . . , zn − xn).

Since the functions fj := (zj − xj)|X are holomorphic on X, the assertion follows.

(iv) Let dimf(x) Y = k and write {f(x)} = N(g1, . . . , gk). Since f is finite, the fiber f −1(f(x)) is finite and hence discrete. So, for each x ∈ f −1(f(x)) we can choose an open neighborhood U of x such that U ∩ f −1(f(x)) = {x}. Shrinking U if necessary, we get

{x} = N(g1 ◦ f . . . , gk ◦ f).

Hence dimx X ≤ k.

(v) The canonical map (id|X|, Red): Xred → X is holomorphic and ﬁnite so

dimx Xred ≤ dimx X for all x ∈ X. ˜ ˜ Now ﬁx an x ∈ X and choose an open neighborhood U of x in X together with f1,..., fk ∈

OXred (U) such that ˜ ˜ {x} = N(f1,..., fk).

Since the sheaf morphism Red: OX → OXred is surjective, there exist an open neighborhood V = V (x) of x in X and sections f1, . . . , fk ∈ OX (U) with

Red(fj) = f˜j ˜ ˜ for 1 ≤ j ≤ k. Clearly, N(f1, . . . , fk) = N(f1,..., fk) holds, and thus we see dimx X ≤ dimx Xred. Since x was arbitrary, the assertion follows.

28 Remark. Note that property (v) of the above lemma enables us to deﬁne the dimension of an analytic set.

Deﬁnition 1.9.15 Let X be a complex space.

(i) The space X is called pure dimensional at x, if the function y 7→ dimy X is constant in a neighborhood of x.

(ii) The space X is called pure dimensional, if

dim X = dimx X,

for all x in X.

Remark. If X is irreducible in x, then X is pure dimensional in x.

Proposition 1.9.16 A point x in a complex space X is smooth if and only if dimx X = embx X.

Proof. By lemma 1.9.14 we know that x smooth implies dimx X = embx X. For the proof of the converse see [GrRe3].

From the above proposition and proposition 1.7.8 one now deduces easily the following corollary.

n Corollary 1.9.17 Let V = V (f1, . . . , fk) be a local model in C of dimension dim V = d. Then ∂f Sing(V ) = x ∈ V ; rk i (x) < n − d . ∂zj

Proposition 1.9.18 Let X be a complex space.

(i) The singular locus Sing(X) is an analytic subset of X.

(ii) If X is pure dimensional then Sing(X) = {x ∈ X; dimx X < embx X}. (iii) If X is reduced, then Sing(X) is thin.

(iv) If X is reduced, then the closure of every connected component of the complex manifold X \ S(X) is an analytic subset of X.

Proof. (i) is a consequence of corollary 1.9.17.

(ii) is a consequence of proposition 1.9.16.

For a proof of (iii) and (iv) see [Fis].

Deﬁnition 1.9.19 Let X be a reduced complex space.

29 (i) If the complex manifold X \ Sing(X) is connected then the space X is called globally irreducible. (ii) The closure of any component of the complex manifold X \ Sing(X) is called an irreducible component of X.

Deﬁnition 1.9.20 (i) A complex space X is called locally irreducible at a point x in X if x is contained in an arbitrarily small reduced and globally irreducible open subspace of X. (ii) A complex space X is called locally irreducible if it is locally irreducible at every point in X.

Remark. It is not hard to see that a complex space X is locally irreducible at a point x in X if and only if the point x is an irreducible point of X. (See deﬁnition 1.7.2).

3 2 2 Example. Let X be the subspace of C given by the equation z1 = z2z3. It goes under the name of Whitney’s umbrella. 2 2 There is a surjective holomorphic map f : C → X deﬁned by f(u, v) = (uv, u, v ) so the space X is locally irreducible at the origin. On the other hand it is easy to see that the space X is not locally irreducible at any point of the form (0, 0, z3) with z3 6= 0 This shows that the condition of being locally irreducible at a point is not an open condition.

1.10 Normal spaces Normal spaces are most important in analytic geometry. They are much more general than the manifolds but share many “nice qualities” with them. For a thorough study of normal spaces see for instance [GrRe3] and [Fis].

Before stating the deﬁnition of a normal space we recall that a point x in a complex space X is called normal if the analytic algebra OX,x is reduced and integrally closed in its full ring of fractions.

Deﬁnition 1.10.1 A complex space X is called normal if every x ∈ X is a normal point.

Theorem 1.10.2 Let X be a normal complex space. Then the inequality

dimx X − dimx Sing(X) ≥ 2 holds for all x ∈ X.

Proof. See [GrRe3] or [Fis].

There is a version of Riemann’s theorems on removable singularities for normal spaces which can be stated as follows.

Theorem 1.10.3 Let X be normal, let A be a thin analytic subset of X and let f be a holomorphic function on X \ A. If either f is bounded near every point a ∈ A or if the inequality dima X −dima A ≥ 2 holds for all a ∈ A, then there exists a unique holomorphic extension of f to X.

30 Proof. See [GrRe3] or [Fis].

Combining the last two theorems we get the following result.

Corollary 1.10.4 Let X be a normal space. Then every holomorphic function on the manifold X \ Sing(X) extends uniquely to a holomorphic function on X.

2 3 w Example. Let X = V (w − z ). The map f(z, w) = z is holomorphic on X \{(0, 0)}, and we have f 2(z, w) = z on X. Therefore, f is continuous on X, but not holomorphic in (0, 0). Hence X is not normal.

By the same way as in subsection 1.9 we get the following result.

Corollary 1.10.5 Let X be a connected normal complex space and let A be a thin analytic subset of X. Then X \ A is connected.

From the last corollary one easily deduces the following corollary.

Corollary 1.10.6 Normal spaces are locally irreducible.

2 Operations on complex spaces

2.1 Inverse images and fibers Let f : X → Y be a holomorphic map. Let B be a complex subspace of Y given by a coherent OY –ideal J . Let I be the OX –ideal generated by the image of J by the sheaf −1 morphism f OY → OX associated with the map f. In other words, we have for each x in X the algebra morphism ˜ fx : OY,f(x) → OX,x and the image of Jf(x) in OX,x by this morphism generates the ideal Ix. The ideal I can be shown to be OX –coherent. (See [GrRe3].) Hence it defines a complex subspace of X which will be denoted by f −1(B) and called the inverse image of B (by f). −1 If B = {y} is a reduced point we usually denote its inverse image by Xy or f (y) instead of f −1({y}) and call it the fiber of f over y.

Examples. 2 −1 √ √ −1 (1) Consider f : C → C, f(z) = z . If a 6= 0, then |f (a)| = { a, − a}. So f (a) is a reduced two point space. The ﬁber over zero is given by −1 2 ∼ f (0) = ({0}, C{z}/hz i) = ({0}, C ⊕ C ε), where ε2 = 0. Thus, f −1(0) is a double point.

31 2 3 −1 (2) Let X = V (w − z ), let f : X → C be the map (z, w) 7→ z. For a 6= 0 the ﬁber f (a) is a reduced two point space, and again

−1 2 3 ∼ 2 f (0) = ({0}, C{z, w}/hw − z , zi) = ({0}, C{w}/hw i)

is a double point.

2.2 Images of complex spaces Let X and Y be complex spaces, let f : X → Y be a holomorphic map. If the image sheaf f∗OX is OY –coherent, then ˜ f : OY → f∗OX has a coherent kernel J and the complex subspace of Y deﬁned by the ideal J is denoted by f(X) and called the image of f. In other words

f(X) := (f(|X|), OY /J ).

Moreover, there is a factorization of the holomorphic map f through its image

f X / Y E O EE EE ι 0 EE f E" f(X) where ι: f(X) → Y denotes the natural inclusion.

If A is a subspace of X such that f(A) is a well deﬁned complex subspace of Y then it is called the image of A by f.

2 2 3 Example. The image of the holomorphic map f : C → C , f(t) = (t , t ) is Neil’s 2 2 3 parabola, i. e. the subspace of C given by the equation w − z = 0.

The following theorem is a special case of the so-called Direct image theorem of Grauert. It is a very deep and extremely important result of analytic geometry.

Theorem 2.2.1 (Grauert) Let f : X → Y be a proper holomorphic map and let S be a OX –coherent sheaf. Then the sheaf f∗S is OY –coherent.

Proof. See [GrRe3].

Remark. The above theorem says in particular that if the holomorphic map f has the topological property of being proper then every subspace of X has a well deﬁned image by f in Y .

32 2.3 Fiber products Suppose we have a commutative diagram of holomorphic maps

p2 Z / X2

p1 g X / Y 1 f

Such a diagram is called a Cartesian square if it has the following universal property.

Universal property for a Cartesian square: For every complex space T and every pair of holomorphic mappings q1 : T → X1 and q2 : T → X2 such that g ◦ q2 = f ◦ q1 there exists a unique holomorphic map q : T → Z satisfying p1 ◦ q = q1 and p2 ◦ q = q2.

This universal property can be visualized by the following diagram

T q2

∃!q

p2 Z / X2 q1 p1 g " X / Y 1 f

Example. Let f : X → Y be holomorphic and let Z be a complex subspace of Y . Then the following diagram is a Cartesian square

f −1(Z) / X

f|f −1(Z) f Z / Y

Now let f : X1 → Y and g : X2 → Y be holomorphic maps and suppose we have two 0 0 0 0 triplets (Z, p1, p2) and (Z , p1, p2), where Z and Z are complex spaces and pk : Z → Xk 0 0 and pk : Z → Xk are holomorphic maps for k = 1, 2, such that both of the following diagrams are commutative

p p0 2 0 2 Z / X2 Z / X2

0 p1 g p1 g X / Y X / Y 1 f 1 f

If in addition each of the above diagrams is a Cartesian square then one deduces easily from the universal property that there exists a biholomorphic map q : Z0 → Z such that 0 0 p1 ◦q = p1 and p2 ◦q = p2. Hence if such a triplet exists it is unique up to a biholomorphic

33 map.

On the other hand we have the following existence theorem.

Theorem 2.3.1 Let f : X1 → Y and g : X2 → Y be holomorphic maps. Then there exist a complex space X1×Y X2 and holomorphic maps p1 : X1×Y X2 → X1, p2 : X1×Y X2 → X2 such that the following diagram is a Cartesian square

p2 X1 ×Y X2 / X2

p1 g X / Y 1 f

Proof. See [Fis].

