mathfe.math.kobe-u.ac.jp/fe/fe_pdf_with_bookmark/fe35-40-en...funkcialajekvacioj, 38 (1995) 121-157...

Funkcialaj Ekvacioj, 38 (1995) 121-157

Foliations on Complex Spaces

By

Akihiro SAEKI(University of Tokyo, Japan)

Contents

§0 Introduction.§1 Coherent foliations on complex spaces.

1.0. Coherent foliations.1.1. Morphisms and pull-back of foliations.1.2. Proper modifications and foliations.1.3. Closed embeddings and foliations.1.4. Quotient singularities.1.5. Foliations of dimension one or codimension one.1.6. Examples.

§2 Coherent modules, their duals and their submodules.2.0. Preliminaries.2.1. The $¥mathcal{O}_{¥mathrm{X}}$-submodule $J_{F}^{¥Gamma}$ .

2.2. The $¥mathcal{O}_{¥mathrm{X}}$-submodule $J_{SF}^{a^{-}}$ .

2.3. The $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi a1}$ .

2.4. The $¥mathcal{O}_{¥mathrm{X}}$-submodule $J_{¥ovalbox{¥tt¥small REJECT}}^{¥varpi}$ .

2.5. Relations of $J_{F}^{¥Gamma}$ , $J_{SF}^{¥Gamma}$ and $J^{¥varpi a1}$ .

2.6. Relations with $J_{¥ovalbox{¥tt¥small REJECT}}^{a^{-}}$ .

Appendices.A.O. Algebraic observations on coherent sheaves on reduced complex

spaces.A.I. Algebraic observations on coherent sheaves on locally irreducible

complex spaces.A.2. Coherent sheaves of rank one.

References.

§0 Introduction

In this paper, we discuss foliations on reduced complex spaces, especiallyon locally irreducible complex spaces. On complex manifolds, foliations aredefined in two ways: as coherent subsheaves of the sheaf $¥Theta$ of germs ofholimorphic vector fields and of the sheaf $¥Omega$ of germs of holomorphic 1-forms,

122 Akihiro SAEKI

satisfying the “integrability conditions”. Foliatios defined by vector fields andby 1-forms correspond with each other (cf. [B-B], [Sul, 2]). In §0.0, wedefine foliations on complex spaces in two ways, using vector fields and 1-forms,as a natural extension of the cases on manifolds (Definition 1.0.0.). Thesetwo definitios are essentially equivalent with each other (Theorem 1.0.5.). Weinvestigate effects of morphisms of complex spaces on foliations on them. Let$X¥rightarrow Y$ be a proper modification of reduced complex spaces. Then foliationson $X$ and on $Y$ are correspondent with each other (Theorem 1.2.4.). Thusfoliations are bimeromorphically invariant. In §1.5, we see that our definitionis an extenstion of the definition by Gomez-Mont [GM1] of foliations bycurves $¥mathrm{i}.¥mathrm{e}$ . the foliation whose leaves are of complex dimension one. Thetechnical bases of these results are given in §2 and appendices.

§1 Coherent foliations on complex spaces

1.0. Coherent foliationsWe define coherent foliations on complex spaces, which is an extension

of the case on manifolds.Let $(X, ¥mathcal{O}_{X})$ be a reduced complex space. We use the following notations

as usual:$¥ovalbox{¥tt¥small REJECT}$? : the sheaf of germs of meromorphic functions on $X$

$¥Omega_{X}$ : the sheaf of germs of holomorphic 1-forms on $X$

$¥Theta_{X}$ : the sheaf of germs of holomorphic vector fields on $X$

By definition, $¥Theta_{X}=¥Omega_{X}^{*}:$ the dual of $¥Omega_{X}$ . Note that $¥Omega_{X}=(¥Omega_{D}/¥mathcal{O}_{D}d¥mathcal{J})|_{X}$

if $X$ is a closed complex subspace of a domain $D¥subset C^{m}$ defined by a coherent$¥mathcal{O}_{¥mathrm{D}}$-ideal $l$ .

For a coherent $¥mathcal{O}_{¥mathrm{X}}$-module $¥Psi$ , we set

Sing $¥Psi:=$ { $x¥in X|¥Psi_{x}$ is not $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -free}.