Definition 2.3.2 The space X1 ×Y X2 is called the fiber product of X1 and X2 over Y . In the case where Y is a reduced point then the fiber product of X1 and X2 over Y is called the Cartesian product of X1 and X2 and is usually denoted by X1 × X2

Remarks.

(i) The explanation of the terminology fiber product is that set theoretically the fiber of −1 −1 the map f ◦p1 = g◦p2 over each point y in Y is the Cartesian product f (y)×g (y). In particular, when Y is a reduced point there is only one fiber which is the Cartesian product X1 × X2. (ii) When we have a Cartesian square

p2 Z / X2

p1 g X / Y 1 f

it is sometimes said that the map p1 is obtained from the map g by the base change f : X1 → Y . Although we are not giving a proof of theorem 2.3.1 let us give a local description of n the fiber product. Suppose X1 is a local model in domain D1 in C with coordinates m z = (z1, . . . , zn) and X2 is a local model in a domain D2 in C with coordinates w = (w1, . . . , wm). Let h1(z) = ··· = hl(z) = 0 be the defining equations for X1 and let k1(w) = ··· = kr(w) = 0 be the defining equations for X2. Then D1 × D2 is a domain in n+m C with coordinates (z, w) and the Cartesian product X1 × X2 is the local model in the domain D1 × D2 given by the equations

h1(z) = ··· = hl(z) = k1(w) = ··· = kr(w) = 0.

34 t Now suppose that Y is also a local model in some domain in C . Then the holomorphic t maps f : X1 → Y and g : X2 → Y can be considered as maps into C . Hence the ﬁber product X1 ×Y X2 is the local model in the domain D1 × D2 given by the equations

h1(z) = ··· = hl(z) = k1(w) = ··· = kr(w) = 0 and f(z) = g(w).

Deﬁnition 2.3.3 Let f : X → Y be a holomorphic map. The space Gf := X ×Y Y is called the graph of f.

Remark. Let pX and pY denote the natural projections of Gf onto X and Y . Since the diagrams

pY f Gf / Y X / Y

pX idY idX idY X / Y X / Y f f are both Cartesian squares there exists a biholomorphic map F : X → Gf such that the diagram f X / Y > }} F } }}py }} Gf commutes. The underlying topological situation is of course the following

|Gf | = {(x, y) ∈ X × Y ; y = f(x)} and F (x) = (x, f(x)).

Deﬁnition 2.3.4 The space ∆X := GidX is called the diagonal of X.

3 Meromorphic functions and maps

3.1 Meromorphic functions Meromorphic functions can be deﬁned on any complex space but to avoid technical diﬃ- culties all complex spaces are assumed to be reduced in this subsection.

Let X be a (reduced) complex space and let f ∈ OX (X). The set of points x ∈ X where f is not a zero divisor is open because it is the complement of the support of the coherent sheaf Ker(OX → OX , gx 7→ gxfx). Thus, we have the following presheaf on X

n f o MX (U) := g ; f, g ∈ OX (U), gx non zero divisor for all x ∈ U , where U ⊂ X is open. The restriction mappings are given by restricting the sections f, g ∈ OX (U). The corresponding sheaf MX is called the sheaf of meromorphic functions

35 on X. An element of MX (U) is called a meromorphic function on U. Notice that in general the presheaf M is not a sheaf. For example M ( ( )) consists X P1(C) P1 C only of the constant functions on P1(C) but the map

P1(C) \{[z1, z2]; z2 6= 0} → C, [z1, z2] 7→ z1 determines a non-constant meromorphic function on ( ) so M ( ( )) 6= ( ( )). P1 C P1(C) P1 C MP1(C) P1 C

Remark. The stalk MX,x is the full ring of fractions of the ring OX,x. See Appendix A.

Theorem 3.1.1 Every ﬁnitely generated OX –submodule of MX is OX –coherent.

Proof See [GrRe3].

Let h ∈ MX (X). The sheaf D deﬁned by

Dx := {β ∈ OX,x; βhx ∈ OX,x} is called the sheaf of denominators of h and the sheaf h·D is called the sheaf of numerators of h. Moreover, we deﬁne

• the zero set of h by N(h) := N(h · D),

• the polar set of h by P (h) := N(D), • the set of indeterminacy of h by N(h) ∩ P (h).

By the above theorem the subsheaf hOX of MX is coherent. Consequently the quotient sheaf hOX /(hOX ∩ OX ) is coherent and since D is the annihilator ideal of this sheaf it is also coherent. It follows that the N(h), P (h) and N(h) ∩ P (h) are analytic sets of X. Notice that P (h) is the support of hOX /(hOX ∩ OX ) in other words

P (h) = {x ∈ X | hx ∈/ OX,x} so P (h) can be described as the smallest subset of X outside of which h is holomorphic. It is easily seen that the polar set of a meromorphic function is a thin analytic set and for the zero set we have the following result.

Proposition 3.1.2 Let X be a complex space and let h ∈ MX (X). Then N(h) is thin −1 if and only if h is a unit in MX (X) and in that case we have N(h ) = P (h) and P (h−1) = N(h).

Proof See [GrRe3].

36 Remark. Let X be an irreducible complex space. Then it is easily seen from proposition 3.1.2 that every meromorphic function on X which is not identically zero is a unit in MX (X). This means that MX (X) is a ﬁeld, called the ﬁeld of meromorphic functions on X.

The pull-back of a meromorphic function by a holomorphic map is not a meromorphic function in general. More precisely, for a holomorphic map f : X → Y the corresponding ˜ ˜ sheaf morphism f : OY → f?OX does not necessarily induce a OY −morphism f : MY → f?MX . For example if f is the natural inclusion of the one point space {0} into C, 1 then there is no way to deﬁne the pull-back of the meromorphic function z 7→ z by f. Nevertheless we have the following result.

Proposition 3.1.3 Let f : X → Y be a holomorphic map having the property that f −1(A) ˜ is a thin analytic subset of X if A is a thin analytic subset of Y . Then f : OY → f?OX ˜ extends uniquely to a OY −morphism f : MY → f?MX .

Proof See [GrRe3].

Remark. Every open holomorphic map satisﬁes the hypothesis of the above proposition.

Examples. (i) Let X = V (zw). Here, the functions z w and z + w z + w are meromorphic and for both of them we have N ∩ P = {(0, 0)} = P . Since the z function w is a zero divisor in (0, 0), the quotient w is not a meromorphic function. 2 3 w (ii) Consider X = V (w −z ). The function h(z, w) = z is continuous and meromorphic on X. We have P (h) = N(h) = {(0, 0)}.

3.2 Modifications Definition 3.2.1 A proper surjective holomorphic map Φ: X → Y is called a (proper) modification if there are closed analytic sets A of X and B of Y such that (i) B = Φ(A),

(ii) Φ|X\A : X \ A → Y \ B is biholomorphic (iii) A and B are analytically rare. If the sets A and B are minimal subject to (i), (ii) and (iii) then the set A is called the exceptional set of Φ, the set B is called the center of Φ and Y is said to be obtained by blowing X down along A. Occasionally such a modiﬁcation is written Φ:(X,A) → (Y,B) instead of Φ: X → Y .

Proposition 3.2.2 Let Φ: X → Y be a modiﬁcation. Then the morphism Φ:˜ OY → Φ?OX is injective and in the case where X and Y are reduced it induces an isomorphism Φ:ˆ MY → Φ?MX .

37 Proof See [Fis].

Remark. Let Φ: (X,A) → (Y,B) be a modiﬁcation. If Y is reduced, then X is also m reduced: Suppose x ∈ A and η ∈ OX,x such that η = 0. Take an open neighborhood U of x in X and a holomorphic function f in OX (U) such that fx = η. Then (shrinking U if necessary) we get f|U\A = 0 because Y is reduced and the open sets U \ A and Φ(U) \ B are biholomorphic. Since the restriction morphism OX (U) → OX (U \ A) is injective this implies f = 0 and consequently fx = 0.

Corollary 3.2.3 Let Φ: X → Y be a ﬁnite modiﬁcation. If Y is a normal space, then Φ is biholomorphic.

Proof. Consider the following diagram

Φ˜ OY / Φ?OX

Φˆ MY / Φ?MX

By proposition 3.2.2 the morphism Φ:ˆ MY → Φ?MX is an isomorphism and the map Φ is ﬁnite so Φ?OX is integral over OY . Now Y is normal so OY is integrally closed in MY and consequently Φ˜ is an isomorphism. Hence Φ is a biholomorphic map.

Remark. Without the ﬁniteness condition the result above is not true in general. For an example see [Pe].

Example. The most important examples of modiﬁcations are the so-called blow-ups. Let X be a complex space and Y be a complex subspace of X with deﬁning ideal I . Put   M k X˜ = Proj  I  . k≥0

See [Ha] for the deﬁnition and elementary properties of Proj as for other notions used in this example. Let π : X˜ → X be the canonical projection induced by

M k OX → I . k≥0

The map π is a proper holomorphic map, called the blow-up of X along Y or the monoidal transformation of X with center Y . The subspace Y˜ := π−1(Y ) is called the exceptional set of π.

The blow-ups have the following properties:

(i) If the deﬁning ideal I is invertible then the map π is biholomorphic. ˜ ˜ (ii) Obviously IX\Y = OX\Y so the map π|X˜\Y˜ : X \ Y → X \ Y is biholomorphic.

38 ! ! L k L k k+1 (iii) Y˜ = Y ×X X˜ = Proj I ⊗ OX /I = Proj I /I k≥0 k≥0

(iv) If Y is a locally complete intersection then

M k k+1 M k 2 I /I = S I /I k≥0 k≥0

˜ so Y is the total space of the normal bundle N |Y |X .

Proposition 3.2.4 (Universal property) Let X be a complex space and let Y be a subspace of X deﬁned by an ideal I . Let π : X˜ → X be a proper holomorphic map having the two following properties:

(i) Y˜ := π−1(Y ) is a hypersurface of X˜,

(ii) for every holomorphic map g : Z → X such that g−1(Y ) is a hypersurface there exists a unique holomorphic map h: Z → X˜ satisfying the condition g = π ◦ h.

g Z / X ~? ~~ h ~~π ~~ X˜ Then the map π is (up to a biholomorphic map) the blow-up of X along Y .

Proof The proof consists of showing that the blow up of X along Y satisﬁes conditions (i) and (ii). See [Fis].

3.3 Meromorphic maps In this subsection all complex spaces are reduced.

Deﬁnition 3.3.1 Let X and Y be complex spaces. A meromorphic map f : X → Y is a map that assigns to every x in X a subset Gf (x) of Y in such a way that

(i) the set Gf := {(x, y) ∈ X × Y | y ∈ Gf (x)} is an analytic subset of X × Y ,

(ii) the canonical projection pX : Gf → X is a modiﬁcation.

The analytic set Gf is called the graph of the meromorphic map f and the center of the modiﬁcation pX : Gf → X is called the set of indeterminacy of f.

Remarks. (i) A meromorphic map is completely determined by its graph and in practice it is usually given by the graph.

(ii) Let f : X → Y be a meromorphic map and let S be its set of indeterminacy. Then f induces a holomorphic map X \ S → Y . More precisely, to every x in X \ S the −1 −1 set Gf (x) = pX (x) is a singleton and f(x) is the unique point in pY (pX (x)).

39 Deﬁnition 3.3.2 Let f : X → Y be a meromorphic map.

(i) The meromorphic map f is said to be surjective if the canonical projection

pY : Gf → Y

is surjective.

(ii) The meromorphic map f is said to be bimeromorphic if the canonical projection

pY : Gf → Y

is a modiﬁcation. In that case the spaces X and Y are said to be bimeromorphically equivalent.

Remark. Let f : X → Y be a bimeromorphic map. Then Gf determines a meromorphic map g : Y → X which will be referred to as the inverse of f. In fact it is in a certain sens the inverse of f as will become clear later.

Examples.

(1) Let f : X → Y be a holomorphic map with X and Y reduced. Then f can be considered as a meromorphic map. The graph of f as a meromorphic map is just the graph of f as a holomorphic map and its set of indeterminacy is empty.

(2) Let f : X → Y be a modification with X and Y reduced. Then f is a bimeromorphic map. Just consider the graph Gf with its two natural projections pX : Gf → X and pY : Gf → Y . The first one is a biholomorphic map and the second one is a modification. Hence the meromorphic map X → Y is f itself in the sense of (1) and the inverse meromorphic map Y → X is the mapping which assigns to each y ∈ Y −1 the set pY (y). The set of indeterminacy of the latter is obviously the center of the modification f.

(3) Let X be a reduced complex space and let h be a meromorphic function on X. Then ˆ ˆ h induces a meromorphic map h: X → P1(C). Roughly h is obtained in the following way. One considers the graph G of the holomorphic map X \ P (h) → C induced by h ¯ and shows that its closure G in X × P1(C) is an analytic subset and that the natural ¯ ˆ ¯ projection pX : G → X is a modiﬁcation. Then one deﬁnes h by Ghˆ := G. See [Fis] for details.

(4) Two Riemann surfaces are bimeromorphically equivalent if and only if they are biholomorphically equivalent. This is an easy consequence of corollary 3.2.3.

(5) The map 2 3 2 3 C → V (w − z ), t 7→ (t , t ), is bimeromorphic, but not biholomorphic.

40 Let us now consider how to deﬁne the pull-back morphism of a meromorphic map and the composition of two meromorphic maps. In both cases we need some restrictions on the meromorphic maps. Let f : X → Y be a meromorphic map and let pX : Gf → X and pY : Gf → Y denote the canonical projections of its graph onto X and Y . If pY has the property that its inverse image of every thin analytic set in Y is a thin analytic set in Gf then by proposition

(3.1.3) there is a well defined pull-back morphism MY → (pY )?MGf induced by pY . In particular we get a pull-back map MY (Y ) → MGf (Gf ) and since pX is a modification it induces an isomorphism MGf (Gf ) → MX (X). By composing these two morphisms we get a pull-back morphism ∗ f : MY (Y ) → MX (X). Notice that we have a well defined pull-back morphism in the two following cases:

• f : X → Y is a surjective meromorphic map and X is irreducible.

• f : X → Y is a bimeromorphic map.

In the latter case the pull-back morphism is an isomorphism. Moreover, if X and Y are irreducible then it is an isomorphism of ﬁelds.

Let f : X → Y and g : Y → Z be two meromorphic maps and let pX : Gf → X, pY : Gf → Y and qY : Gg → Y be the corresponding projections associated with their graphs. Let A be the set of indeterminacy of f, in other words the center of the modiﬁcation pX : Gf → X and let B be the set of indeterminacy of g, in other words the center of the modiﬁcation qY : Gg → Y . The meromorphic maps f and g then induce holomorphic maps X \ A → Y and Y \ B → Z. By composition we then get a holomorphic map

−1 h: X \ A ∪ pX (pY (B)) → Z.

−1 Proposition 3.3.3 Assuming the hypotheses above and assuming also that pY (B) is a thin analytic subset of Gf . Then there exists a unique meromorphic map g ◦ f : X → Z −1 which coincides with the holomorphic map h on X \ A ∪ pX (pY (B)).

Proof. We have to show that there exists an analytic subset G of X × Z satisfying the following properties:

• the canonical projection G → X is proper,

• G = Gh ∪ A where A is a thin analytic subset of G.

Consider ﬁrst the ﬁber product Gf ×Y Gg. The canonical projection

Gf ×Y Gg → X × Z is clearly proper so its image, which will be denoted by G0, is an analytic subset of X × Z. Now, a point (x, z) belongs to G0 if and only if there exists a point y in Y such that −1 (x, y) ∈ Gf and (y, z) ∈ Gg. It follows that for each x∈ / A ∪ pX (pY (B)) and each z ∈ Z 0 0 the point (x, z) belongs to G if and only if it belongs to Gh, in other words G and Gh −1 coincide over X \ A ∪ pX (pY (B)).

41 0 It is not hard to see that G is proper over X and obviously Gh is the complement of an analytic subset of G0. But the problem is that this analytic subset need not be thin so we have to shrink G0 to get the desired result. Suppose ﬁrst that X is irreducible. Then Gh is contained in exactly one of the irreducible components of G0 and we take G to be that component. In the general case we associate in the same way with each irreducible component of X a unique irreducible component of 0 G and then we let G be the union these irreducible components.

Deﬁnition 3.3.4 The meromorphic map g ◦ f is called the composition of f and g.

Remark. If f : X → Y is a bimeromorphic map with inverse g : Y → X, then g◦f = idX and f ◦ g = idY .

Corollary 3.3.5 The composition of two meromorphic maps f : X → Y and g : Y → Z is well deﬁned in the two following cases

• g is holomorphic,

• X is irreducible and f is surjective.

Proof. Obvious.

Proposition 3.3.6 Let f : X → Y and g : Z → W be meromorphic maps.

(i) The product map f × g : X × Z → Y × W is a well deﬁned meromorphic map.

(ii) If X = Z then (f, g): X → Y × W is a well deﬁned meromorphic map.

Proof.

(i) We let Gf×g be the image of Gf × Gg by the obviously biholomorphic map

(X × Y ) × (Z × W ) → (X × Z) × (Y × W ).

Then it is easily seen that the canonical projection pX×Z : Gf×g → X × Z is a modiﬁcation.

(ii) Let j : X → X × X be the diagonal mapping. To show that (f, g) = (f × g) ◦ j is a well deﬁned composition of meromorphic maps it is enough to show that the inverse image of the set of indeterminacy of f ×g by j is a thin analytic subset of X. Denote by A and B the sets of indeterminacy of f and g. Then (A × X) ∪ (X × B) is the set of indeterminacy of f × g and j−1((A × X) ∪ (X × B)) = A ∪ B is a thin analytic set of X.

4 Diﬀerential forms

In this section we suppose that the reader is familiar with diﬀerential forms on complex manifolds.

42 4.1 Differential forms on complex spaces m ∞ Let X be a complex space. We will define the sheaf AX of C differential forms of degree p,q ∞ m on X and the sheaf AX of C differential forms of type (p, q) on X. We will also m m+1 p,q p+1,q p,q p,q+1 introduce the differentials d: AX → AX , ∂ : AX → AX and ∂ : AX → AX and establish a natural decomposition

m M p,q AX = AX p+q=m such that d = ∂ + ∂. n Suppose first that X is a subspace of an open set U in C defined by an ideal sheaf I . m ∞ p,q Let AU be the sheaf of C differential forms of degree m on U and let AU be the sheaf of C∞ differential forms of type (p, q) on U. In general we will denote the sheaf of C∞ ∞ 0 ¯ ¯ functions on U by CU instead of AU . Put I := {f | f ∈ I } where f is the complex conjugate of f, and put ∞ ∞ IX := (I + I )CU . ∞ ∞ Then the sheaf CX of C functions on X is defined by ∞ ∞ ∞ CX := CU /IX . m 0 ∞ The sheaves of AX are then defined by putting AX := CX and for m ≥ 1 put m m ∞ m ∞ m−1 AX := AU /(IX AU + dIX ∧ AU ), ∞ m ∞ m where IX AU is the CU −submodule of AU generated by all germs of the form fω with ∞ m ∞ m−1 ∞ m f ∈ IX and ω ∈ AU , and dIX ∧ AU is the CU −submodule of AU generated by all ∞ m−1 germs of the form df ∧ ω with f ∈ IX and ω ∈ AU . p,q p,q p+q p+q Let AX be the image of AU under the canonical mapping AU → AX and define the differentials d, ∂ and ∂ as being the mappings induced by the usual differentials defined for the C∞ forms on U. Then it is easy to see that the required conditions are satisfied. m m Notice that the complex conjugation on AU induces a C−antilinear involution on AX . We will say that a C∞ form on X is a real differential form if it is invariant by this involution. Let us now consider the general case. It is not hard to prove that the above construction n is independent of the embedding of the complex space X into an open set of some C . Keeping this in mind let X be any complex space and take an open covering of X having n the property that each of its sets can be embedded into an open set of some C . Then the sheaves obtained by the above construction on these open sets will paste together. Observe that the notion of a real differential form is well defined for any complex space.

4.2 K¨ahler–spaces We begin by recalling a few basic notions.

(i) Let V be a complex vector space. Then there is a one–one correspondence between

• Hermitian forms on V , in other words mappings H : V × V → C which satisfy

H(sv1 + tv2, w) = sH(v1, w) + tH(v2, w) and H(v, w) = H(w, v),

43 • exterior 2–forms on V invariant under multiplication by i, in other word mappings ω : V × V → R which satisfy

ω(sv1+tv2, w) = sω(v1, w)+tω(v2, w), ω(v, w) = ω(w, v) and ω(iv, iw) = ω(v, w).

This correspondence is given by ω := ImH, H(v, w) := ω(iv, w) + iω(v, w). The forms ω are real (1, 1)–forms on V .