For a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{a^{-}}$ of $¥Psi$ , we use the notation:

$S(J^{¥varpi}):=$ Sing $¥ovalbox{¥tt¥small REJECT}¥cup$ Sing $(¥Psi/J^{¥varpi})$ .

Note that

Sing $X=$ Sing $¥Omega_{X}$

holds, where Sing $X$ is the singular locus of the complex space $X$ . We denoteby $¥mathrm{s}¥mathrm{p}$

$X$ the underlying topological space of the complex space $(X, ¥mathcal{O}_{X})$ .

Definition 1.0.0. We define coherent foliations in two ways.. Definition $¥mathrm{a}$). (by 1-forms).0) A coherent foliation on $X$ is a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $F$ of $¥Omega_{X}$ satisfying

Foliations on Complex Spaces 123

(1.0. 1) $dF_{x}¥subset F_{X}¥wedge¥Omega_{X,x}$

at any $x¥in X-S(F)$ . This condition is called the integrability condition.We call $S(F)$ the singular locus of the foliation $F$ .

1) A coherent foliation $F$ $¥subset¥Omega_{X}$ is said to be reduced if, for any opensubspace $U¥subset X$ , that $¥xi¥in¥Gamma(U, ¥Omega_{X})$ and that $¥xi|_{US(F)}¥_¥in¥Gamma(U-S(F), F)$ implythat $¥xi¥in¥Gamma(U, F)$. Definition $¥mathrm{b}$ ). (by vector fields).

0) A coherent foliation on $X$ is a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $E$ of $¥Theta_{X}$ satisfying

(1.0.2) $[E_{x}, E_{x}]¥subset E_{X}$

at any $x¥in X-$ ($ S(E)¥cup$ Sing $X$). This condition is called the integrabilitycondition. We call $ S(E)¥cup$ Sing $X$ the singular locus of the foliation $E$ .

1) A coherent foliation $E¥subset¥Theta_{X}$ is said to be reduced if, for any opensubspace $U¥subset X$, that $v¥in¥Gamma(U, ¥Theta_{X})$ and that $v|_{U-(S(E)¥cup ¥mathrm{S}¥mathrm{i}¥mathrm{n}¥mathrm{g}X)}¥in¥Gamma(U-(S(E)$

$¥cup$ Sing $X$ ), $E$ ) imply that $v¥in¥Gamma(U, E)$ .

Remark.0) If the complex space $(X, ¥mathcal{O}_{X})$ is a complex manifold, then Sing $X=$

Sing $¥Omega_{X}$ is empty and the above definitions are those of holomorphicfoliations with singularities in [Sul, 2] and [B-B].

1) The reducedness of foliations $F¥subset¥Omega_{X}$ and $E¥subset¥Theta_{X}$ is just the fullness(§2 below) of the coherent submodules $F$ and $E$ in $¥Omega_{X}$ and $¥Theta_{X}$ ,respectively.

2) The above integrability condition (1.0.1) for $F$ is equivalent with thefollowing:

(1.0.3) $¥xi¥in F_{x}$ , $u¥in F_{x}^{a}$ , $v¥in F_{X}^{a}=d¥xi(u, v)=0¥in ¥mathcal{O}_{X,¥mathrm{x}}$

holds at any $¥mathrm{x}¥in X$ .

3) If $E$ $¥subset¥Theta_{X}$ is a reduced coherent foliation, then the condition (1.0.2)holds at any $x¥in X$ . This is a direct consequence of that $E$ is full in $¥Theta_{X}$ .

There are natural correspondences between foliations defined by the abovetwo definitions. Namely,

DEFINITION 1.0.4.0) For a coherent foliation $F¥subset¥Omega_{X}$ , we define a coherent foliation $F^{a}¥subset¥Theta_{X}$

by$F_{x}^{a}:=$ { $v¥in¥Theta_{X,x}|¥langle v$, $¥xi¥rangle=0$ for all $¥xi¥in F_{x}$ }.

We will see that $F^{a}$ is a reduced coherent foliation in §2.1) For a coherent foliation $E$ $¥subset¥Theta_{X}$ , we define a coherent foliation $E^{¥perp}¥subset¥Omega_{X}$

by

124 Akihiro SAEKI

$E_{x}^{¥perp}:=$ { $¥xi¥in¥Omega_{X,x}|¥langle v$ , $¥xi¥rangle=0$ for all $v¥in E_{x}$ }.