(ii) Every complex manifold has a canonical orientation obtained in the following way. The complex plane C is canonically oriented by the volume forme i dz ∧ dz = dx ∧ dy 2 n where z = x + iy. Then C is canonically oriented as a Cartesian product by the volume form i n dz ∧ dz ∧ · · · ∧ dz ∧ dz 2 1 1 n n n where (z1, . . . , zn) are the standard coordinates on C . It follows that open sets in n C are canonically oriented and it is easy to see that this orientation is conserved by n holomorphic maps between open sets in C . Now let M be a complex manifold and let a be a point in M. Then all charts of M that contain a induce the same orientation on M at a so M is canonically oriented by its charts.

Let M be a real oriented C∞ manifold of dimension m and let ω be a volume form on M. Then for every real diﬀerential form ϕ of degree m on M there exists a unique real valued C∞ function f on M such that ϕ = fω. The form ϕ is said to be positive if the function f is positive at every point in M. Observe that this notion of positivity does not depend on the choice of the volume form but depends only on the orientation of the manifold. Now suppose M is a complex manifold of (complex) dimension n. Then M is canonically oriented and a real C∞ form of degree 2n on M is said to be positive if it is positive with respect to that orientation. n ∞ Let U be an open set in C and let ϕ be a real C form of type (q, q) on U. Then ϕ is said to be positive if for every embedding of a complex manifold Y of dimension q into an open subset of U the pull back of ϕ onto Y is positive. If ϕ is a positive form of type (1, 1) on U then it is easily seen that ϕk := ϕ ∧ · · · ∧ ϕ is a positive form on U for k = 1, . . . , n.

Deﬁnition 4.2.1 Let M be a complex manifold and let TM be the holomorphic tangent bundle of M.A Hermitian metric on M is a C∞ mapping

h: TM ×M TM → C that induces a positive deﬁnite Hermitian form on TM,a for all a in M.

44 1 Suppose h is a Hermitian metric on a complex manifold M and put ω := − 2 Imh. Then it is not hard to see that ω is a positive (1, 1)–form on M. It is called the (1, 1)–form associated with the Hermitian metric h. A positive C∞ form ω of type (1, 1) on M is called a Hermitian form, and such a form is the associated form of exactly one Hermitian metric h. If the form ω is closed, i. e.

dω = 0, then we say that it is a K¨ahlerform and that h is a K¨ahlermetric.

By a partition of unity argument one shows easily that every complex manifold can be endowed with a Hermitian metric, but a complex manifold that can be endowed with a K¨ahler metric is called a K¨ahlermanifold.

Deﬁnition 4.2.2 Let X be a complex space.

(i) A real C∞ form ϕ of type (q, q) on X is said to be positive if there exists an open covering {Uα} of X having the following properties for each α

n • there exist an embedding fα : Uα → Wα where Wα is an open set in some C , ∗ • there exists a positive form ψα on Wα such that ϕ|Uα = fαψα. (ii) A positive C∞ form of type (1, 1) on X is called a Hermitian form.

(iii) A Hermitian form on X is called a K¨ahler form if all the forms ψα in (i) can be taken to be closed.

Definition 4.2.3 A reduced complex space X on which there exists a Kählerform is called a Kähler space.

Remark. The notion of a K¨ahlerspace exists also for non reduced spaces, but it is quite technical and will not be given here. See [Bin] and [Va2].

Proposition 4.2.4 Let X be a K¨ahler space.

(i) If X is smooth then X is a K¨ahlermanifold.

(ii) Every reduced subspace of X is a K¨ahlerspace.

(iii) If f : Y → X is a ﬁnite holomorphic map then Y is a K¨ahlerspace.

(iv) Every blow-up of X is a K¨ahlerspace.

Proof. (i) and (ii) are easy to prove. For (iii) and (iv) see [Bin] and [Fu2].

Examples.

(1) The Fubini–Study metric on Pn(C) is a K¨ahler metric. Hence every (reduced) projective variety is a K¨ahlerspace.

n n (2) Let Γ denote a lattice in C . Then the complex torus T := C /Γ is K¨ahler manifold.

45 4.3 Integration on complex spaces Theorem 4.3.1 (Lelong) Let X be a complex manifold, let Z be an analytic subset of pure dimension n in X and let φ be a continuous diﬀerential form of type (n, n) on X with compact support. Then the improper integral Z φ

Z\Sing(Z) is convergent. Moreover it has the following properties.

(i) The complex valued functional Z φ 7→ φ

Z\Sing(Z)

is continuous on the space of continuous diﬀerential forms of type (n, n) on X.

(ii) R φ ≥ 0, if φ is positive. Z\Sing(Z)

(iii) R φ = 0, if φ is exact. Z\Sing(Z)

Proof. A proof can be found in Lelong’s original paper [Le]. See also [Ba3] and [De].

Remarks.

(i) In the situation of the above theorem one always writes R φ instead of R φ. Z Z\Sing(Z) (ii) Under the assumption that the manifold X is of pure dimension n + p one can formulate (i), (ii) and (iii) of the theorem by saying that integration on Z deﬁnes a positive closed current of type (p, p) on X. This current is called the current of integration deﬁned by Z and usually denoted by [Z].

(iii) In the case when Z is not reduced one deﬁnes integration on Z by putting Z Z φ := φ

Z Zred

(iv) By considering an appropriate partition of unity one sees that the theorem is essen- tially local. Now, let Z be a complex space of pure dimension n then any diﬀerential form on Z of type (n, n) can be integrated on Z because locally the space Z can be embedded into a complex manifold where the form is the restriction of a globally deﬁned form of the same type.

46 5 Flatness

5.1 Flat maps Let R be a commutative ring with unity.

Definition 5.1.1 A R–module M is called R–flat (or simply flat) if for every injective morphism of R–modules N1 → N2 the induced morphism N1 ⊗ M → N2 ⊗ M is injective.

Proposition 5.1.2 Let R be a principal ideal domain. An R–module M is R–ﬂat if and only if it is torsion free.

Proof. See [Dou2].

Let X and Y be complex spaces.

Definition 5.1.3 A holomorphic map f : X → Y is called flat at x ∈ X if OX,x is ˜ OY,f(x)–flat (via fx : OY,f(x) → OX,x). The map f is called flat if it is flat at all x ∈ X.

Definition 5.1.4 Let f : X → Y be a holomorphic map. A coherent OX –module S is called flat over Y or f−flat if Sx is OY,f(x)–flat for all x ∈ X.

Lemma 5.1.5 Let f : X → Y be finite. Then f is flat if and only if f∗OX is a locally free OY –module. Moreover, if Y is reduced, then f is flat if and only if the function X y 7→ νf (y) := dimC OXy,x x∈|Xy| is locally constant.

Proof. See [Dou2].

Theorem 5.1.6 Let f : X → Y be a holomorphic map.

(i) If f is ﬂat, then dimx X = dimf(x) Y + dimx Xf(x) for all x ∈ X.

(ii) If f is ﬂat, then f is open.

(iii) If X and Y are complex manifolds, then f is ﬂat if and only if f is open.

Proof. A proof can be found in [Fis].

47 Examples. (Douady)

(1) Consider the following complex space consisting of two 2–planes intersecting transver- 4 sally at the origin in C

4 X := V (z3, z4) ∪ V (z3 − z1, z4 − z2) ⊂ C ,

4 2 where (z1, z2, z3, z4) are the standard coordinates of C . Let f : X → C be the projection f(z1, z2, z3, z4) = (z1, z2). Then we have νf (a, b) = 2, if (a, b) 6= (0, 0). The ﬁber X(0,0) is given by the ideal

2 2 a := hz1, z2, z3, z3z4, z4i,

so we have

= {z . . . , z }/a ∼ {z , z }/hz2, z z , z2i. OX(0,0),(0,0) C 1 4 = C 3 4 3 3 4 4

This analytic algebra is a complex vector space of dimension 3, so νf (0, 0) = 3. Thus, f is not ﬂat. Notice however that the mapping f is open.

3 (2) Let (x, y, z) denote the standard coordinates of C and consider the space

3 C ⊃ X := V (xz − y),

together with the map

2 f : X → C , (x, y, z) 7→ (x, y).

2 For (a, b) ∈ C the ﬁber X(a,b) is deﬁned by the ideal

 hx − a, y − b, z − b i : a 6= 0  a hx − a, y − b, xz − yi = hx, yi :(a, b) = (0, 0) .  h1i : a = 0, b 6= 0

This implies that the ﬁbers X(a,b) are given by

 {(a, b, b )} : a 6= 0  a X(a,b) = z–axis : (a, b) = (0, 0) .  ∅ : a = 0, b 6= 0

Hence f is not open and consequently not ﬂat.

The following result is usually described by saying that ﬂatness is preserved by base change.

Theorem 5.1.7 A holomorphic map which is obtained from a ﬂat holomorphic map by base change is ﬂat.

48 Proof. Consider a Cartesian square

p X1 / X

f1 f Y1 q / Y where f is a ﬂat holomorphic map. Then one has to prove that f1 is also a ﬂat map. A proof of this can be found in [Fis].

5.2 Flat families of compact subspaces Definition 5.2.1 Let X be a complex space. A flat family of compact subspaces of X is a couple of complex spaces (S, Z) such that Z is a subspace of S × X and such that the natural projection π : Z → S is proper and flat.

Remarks. (i) Notice that none of the complex spaces in the above deﬁnition is assumed to be reduced; not even the parameter space.

(ii) The terminology ﬂat family of compact subspaces needs an explanation. If we put −1 Zs := π (s) for all s in S, then we have a family (Zs)s∈S of subspaces of X parameterized by S. Since π is proper these subspaces are all compact so we have a family of compact subspaces. We think of Z as the graph of the family because set theoretically we have

|Z| = {(s, x) ∈ S × X ; x ∈ Zs}.

To say that Z is a complex subspace of S × X amounts to saying that the family depends holomorphically on the parameter s. But experience shows that this condition is not strong enough to guarantee “reasonable behavior” of the family. The properness condition means that for a given point s0 in S and an open neigh-

borhood U of Zs0 there exists an open neighborhood S0 of s0 in S such that Zs is contained in U for all s in S0. The ﬂatness condition is harder to understand but we would like to point out that the dimension formula in theorem 5.1.6 is particularly important. But it is a kind of mystery that this algebraic condition ensures a good geometric situation.

2 Example. For each (s, t) in C let X(s,t) be the subspace of the complex zw−plane consisting of the points (0, 0) and (s, t). Then X(s,t) is given by the equations

z(z − s) = 0, w(z − s) = 0, z(w − t) = 0 and w(w − t).

The graph of the family (X ) 2 is the subspace given by the above equations in the (s,t) (s,t)∈C complex stzw−space and the projection onto the parameter space is the restriction of the natural projection 4 2 C → C , (s, t, z, w) 7→ (z, w).

49 This is exactly the situation which was under consideration in example 1 in subsection 5.1 where we showed that this projection is not ﬂat. Thus the family (X ) 2 is not (s,t) (s,t)∈C ﬂat.

6 The Douady space of a complex space

Notation. For a product X × Y of complex spaces we will henceforth denote by π the projection onto the ﬁrst factor and by p the projection onto the second factor, i. e.

p X × Y / Y

π X

If more then one product is under consideration we will write πX and pY .

Theorem 6.0.2 [Douady] Let X be a complex space and let H be a coherent OX – module. Then there exist a complex space D = D(H ) and a sheaf R on D × X having the following properties:

(i) R is a quotient of p∗H ,

(ii) R is ﬂat over D and π| Supp(R) is proper,

(iii) (universal property) if S is a complex space and F is a coherent quotient of p∗H on S × X such that F is ﬂat over S and πS| Supp(F ) is proper, then there exists a unique holomorphic map f : S → D such that

∗ F = (f × idX ) R.

Proof. See [Dou1].

Remark. With obvious identiﬁcations the above theorem says that, for every coherent quotient F of H with compact support, there exists a unique point s in S such that the analytic restriction of the sheaf R to {s} × X is equal to F . Thus the complex space D parameterizes all the coherent quotients of H that have compact support.

Corollary 6.0.3 Let X be a complex space. Then there exists a complex space D = D(X) and a subspace X ⊂ D × X, called the universal (ﬂat) family, having the following properties:

(i) X is ﬂat over D and π|X is proper,

(ii) (universal property) if S is a complex space and if Z is a subspace of S × X such that Z is ﬂat over S and πS|Z is proper, then there exists a unique holomorphic map f : S → D such that Z = S ×D X .

50 Proof. First we apply theorem 6.0.2 to the case where H = OX and obtain the complex ∗ space D = D(H ) and the coherent quotient R of p OX = OD×X . Then we put |X | := Supp(R) and let OX be the restriction of R to its support.

Deﬁnition 6.0.4 The complex space D = D(X) is called the Douady space of X.

Remarks.

(i) The space D parameterizes all compact complex subspaces of X.

(ii) If X is projective, then D(X) coincides with the analytic space associated with the Hilbert scheme Hilb(X) as constructed by Grothendieck. For more informations about the Hilbert scheme see for instance [Ha] or [Gro2]

(iii) The theorem 6.0.2 as its corollary is usually referred to as the Theorem of Douady.

7 The space of holomorphic maps

7.1 The universal holomorphic map Let X and Y be complex spaces with X compact. By identifying each holomorphic map X → Y with its graph we can think of Hol(X,Y ) as a subset of D(X × Y ). In virtue of this identiﬁcation we have

Hol(X,Y ) = {t ∈ D(X × Y ); Xt is a graph of a holomorphic map X → Y }, where X is the universal family of D(X × Y ) × X × Y .

Theorem 7.1.1 (Douady) Let X and Y be complex spaces with X compact.

(i) Hol(X,Y ) is an open subset of D(X × Y ) and as such inherits a complex structure.

(ii) The space X ∩ (Hol(X,Y ) × X × Y ) is the graph of a holomorphic map, called the universal holomorphic map

Φ: Hol(X,Y ) × X → Y.

(iii) (universal property) If S is a complex space and if f : S × X → Y is a holomorphic map, then there exists a unique holomorphic map g : S → Hol(X,Y ) such that

f = Φ ◦ (g × idX ).

(iv) If the complex space X is reduced, then the topology of Hol(X,Y ) is the topology of compact convergence.

Proof. See [Dou1].

51 Remark. It is easy to see that the underlying continuous mapping of the universal holomorphic map Φ: Hol(X,Y ) × X → Y is the evaluation mapping (f, x) 7→ f(x).

Let X, Y and Z be complex spaces with X and Y compact and let

Φ: Hol(X,Y ) × X → Y and Ψ: Hol(Y,Z) × Y → Z be the corresponding universal holomorphic maps. Then we get a holomorphic map

Ψ ◦ (id × Φ): Hol(Y,Z) × Hol(X,Y ) × X → Z to which, according to the universal property, corresponds exactly one holomorphic map

Hol(X,Y ) × Hol(Y,Z) → Hol(X,Z).

It is readily seen that the underlying continuous mapping of this holomorphic map is the composition (f, g) 7→ g ◦ f.

7.2 The group of automorphisms From the last theorem and its remark it is relatively easy to deduce the following theorem.

Theorem 7.2.1 Let X be a compact complex space and Aut(X) be the group of automorphisms of X. Then Aut(X) is an open subset of Hol(X,X) and as such inherits a complex structure. Furthermore the law of composition makes it a complex Lie group.

Proof. See [Dou1].

Remark. What Douady actually proves is that the group operations on Aut(X) deﬁned by the composition are holomorphic maps. But by deﬁnition a complex Lie group is a complex manifold, so to prove that the complex space Aut(X), endowed with the group operation of composition, is a complex Lie group one has to show that it is smooth at every point. In fact this is true for any complex space endowed with holomorphic group operations. More precisely, let G be a complex space together with a distinguished element e ∈ G called the neutral element, and two holomorphic maps

C : G × G → G (multiplication),

I : G → G (inversion) having the following properties

C ◦ (C × idG) = C ◦ (idG ×C) (associative law),

C ◦ (ιe × idG) = C ◦ (idG ×ιe) = idG, where ιe is the inclusion map {e} → G (e is the neutral element),

C ◦ (I × idG) = C ◦ (idG ×I) = ιe (every element has an inverse). Then G is a complex Lie group. For the proof see [Car1]. In the light of this result it is interesting to have another look at example 4b in subsection 1.4.

52 8 Ramiﬁed coverings

8.1 Analytic coverings n+p Let X be a reduced complex subspace of pure dimension n in an open set W in C and ﬁx a point x ∈ X.

n p Then we can choose open polydiscs U ⊂⊂ C and B ⊂⊂ C such that the following conditions are fulﬁlled:

(i) x ∈ U × B,

(ii) U × B ⊂ W and

(iii) X ∩ (U × (bd B)) = ∅.

We put X0 := X ∩ (U × B) and let π : X0 → U be the canonical projection.

Then the map π : X0 → U is a so-called analytic covering, in other words there exists a thin analytic subset T ⊂ U such that

π|X0 \ π−1(T ): X0 \ π−1(T ) → U \ T is a k–sheeted covering map with k = νπ(y) for all y ∈ U \ T . The integer k is called the degree of the analytic covering. For a more detailed discussion see for instance [GrRe3].

k Let Sk be the k–th symmetric group. The group Sk acts on B = B × · · · × B by permutation β(b1, . . . , bk) := (bβ(1), . . . , bβ(k)). We denote the orbit space of this action by Symk(B), i. e.

k k Sym (B) := B /Sk.

k k Let π : B → Sym (B) be the canonical map and put [b1, . . . , bk] := π(b1, . . . , bk) for all k (b1, . . . , bk) ∈ B . From a theorem of H. Cartan the orbit space can be made into a complex space in such a way that the canonical map π is holomorphic. (See [KK]).

l p For every positive integer l let S (C ) be the l−th symmetric product of the complex vector p p k l p space C and let σl :(C ) → S (C ) be the l−th elementary symmetric polynomial. One p can interpret this mapping in the following way. Think of the elements in C as C–linear p ? p k forms on the dual space (C ) and, for each (x1, . . . , xk) ∈ (C ) , think of σl(x1, . . . , xk) p ? as a symmetric C–linear l–form on (C ) . By the theorem of Cartan mentioned above the k p orbit space Sym (C ) is a complex space.

53 Proposition 8.1.1 The mapping

k k p M l p S : Sym (C ) → S (C ) l=1 induced by (σ1, . . . , σk) is a proper holomorphic embedding.

Proof. See [Ba1] or [W].

k p It follows from the above proposition that Sym (C ) is an aﬃne variety and it is not hard k k p to see that it is normal and that Sym (B) is isomorphic to its (open) image in Sym (C ).

For the singular locus of Symk(B) one clearly has the the inclusion

k Sing(Sym (B)) ⊂ {[b1, . . . , bk]; bi = bj for some i 6= j}.

If p = 1, then B is an open disk in the complex plane and it is not hard to prove that in this case the complex space Symk(B) is a k–dimensional manifold. If p > 1, then the inclusion is in fact an identity.

Now let us reconsider the k–sheeted covering map

π|X0 \ π−1(T ): X0 \ π−1(T ) → U \ T.

For each trivializing open subset V in U \ T let s1, . . . , sk be the diﬀerent sections of the covering map over V . Then the mapping

k V → Sym (B) x 7→ [s1(x), . . . , sk(x)] is obviously holomorphic. These maps glue together and induce a holomorphic map

U \ T → Symk(B) and it extends to a holomorphic map

U → Symk(B) as can be seen from Riemann’s theorem on removable singularity.

8.2 Ramiﬁed coverings Let us now consider a more general situation and assume we have a ﬁnite number of globally irreducible analytic subsets X1,...,Xm of dimension n in an open set W in n+p C . Suppose also that to each Xj there is associated a positive integer nj, called the multiplicity of Xj. For every point x in the union X1 ∪ · · · ∪ Xm we can choose polydiscs n p U ⊂⊂ C and B ⊂⊂ C satisfying the following conditions: (i) x ∈ U × B,

54 (ii) U × B ⊂ W ,

(iii) Xj ∩ (U × (bd B)) = ∅, for all j.

Then for each j we obtain an analytic covering of U of a certain degree kj. (If x∈ / Xj, then kj = 0 is not impossible). In this situation we say that we have a ramified covering m P of U of degree k := njkj. Such a ramified covering defines a holomorphic map j=1

U → Symk(B), called the holomorphic map associated with the ramiﬁed covering. This map is constructed more or less in the same way as the holomorphic map associated with an analytic covering. One considers open sets that are trivializing for all the analytic coverings simultaneously and each local section over such a set is counted nj times if it belongs to the analytic covering induced by Xj.

9 The Barlet space of a (reduced) complex space

9.1 Analytic families Let X be a reduced complex space.

Deﬁnition 9.1.1 (n–cycle) An n–cycle in X is a ﬁnite linear combination X Z = niZi i∈I

∗ with coeﬃcients in N , where the Zi are globally irreducible complex subspaces of X of dimension n such that Zi 6= Zj holds for i 6= j. The set [ |Z| := |Zi| i∈I is called the support of Z. The n–cycle Z is called compact if its support |Z| is compact.