We will see that $E^{¥perp}$ is a reduced coherent foliation in §2.

Remark. The coherence of $F^{a}$ and $E^{¥perp}$ is proved in §2. The integrabilityconditions (1.0.2) and (1.0.3) are directly verified. Moreover, we have

Theorem 1.0.5.0) For a coherent foliation $F¥subset¥Omega_{X}$ , the $¥mathcal{O}_{X^{-}}$submodules $F_{F}(=F_{SF}=F^{a¥perp})$ of

$¥Omega_{X}$ and $F^{a}$ of $¥Theta_{X}$ are reduced coherent foliations.1) For a coherent foliation $E¥subset¥Theta_{X}$ , the $¥mathcal{O}_{X^{-}}$submodules $E_{F}(=E_{SF}=E^{¥perp a})$ of

$¥Theta_{X}$ and $E^{¥perp}of$ $¥Omega_{X}$ are reduced coherent foliations.2) These correspondences

$F$ $¥subset¥Omega_{X}¥rightarrow F^{a}¥subset¥Theta_{X}$ and $E¥subset¥Theta_{X}¥rightarrow E^{¥perp}¥subset¥Omega_{X}$

restricted to reduced coherent foliations are inverse of each other.Proof. Except for the integrability conditions, these are direct con-

sequences of Theorem 2.5.6. in §2 below. If a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $F$ of$¥Omega_{X}$ satisfies the integrability condition (1.0.1), it satisfies (1.0.3). Since $S(F)$ isthin in $X$ , $ F|_{XS(F)}¥_=F^{a¥perp}|_{XS(F)}¥_$ and $F^{a}=(F^{a¥perp})^{a}$ , the $¥mathcal{O}_{¥mathrm{X}}$-module $F_{F}=F^{a¥perp}$ alsosatisfies (1.0.3). On the complex manifold $X-S(F)$ , $F^{a}|_{X-S(F)}=(F|_{X-S(F)})^{a}$

satisfies the integrability condition (1.0.2) on $X-S(F)$ . Since $F^{a}$ is full in$¥Theta_{X}$ by Theorem 2.5.6., $F^{a}$ satisfies (1.0.2) on $X$ . Assume that $E¥subset¥Theta_{X}$ is acoherent foliation on $X$ . Then $E^{¥perp}|_{X(S(E)¥cup ¥mathrm{S}¥mathrm{i}¥mathrm{n}¥mathrm{g}X)}¥_=(E|_{X-(S(E)¥cup ¥mathrm{S}¥mathrm{i}¥mathrm{n}¥mathrm{g}X)})^{¥perp}$ satisfies(1.0.3) on $X-$ ($ S(E)¥cup$ Sing $X$ ). Since $ S(E)¥cup$ Sing $X$ is thin in $X$, $E^{¥perp}$ satisfies(1.0.3) on X. $¥blacksquare$

Remark. On a reduced complex space $X$, the sheaf $¥ovalbox{¥tt¥small REJECT} c$ of active germssatisfies the condition (A.O.8) for $¥Theta_{X}$ . Thus Proposition A.0.9 can be appliedto $¥Theta_{X}$ . This is very important as we see in 1.2 and 1.5.

We examine coherent foliations in several situations.

1.1. Morphisms and pull-back of foliations

Let $f:X¥rightarrow Y$ be a morphism of reduced complex spaces. For a coherent$¥mathcal{O}_{¥mathrm{Y}}$ -submodule $F$ of $¥Omega_{¥mathrm{Y}}$ , a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $f^{¥star}F$ of $¥Omega_{X}$ , is determinedby means of pulling back of 1-forms. Note that $f^{-1}(S(F))$ is an analytic setin $X$ and that

Sing $X=$ Sing $¥Omega_{X}¥subset S(f^{¥star}F)¥subset f^{-1}(S(F))$

holds. $S(f^{¥star}F)$ is a thin analytic set in $X$ . In the case that $f^{-1}(S(F))$ is thinin $X$, it follows from the equivalence of (1.0.1) with (1.0.3) that


Proposition and Definition 1.1.0. Let $f:X¥rightarrow Y$ be a morphism of reducedcomplex spaces and $F$ $¥subset¥Omega_{¥mathrm{Y}}$ a coherent foliation on Y. Assume $f^{-1}(S(F))$ isthin in X. Then the $¥mathcal{O}_{X^{-}}$submodule $f^{¥star}F¥subset¥Omega_{X}$ is a coherent foliation on $X$ . Wecall it the pulled-back of $F$ by $f$.