P Remark. An n–cycle Z = i∈I niZi is compact if and only if each Zi is compact.

Deﬁnition 9.1.2 A scale of X is a triplet E = (U, B, f) having the following properties:

n p (i) U ⊂⊂ C and B ⊂⊂ C are open polydiscs,

(ii) f is a holomorphic embedding of an open subset XE of X into an open neighborhood n+p of U × B in C . The scale E is said to be adapted to an n–cycle Z if

f(|Z| ∩ XE) ∩ (U × (bd B)) = ∅.

55 Assume E = (U, B, f) is a scale of a complex space X adapted to an n–cycle Z in X. Then Z induces a ramiﬁed covering of a certain degree, which we shall denote by kE = degE Z, of an open neighborhood of U. Hence, Z induces a holomorphic map

U → SymkE (B).

More generally let (Zt)t∈T be a family of n−cycles in X and suppose the scale E is adapted to all of the cycles in the family. Then for each t in T we have a ramified covering of an open neighborhood of U. If all of these ramified coverings have the same degree kE then we get a well defined mapping

kE gE : T × U → Sym (B)

kE such that gE(t, ·): U → Sym (B) is the holomorphic map induced by Zt.

Deﬁnition 9.1.3 Let S be a reduced complex space and let (Zs)s∈S be a family of n–cycles in X.

(i) The family (Zs)s∈S is called analytic if for every s0 ∈ S and every scale E =

(U, B, f), adapted to the cycle Zs0 , there exists an open neighborhood SE of s0 in S such that

(a) E is adapted to Zs for all s ∈ SE,

(b) degE Zs = degE Zs0 for all s ∈ SE, kE kE (c) the map gE : SE×U → Sym (B) is holomorphic, where gE(s, ·): U → Sym (B) is the holomorphic map induced by Zs.

(ii) The family (Zs)s∈S is called an analytic family of compact n–cycles if it is analytic,

every cycle is compact and for every so ∈ S and every neighborhood W of |Zs0 | there 0 0 exists an open neighborhood S such that |Zs| ⊂ W for all s ∈ S .

Deﬁnition 9.1.4 Let (Zs)s∈S be a family of n−cycles in X. Then the graph of the family (Zs)s∈S is deﬁned by GS := {(s, x) ∈ S × X; x ∈ |Zs|}.

Theorem 9.1.5 (Barlet) The graph GS of an analytic family (Zs)s∈S in a complex space X is an analytic subset of S × X. For the proof of the theorem we need the following lemma.

Lemma 9.1.6 Let k be a positive integer. Then the set

k n k o Sym (B)#B := ([x1, . . . , xk], y) ∈ Sym (B) × B ; y ∈ {x1, . . . , xk} is an analytic subset of Symk(B) × B.

56 k Proof. For 1 ≤ j ≤ k let pj : B → B be the j–th canonical projection, and consider the holomorphic map k dj := pj × idB : B × B → B × B.

The diagonal, ∆B, of B × B is an analytic subset of B × B so k [ −1 k {((x1, . . . , xk), y); y ∈ {x1, . . . , xk}} = dj (∆B) j=1 is an analytic subset of Bk × B. Now, Symk(B)#B is the image of the above analytic set by the ﬁnite holomorphic map k k π × idB : B × B → Sym (B) × B so Symk(B)#B is an analytic subset of Symk(B) × B.

Proof of the theorem. Let (s0, z0) ∈ S × X. Choose a scale E = (U, B, f) of X adapted to Zs0 at z0 and let SE be an open neighborhood of s0 such that E is adapted k to Zs for all s ∈ SE. Let g : SE × U → Sym (B) be the holomorphic map associated with −1 the induced ramiﬁed coverings (See 9.1.3). Put V := f (U × B) and let GS denote the graph of the family. Then it is enough to show that GS ∩ (SE × V ) is an analytic subset of SE × V . Notice ﬁrst that a point (s, x) in SE × V belongs to GS if and only if the point (s, f(x)) belongs to the set {(s, u, b) ∈ SE × U × B ; b ∈ g(s, u)} . Now, from lemma (9.1.6) we know that the set Symk(B)#B is an analytic subset of Symk(B) × B and since the map k g × idB : SE × U × B → Sym (B) × B is holomorphic the set −1 k A := (g × idB) (Sym (B)#B) = {(s, u, b) ∈ SE × U × B ; b ∈ g(s, u)} −1 is an analytic subset of SE ×U ×B. It then follows that GS ∩(SE ×V ) = (idSE ×f|V ) (A) is an analytic subset of SE × V .

Remark. The topological condition in (ii) of deﬁnition (9.1.3) is satisﬁed if and only if the graph of the family (Zs)s∈S is proper over S.

Theorem 9.1.7 (Barlet) Let X be a complex space. S (i) Let (Zs)s∈S be an analytic family of compact n–cycles in X and let GS := Gi be i∈I the decomposition of its graph into irreducible components. Then for each i ∈ I there exists an ni > 0 such that for generic s ∈ S each irreducible component of |Zs| which is contained in Gi has multiplicity ni. P (ii) Let S be a normal space and let G := niGi be a cycle in S × X such that the i∈I projection Gi → S is proper and surjective with all ﬁbers of pure dimension n. Then G deﬁnes an analytic family of compact n–cycles in X.

57 Proof. See [Ba1].

Examples. Let us reconsider the family in the example of subsection (5.2). To each space X(s,t) of this family we can associate the 0–cycle {(0, 0)} + {(s, t)}. (This procedure 2 will be discussed in section 10.) Hence a family of 0–cycles parameterized by C . From 2 (ii) in the above theorem we see that this family is analytic since the space C is normal.

Corollary 9.1.8 Let S be a normal complex space and let f : X → S be a proper surjective holomorphic map with all ﬁbers of pure dimension n. Then f induces an analytic family of n–cycles (Zs)s∈S such that Zs is the n–cycle associated to the ﬁber Xs for generic s ∈ S.

Proof. See [Ba1].

Remark. Assuming the hypotheses of the corollary the family of compact subspaces −1 (f (s))s∈S is not ﬂat unless the map f is ﬂat.

9.2 The Barlet space For a complex space X the set of all compact n–cycles in X will always be denoted by Cn(X).

Theorem 9.2.1 (Barlet) The set Cn(X) carries a reduced complex structure such that the following conditions are fulﬁlled:

(i) the family (Zs)s∈Cn(X) is an analytic family of compact n–cycles,

(ii) (universal property) for every analytic family (Zs)s∈S of compact n–cycles the map

S → Cn(X), s 7→ Zs

is holomorphic.

Proof. See [Ba1].

Deﬁnition 9.2.2 The complex space [ C (X) := Cn(X) n≥0 is called the Barlet cycle space of X, the Barlet space of X or the cycle space of X.

Theorem 9.2.3 If X is a projective variety, then the Barlet space of X coincides with the analytic space associated with the Chow scheme of X.

58 Proof. See [Ba1]. For the deﬁnition and fundamental properties of the Chow scheme see for instance [AN] or [HP].

Proposition 9.2.4 Let X be a complex space.

(i) For each complex subspace Y of X the canonical map Cn(Y ) → Cn(X) is a holomorphic embedding.

(ii) If A is an analytic subset of X, then the set

{Z ∈ Cn(X); A ∩ |Z|= 6 ∅}

is an analytic subset of Cn(X).

Proof. (i) See [Ba1].

(ii) Let p and π denote the projections of the graph onto X and Cn(X) respectively and −1 −1 observe that π(p (A)) = {Z ∈ Cn(X); A ∩ |Z|= 6 ∅}. Since π is proper π(p (A)) is an analytic subset of Cn(X).

10 The morphism Douady to Barlet

10.1 Fundamental cycles

Let Z be a complex space of pure dimension n such that Zred has only ﬁnitely many irreducible components. Then we write [ Zred = Zi, i∈I where the Zi are the irreducible components of Zred, Zi 6= Zj for i 6= j and |I| < ∞. Then Z˚ := Z \ Sing(Z ) is a connected manifold and we have = Red( ) for all i i red OZ˚i,z OZ,z z ∈ Z˚i. For each z ∈ Z˚i there exists a section u O ˚ OZ,z can / Zi,z , which makes into a coherent –module. These modules have a generic rank on OZ,z OZ˚i,z Z˚i which does not depend on the choice of the section and it will be denoted by ni. This rank is called the multiplicity of Zi in Z.

Deﬁnition 10.1.1 The n–cycle X [Z] := niZi i∈I is called the n–cycle associated with the complex space Z or the fundamental cycle of Z.

59 Examples.

3 2 (1) Let Z be the complex space given by the equation y x = 0 in C . This space has two irreducible components, Z1 deﬁned by x = 0 and Z2 deﬁned by y = 0. The component Z1 has multiplicity 1 and the component Z2 has multiplicity 3. In other words [Z] = Z1 + 3Z2.

2 2 (2) Let Z be the complex space given by the equations xy = 0 and y = 0 in C . This space is not reduced and has only one irreducible component Z1 given by the equation y = 0. This component has multiplicity 1 since Z is generically reduced. In other words [Z] = Z1.

10.2 Deﬁnition of the morphism Let X be a complex space. To each compact subspace Z of pure dimension in X corresponds the fundamental cycle [Z]. Hence a mapping from the subset of the Douady space of X, consisting of the pure dimensional subspaces of X, to the Barlet space of X. This subset of the Douady space is not easy to cope with. The reason for this is that there can be subspaces, which are not of pure dimension, arbitrarily near to a pure dimensional subspace. This can happen if the pure dimensional subspace has a so called embedded component. We will not give a deﬁnition of an embedded component here since it is rather technical but the following example explains what kind of a phenomenon this is.

Example. In the complex zw−plane consider the family (Xs)s∈C where Xs consists of the z−axis and the point (0, s). The z−axis is defined by the equation w = 0 and the point (0, s) is defined by the equations z = 0, w − s = 0 so the subspace Xs is defined by the equations zw = 0 and w(w − s) = 0. The graph of the family is given by the above equations in the complex szw−space and it consists of the sz−plane and the line given by w = s in the sw−plane. Now, the family

(Xs)s∈C is flat if and only if the projection from its graph to the s−axis is a flat map. But outside the origin the projection is flat thanks to theorem 5.1.6 and at the origin it is also flat in virtue of proposition 5.1.2 so the family is flat. Notice that the subspace X0 is pure dimensional but for all s 6= 0 the subspace Xs is not pure dimensional. This phenomenon is due to the fact that the double point, given by the 2 equations z = 0 and w = 0, is an embedded component of X0.

Let Dn(X) denote the set of all compact complex subspaces of X of pure dimension n without embedded components. It can be shown that the set Dn(X) is both open and closed in D(X) and as such inherits the complex structure from D(X). Then we have a well deﬁned map Cˆ : Dn(X)red → Cn(X),Z 7→ [Z].

Theorem 10.2.1 (Barlet) The map Cˆ : Dn(X)red → Cn(X) is holomorphic.

Proof. See [Ba1].

60 10.3 Some special cases We will now consider the morphism Douady to Barlet in some special cases. We start with a remarkable result concerning zero dimensional subspaces.

Let A be a 0–dimensional compact subspace of X. Then the integer X dim OA,a. a∈A

[k] is called the length of A. Let X denote the set of all A in D0(X) which have length k. The restriction of the map Cˆ to X[k] is the map

[k] k X ρ : X → Sym (X) ⊂ C0(X),A 7→ naa, a∈A

k where na = dim OA,a. Moreover, Sym (X) is open and closed in C0(X).

Theorem 10.3.1 (Fogarty) Let X be a complex projective surface. Then we have the following:

(i) X[k] is a 2k–dimensional manifold.

(ii) The map ρ: X[k] → Symk(X) induces a biholomorphic map

X[k] \ ρ−1(D) → Symk(X) \ D,

k where D = {[x1, . . . , xk] ∈ Sym (X); xi = xj for some i and j with i 6= j}. (iii) The set D is the singular locus of Symk(X).

Proof. (ii) is obviously true for any complex space X and (iii) is true if X is a complex manifold of dimension greater than or equal to 2. For a proof of (i) see [Fog].

For cycles of codimension 1, so-called eﬀective Weil divisors, in a complex manifold we have the following result.

Theorem 10.3.2 (Barlet) Let X be a complex manifold of pure dimension n+1. Then the map Cˆ : Dn(X)red → Cn(X) is biholomorphic.

Proof. See [Ba1]

At a point in the Douady space corresponding to a smooth subspace we have the following result.

Theorem 10.3.3 (Barlet) Let X be a complex space. If Z ∈ Dn(X) is a manifold, then the map Cˆ : Dn(X)red → Cn(X) is biholomorphic in a neighborhood of Z in Dn(X)red.

Proof. See [Ba1].

61 Remark. Under the weaker assumption that Z is reduced instead of smooth the theorem is generally wrong. For an example see [Ba1].

11 Compact subsets of Barlet spaces

Let X be a projective variety. Then by theorem 9.2.3 each connected component of Cn(X) is projective and in particular compact. In general this is not the case; there exist compact complex spaces having Barlet spaces whose connected components are not all compact.

11.1 The theorem of Bishop and its consequences In 1964 E. Bishop proved a theorem which plays a key role in characterizing compact subsets of Barlet spaces, but before we state this theorem we need to introduce the notion of a Hausdorﬀ-metric.

Let (X, d) be a bounded metric space and let C be the set of all closed sets in X. For all (A, B) ∈ C × C put D(A, B) := sup inf d(a, b) a∈A b∈B Then we deﬁne a mapping DH : C × C → R+ by

DH (A, B) := max{D(A, B),D(B,A)}.

It is not hard to see that DH is a metric on C; called the Hausdorﬀ-metric of X.

n Theorem 11.1.1 (Bishop) Let U be an open set in C and let (Aj)j be a sequence of analytic subsets of U each of pure dimension k. If the sequence converges to a closed non empty set A in the Hausdorﬀ-metric on closed subsets of U and if there exists a constant b such that vol(Aj) ≤ b for all j, where vol(Aj) denotes the Euclidean 2k−volume of Aj, then A is an analytic set of pure dimension k in U.

Proof. See [Bi]. Deﬁnition 11.1.2 Let X be a complex space with a Hermitian metric h and let ω be the P (1,1) form associated with h. For any n−cycle Z = niZi ∈ Cn(X) the number i∈I Z X n volh(Z) := ni ω . i∈I Zi is called the h−volume of Z, where ωn = ω ∧ · · · ∧ ω.

The following theorem is a consequence of Bishop’s theorem.

Theorem 11.1.3 (Liebermann) A set S ⊂ C (X) is relatively compact if and only if there exists a compact set K in X, a Hermitian metric h on X and a constant b such that the following two conditions are satisﬁed for every cycle Z ∈ S:

62 (i) Z is contained in K,

(ii) volh(Z) ≤ b.

Proof. See [Li]. Compare also [Ba2].

Remark. Condition (i) of the above theorem is automatically satisﬁed in the case where X is a compact complex space.

Corollary 11.1.4 Let X be a complex space that is countable at inﬁnity. Then its Barlet space C (X) is also countable at inﬁnity.

Proof. Let K0 ⊂ K1 ⊂ · · · be a sequence of compact subsets in X whose union is X. For each n let Sn be the set of all cycles in C (X) that are contained in Kn and whose volume is bounded by n. Obviously these sets form a covering of C (X) and by the previous theorem they are compact. Hence C (X) is countable at inﬁnity.

Corollary 11.1.5 If X is a compact K¨ahlerspace, then the connected components of C (X) are compact.

Proof. Suppose ﬁrst that X is a manifold and let ω be a K¨ahler-form on X. If Z1 and Z2 are two n−cycles in X belonging to the same homology class in H2n(X, Z) and W is a cycle having Z2 − Z1 as boundary, then Stokes’ formula implies Z Z n n 0 = dω = ω = volh(Z2) − volh(Z1). W ∂W

Now two n−cycles belonging to the same connected component of Cn(X) are easily seen to be homologous so the function Z 7→ volh(Z) is constant on every connected component of Cn(X). Theorem 11.1.3 then implies that every connected component of Cn(X) is compact. To prove the theorem in the general case one needs the Stokes’ formula for semianalytic sets which we will not discuss here. See [Fu1].

Remark. For cycles of codimension 1, i. e. the eﬀective Weil divisors, one can show without the K¨ahler assumption that the irreducible components are compact; more precisely if X is a compact complex space of pure dimension n + 1, then the irreducible components of Cn(X) are compact.

We ﬁnish this subsection with a remarkable result.

Theorem 11.1.6 (Varouchas) The Barlet cycle space of a K¨ahlerspace is a K¨ahler space.

Proof. A weaker version of this theorem is proved in [Va2], but there it is explained what kind of result is needed to prove the stronger version. That result is then proved in [BV].

63 11.2 The Fujiki class Deﬁnition 11.2.1 The Fujiki class C consists of all reduced complex spaces X such that there exists a surjective holomorphic map f : X˜ → X, where X˜ is a compact K¨ahler manifold.

Theorem 11.2.2 The class C has the three following properties.

(i) If X and Y belong to C then so does X × Y .

(ii) If f : X → Y is a surjective holomorphic map where X belongs to C, then Y also belongs to C.

(iii) If X belongs to C then so does every subspace of X.

Proof. See [Ca].

Theorem 11.2.3 (Campana) If X belongs to C, then every irreducible component of C (X) belongs to C.

Proof. See [Ca].

Remark. The theorem says in particular that the irreducible components of C (X) are compact if X belongs to C. But this is not true for the connected components; there are complex spaces which belong to the Fujiki class whose Barlet spaces have non compact connected components as can be seen from the following example.

Example. Let L be a projective line and let C be a smooth conic in P3(C) such that L ∩ C = ∅. Choose a biholomorphic map f : L → C and let Z be the quotient space of P3(C) obtained by gluing L and C together via f. This space has a natural structure of a complex space such that the quotient mapping

q : P3(C) → Z is a holomorphic map. (See [KK]). It is easy to construct an analytic family of 1–cycles

(Xs)s∈C in P3(C) such that X0 = 2L and that X1 = C. The images of these cycles by q form an analytic family (Ys)s∈C such that Y0 = 2q(L) and Y1 = q(C) = q(L). Hence q(L) and 2q(L) belong to the same connected component of C (Z). Consequently the set {nq(L); n ≥ 1} is contained in exactly one of the connected components of C (Z). From theorem 11.1.3 it then follows that this connected component is not compact.

The following theorem gives an important characterization of the Fujiki class.

Theorem 11.2.4 (Varouchas) A complex space X belongs to C if and only if X is bimeromorphic to a compact K¨ahlermanifold.

Proof. See [Va1].

64 11.3 Corresponding results for the Douady space

Let X be a complex space and let {Dj | j ∈ J} be the collection of all those irreducible components of the Douady space D(X) that contain at least one point that corresponds to a reduced subspace of pure dimension n in X. Put

0 [ Dn(X) := Dj. j∈J

Theorem 11.3.1 (Fujiki) Let X be a compact K¨ahler space. Then the map Cˆ : Dn(X)red → Cn(X) induces a proper holomorphic map

0 Dn(X) → Cn(X).

Proof. A proof can be found in [Fu1]. Its main ingredient is the important Flattening theorem of Hironaka. We will not discuss it here but for a short introduction to the Flattening theorem and some of its consequences see [Pe].

The above theorem combined with corollary 11.1.5 now gives the following result.

0 Corollary 11.3.2 For every compact K¨ahlerspace X the connected components of Dn(X) are compact.

Theorem 11.3.3 (Fujiki) The Douady space of a complex space has only countably many irreducible components.

Proof. See [Fu2].

It seems natural to ask whether the Douady version of theorem 11.1.6 holds.

Problem. Let X be a K¨ahlerspace. Is the Douady space of X also a K¨ahler space?

As a matter of fact this problem was raised by Hironaka in 1977 for compact K¨ahler spaces well before Varouchas proved his theorem. See [Hir]. It is still open.

A Algebraic supplements

A.1 Elementary theory of rings Throughout this appendix all rings are commutative and possess a unity. Moreover, all homomorphisms of rings (or ring morphisms) are assumed to preserve the unity.

65 Notation. Rings will be denoted by R, S, . . . , their unities by 1R, 1S,... and their zeros by 0R, 0S. . . . If no confusion is possible we will simply write 1 and 0 instead of 1R and 0R.

Deﬁnition A.1.1 An element x ∈ R is called

(i) a unit if there exists an element y ∈ R such that

xy = yx = 1,

(ii) a zero divisor if there exists an element y ∈ R \{0} such that

xy = 0,

(iii) nilpotent if there exists a positive integer n such that

xn = 0.