Remark. Even if a coherent foliation $F$ $¥subset¥Omega_{¥mathrm{Y}}$ is reduced the pulled-back$f^{¥star}F$ is not always reduced. For example, let $F:=¥mathcal{O}_{¥mathrm{Y}}¥xi¥subset¥Omega_{¥mathrm{Y}}$ , where $¥xi=ydx$

$-xdy¥in¥Omega_{Y}(Y)$ on $Y=C^{2}$ with a coordinate system $(x, y)$ . Consider theblowing up $¥sigma:X¥rightarrow Y$ at 0 and the pulled-back $¥sigma^{¥star}$ F. $X=X_{¥alpha}¥cup X_{¥beta}$ , where$X_{¥alpha}=C^{2}$ and $X_{¥beta}=C^{2}$ with coordinate systems $(s_{¥alpha}, t_{a})$ and $(s_{¥beta}, t_{¥beta})$ , respectively,patched together at $X_{¥alpha}¥cap X_{¥beta}=¥{(s_{a}, t_{¥alpha})|t_{a}¥neq 0¥}¥subset X_{¥alpha}$ and $X_{¥alpha}¥cap X_{¥beta}=¥{(s_{¥beta}, t_{¥beta})|t_{¥beta}$

$¥neq 0¥}¥subset X_{¥beta}$ by $s_{a}=s_{¥beta}t_{¥beta}$ and $t_{a}=t_{¥beta}^{-1}¥cdot¥sigma:X¥rightarrow Y$ is defined by

$¥sigma|_{X_{¥alpha}}$ $¥sigma|_{X_{¥beta}}$

$X_{a}=C^{2}¥rightarrow Y=C^{2}$ and $X_{¥beta}=C^{2}¥rightarrow Y=C^{2}$

$(s_{¥alpha}, t_{a})¥mapsto$ $(x, y)$ $(s_{¥beta}, t_{¥beta})¥mapsto$ $(x, y)$

$x=s_{a}$ $x$ $=s_{¥beta}t_{¥beta}$

$y=s_{a}t_{a}$ $y=s_{¥beta}$ .

$¥xi¥in¥Omega_{¥mathrm{Y}}(Y)$ defines $¥sigma^{*}¥xi¥in¥Omega_{X}(X)$ defines $¥sigma^{*}¥xi¥in¥Omega_{X}(X)$ and we have $¥sigma^{¥star}F=¥mathcal{O}_{X}¥cdot¥sigma^{*}¥xi$

$¥subset¥Omega_{X}$ . Since

$¥sigma^{*}¥xi|_{X_{¥alpha}}=-s_{¥alpha}^{2}dt_{a}$ and $¥sigma^{*}¥xi|_{X_{¥beta}}=s_{¥beta}^{2}dt_{¥beta}$ ,

$¥sigma^{¥star}F$ is not reduced. Examples on a complex space with non-trivial singularlocus are given in §1.6.

1.2. Proper modifications and foliationsNow we investigate the case that $X$ and $Y$ are reduced and that $f:X¥rightarrow ¥mathrm{Y}$

is a proper modification. First, we recall the definitions.

Definition 1.2.0.0) A morphism $(f,¥tilde{f}):(X, ¥mathcal{O}_{X})¥rightarrow(Y, ¥mathcal{O}_{¥mathrm{Y}})$ of complex spaces is said to be

proper if the underlying continuous map $f:¥mathrm{s}¥mathrm{p}$ $X¥rightarrow ¥mathrm{s}¥mathrm{p}Y$ is a proper map.1) A morphism $(f,¥tilde{f}):(X, ¥mathcal{O}_{X})¥rightarrow(Y, ¥mathcal{O}_{¥mathrm{Y}})$ of complex spaces is said to be a

modification of $Y$ if there is a thin analytic set $A$ in $Y$ satisfyinga) $f^{-1}(A)$ is a thin analytic set in $X$ andb) $f|_{X-f^{-1}(A)}$ : $X-f^{-1}(A)¥rightarrow Y-A$ is an isomorphism of complex spaces.