Remarks.

(i) If x ∈ R is a unit, then there is a unique element y ∈ R such that xy = yx = 1. It is called the inverse of x and denoted by x−1.

(ii) The units in R form a multiplicative group. A ring R is a ﬁeld if 0R 6= 1R and every element other than 0R is a unit. (iii) Every nilpotent element x ∈ R is a zero divisor. The converse need not be true.

Deﬁnition A.1.2 We call an additive subgroup a of R an ideal in R if

rx ∈ a for all r ∈ R and all x ∈ a.

Remarks.

(i) For every ideal a ⊂ R the quotient R/a is a commutative ring. If a ( R then the quotient R/a is a ring with unity 1R + a 6= 0 and the canonical map π : R → R/a is a morphism of rings.

(ii) All nilpotent elements in R form an ideal, called the nilradical of R, and denoted by n = nR. If nR = {0}, then the ring R is said to be reduced. The quotient R/n is a reduced ring, called the reduction of R.

Deﬁnition A.1.3 A ring R is called an integral domain if it contains no other zero divisors than the zero element.

66 Notation Let x1, . . . , xk be elements in a ring R. The ideal generated by the elements will be denoted by hx1, . . . , xki, more precisely

hx1, . . . , xki := R · x1 + ··· + R · xk := {r1x1 + ··· + rkxk; r1, . . . , rk ∈ R}.

Deﬁnition A.1.4 An ideal a ⊂ R is called ﬁnitely generated if there exist elements x1, . . . , xk in R such that a = hx1, . . . , xki.

Deﬁnition A.1.5 An ideal a ⊂ R is called maximal if a 6= R and for every ideal b in R such that a ( b we have b = R.

Lemma A.1.6 An ideal a ⊂ R is maximal if and only if the quotient ring R/a is a ﬁeld.

Proof. See [AMcD].

Lemma A.1.7 Let R 6= 0 be a ring. Then R has at least one maximal ideal and each ideal a ( R is contained in a maximal ideal. In particular, every non–unit in R is contained in a maximal ideal.

Proof. See [AMcD].

Deﬁnition A.1.8 An ideal a ⊂ R is said to be prime if it satisﬁes the following condition

x, y ∈ R and xy ∈ a implies x ∈ a or y ∈ a.

Lemma A.1.9 An ideal a is prime if and only if the quotient ring R/a is an integral domain.

Proof. See [AMcD].

Deﬁnition A.1.10 Let a ⊂ R be an ideal. Then

√ n a := {r ∈ R; r ∈ a for some n ∈ N0} is called the radical of a.

Remark. It is easy to see that the radical of an ideal in R is also an ideal in R.

A.2 The full ring of fractions Deﬁnition A.2.1 A subset G ⊂ R of a ring R is called multiplicative, if the following conditions are satisﬁed.

(i) 1 ∈ G and 0 ∈/ G.

(ii) For all g1, g2 ∈ G we have g1g2 ∈ G.

67 Example. The ring R is an integral domain if and only if the set R\{0} is multiplicative.

Now let G be a ﬁxed multiplicative subset of R and deﬁne on R ×G the following relation:

(r1, g1) ∼ (r2, g2):⇐⇒ ∃g ∈ G : g(g2r1 − g1r2) = 0.

It is elementary to check that this relation is an equivalence relation.

Deﬁnition A.2.2 The set QG(R) := (R × G)/ ∼ is called the ring of fractions of R with r respect to G. We denote the equivalence class of (r, g) in QG(R) by g .

Lemma A.2.3 If addition and multiplication are deﬁned on QG(R) by putting r r r g + r g 1 + 2 := 1 2 2 1 g1 g2 g1g2 and r r r r 1 · 2 := 1 2 , g1 g2 g1g2 then QG(R) equipped with these operations becomes a commutative ring with identity 1 = 1R 1 . Moreover, the mapping R r R → QG(R), r 7→ . (∗) 1R is a ring homomorphism.

Warning. In general the homomorphism (∗) is not injective.

Deﬁnition A.2.4 Let R be a reduced ring and let G be the set of all non–zero–divisors in R. Then Q(R) := QG(R) is called the full ring of fractions of the ring R.

Example. If R is an integral domain, then R is automatically reduced and the set of non–zero–divisors in R is equal to R \{0}. In this case Q(R) is the ﬁeld of fractions of R. For example, if R = Z we have Q(R) = Q.

Deﬁnition A.2.5 Let S be a ring extension of a ring R, i. e. S is a ring containing R as a subring.

(i) An element s ∈ S is called integral over R if there exists an integer k ≥ 1 and and elements r0, . . . , rk−1 ∈ R such that

k−1 k r0 + r1s + ··· + rk−1s + s = 0.

(ii) The ring S is called integral over R if each element of S is integral over R.

(iii) The ring R is called integrally closed in S if every element in S that is integral over R already belongs to R.

Deﬁnition A.2.6 A ring R is called normal if R is reduced and integrally closed in Q(R).

68 A.3 Local rings and Noetherian rings Deﬁnition A.3.1 A ring R is called local, if it is commutative and has a unique maximal ideal m = mR.

Example. A ﬁeld is a local ring having {0} as a maximal ideal.

Deﬁnition A.3.2 Let R and S be local rings. A ring morphism φ: R → S is called local if the inclusion φ(mR) ⊂ mS holds.

Lemma A.3.3 A ring R is local if and only if the subset of non–units in R is an ideal in R. In that case we have mR = {r ∈ R; r is not a unit}.

Proof. See [AMcD].

Definition A.3.4 A local ring A is called a local C–algebra if A is a C–algebra and the canonical morphisms ι: C → A, ι(z) := z · 1A and ε: A → A/mA have the property that the composition ε ◦ ι: C → A/mA is an isomorphism of fields. The fields C and A/mA will be identified via the isomorphism ε ◦ ι and the map ∼ ε: A → A/mA = C will be called the evaluation map.

Remarks.

(i) In virtue of the morphism ι we consider C as a subalgebra of A.

(ii) As a complex vector space A is isomorphic to the direct sum C ⊕ mA. Every a in A has a unique decomposition

a = z + x ∈ C ⊕ mA

with ε(a) = z.

Example. Let X be a topological space and let CX denote the sheaf of germs of continuous functions X → C. The elements of a stalk CX,x, in other words the germs, are equivalence classes [f] of continuous functions f ∈ C (U) where U ⊂ X is an arbitrary neighborhood of x ∈ X. Here two such functions fi ∈ C (Ui), i = 1, 2, are called equivalent if there exists a neighborhood W ⊂ U1 ∩ U2 such that

f1|W = f2|W.

69 For λ ∈ C and fi ∈ C (Ui), i = 1, 2 we choose an open neighborhood W of x in U1 ∩ U2 and deﬁne

[f1] + [f2] := [(f1 + f2)|W ], λ[f1] := [λf1], [f1][f2] := [(f1f2)|W ].

Having checked that these operations are well–deﬁned, we see that each stalk CX,x is a commutative C–algebra with unity [1] where 1 denotes the constant function y 7→ 1.

Claim. The C−algebra CX,x is a local algebra with maximal ideal

mx := {[f] ∈ CX,x; f(x) = 0}.

Proof. By lemma A.3.3 it is enough to show that mx is the set of non–units in CX,x. Obviously every element of mx is a non–unit. Now suppose [g] ∈ CX,x \ mx. Then g(x) 6= 0 and the continuity of g implies that there exists a neighborhood U ⊂ X of x such that

g(y) 6= 0

h 1 i for all y ∈ U. Therefore the germ g ∈ CX,x is well deﬁned and we clearly have

1 [1] = [g]. g

Consequently [g] is a unit in CX,x.

n Example. Let D be a domain in C , let OD denote the sheaf of germs of holomorphic functions on D. Exactly the same argumentation as above shows that each stalk OD,x is a local algebra.

Lemma A.3.5 Let A and B be local C–algebras. Then every C–algebra morphism φ: A → B is local.

Proof. Let φ: A → B be a C–algebra morphism. Then the composite morphism ∼ εB ◦φ: A → B/mB = C is surjective and consequently φ(mA) ⊂ mB.

Lemma A.3.6 Let A be a local C–algebra with maximal ideal m and let a ( A be an ideal. Then the quotient algebra A/a is again a local C–algebra. Moreover, m/a is the maximal ideal of A/a.

Proof. Obvious.

Theorem A.3.7 (Nakayama’s lemma) Let R be a local ring with maximal ideal m and let M be a ﬁnitely generated R–module. Then M ⊂ mM implies M = 0.

70 Proof. See [AMcD].

Corollary A.3.8 Let R be a local ring and let φ: M → N be a morphism of ﬁnitely generated R–modules. Then φ is surjective if and only if the induced morphism of C– vector spaces M/mM → N/mN is surjective.

Proof. Left as exercise.

Deﬁnition A.3.9 A ring R is called Noetherian if every ideal in R is ﬁnitely generated.

Theorem A.3.10 (Krull’s intersection lemma) Let R be a Noetherian local ring with maximal ideal mR and let M be a ﬁnitely generated R–module. Then for every submodule N of M the following holds: ∞ \ (N + mjM) = N. j=1

Proof. See [AMcD].

From Krull’s theorem its not hard to deduce the following result.

Corollary A.3.11 Let A and B be local C–algebras and let B be Noetherian. Let x1, . . . , xk generate the maximal ideal mA of A. If φ, ψ : A → B are C–algebra homomorphisms such that φ(xj) = ψ(xj) for all 1 ≤ j ≤ k, then φ = ψ.

A.4 The ring C{z1, . . . , zn}

Let C{z1, . . . , zn} denote the set of convergent power series in z1, . . . , zn. Obviously C{z1, . . . , zn} is a commutative C–algebra.

Theorem A.4.1 The algebra C{z1, . . . , zn} is local with maximal ideal m = hz1, . . . , zni.

n Proof. Since each germ [f] ∈ OC ,0 has a well–deﬁned expansion into a power series n which converges in a neighborhood of 0 ∈ C , we have a well deﬁned mapping

n OC ,0 → C{z1, . . . , zn}.

It is easy to check that it is a isomorphism of C–algebras and the assertion follows from our considerations in section A.3.

Theorem A.4.2 The ring C{z1, . . . , zn} is a unique factorization domain.

71 Proof. See [H¨o].

Theorem A.4.3 (Hilbert, Ruckert)¨ The ring C{z1, . . . , zn} is Noetherian.

Proof. See [H¨o].

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