2) A morphism $(f,¥tilde{f}):(X, ¥mathcal{O}_{X})¥rightarrow(Y, ¥mathcal{O}_{¥mathrm{Y}})$ of complex spaces is said to bea proper modification of $Y$ if it is a proper morphism and if it is amodification of $Y$.

We use the following theorems. (See $¥mathrm{e}¥mathrm{g}$ . $[¥mathrm{G}$ -R2].)

126 Akihiro SAEKI

Theorem 1.2.1. (Direct image theorem) ( $[¥mathrm{G}$-R2] p. 207) Let $f:X¥rightarrow Y$ be $a$

proper morphism. For any coherent $¥mathcal{O}_{X^{-}}$module $l$, all direct image sheaves$ f_{(j)}¥Psi$ , $j=0,1$ , 2, $¥cdots$ , are coherent $¥mathcal{O}_{¥mathrm{Y}^{-}}$modules.

Theorem 1.2.2. (Lifting lemma) ( $[¥mathrm{G}$-R2] pp. 154-155) Assume that$f:X¥rightarrow Y$ is a proper modification of a reduced complex space Y. Then $f$

induces an isomorphism $¥tilde{f}:¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}¥rightarrow f_{*}¥ovalbox{¥tt¥small REJECT}_{X}$ which makes the following diagramcommute:

$¥mathcal{O}_{¥mathrm{Y}}¥rightarrow¥tilde{f}f_{*}¥mathcal{O}_{X}$

$¥cap$ $¥cap$

$¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}^{¥simeq}¥vec{¥tilde{f}}f_{*}¥ovalbox{¥tt¥small REJECT}_{X}$

Let $f:X¥rightarrow Y$ be a proper modification of a reduced complex space $Y$.

The above theorems tell us that the direct image $f_{*}¥mathcal{O}_{X}$ is an $¥mathcal{O}_{¥mathrm{Y}}$ -coherent $¥mathcal{O}_{¥mathrm{Y}^{-}}$

submodule of $f_{*}¥ovalbox{¥tt¥small REJECT}_{X}¥simeq¥ovalbox{¥tt¥small REJECT}_{Y}$ . On $U¥subset Y$, a section $v¥in¥Gamma(U, f_{*}¥Theta_{X})=¥Gamma(f^{-1}(U)$ ,$¥Theta_{X})$ defines a derivation $¥mathcal{O}_{Y}|_{U}¥rightarrow f_{*}¥mathcal{O}_{X}|_{U}$ , $¥mathrm{i}.¥mathrm{e}$ . the composite with $¥tilde{f}:¥mathcal{O}_{¥mathrm{Y}}|_{U}¥rightarrow f_{*}¥mathcal{O}_{X}|_{U}$ .

This, followed by $f_{*}¥mathcal{O}_{X}¥subset f_{*}¥ovalbox{¥tt¥small REJECT}_{X}¥simeq¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}$ , determines a unique section $f_{*}v$ on $U$

of $¥ovalbox{¥tt¥small REJECT}_{om_{¥mathcal{O}_{¥mathrm{Y}}}}$ $(¥Omega_{¥mathrm{Y}}, ¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}})¥simeq¥Theta_{¥mathrm{Y}}¥mathrm{O}¥times(¥emptyset_{¥mathrm{Y}}¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}$ . Since $¥ovalbox{¥tt¥small REJECT}_{am_{¥mathcal{O}_{Y}}}$ $(¥Omega_{¥mathrm{Y}}, ¥cdot)$ is left exact, this$¥mathcal{O}_{¥mathrm{Y}}$ -morphism $¥ovalbox{¥tt¥small REJECT}_{om_{¥mathcal{O}_{¥mathrm{Y}}}}$ $(¥Omega_{¥mathrm{Y}}, f_{*}¥mathcal{O}_{X})¥rightarrow¥ovalbox{¥tt¥small REJECT}_{¥rho m_{¥mathcal{O}_{Y}}}$ $(¥Omega_{¥mathrm{Y}}, ¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}})¥simeq¥Theta_{¥mathrm{Y}}¥otimes_{(¥emptyset_{Y}}¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}$ is injective.Thus an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $E$ of $¥Theta_{X}$ determines a unique $¥mathcal{O}_{¥mathrm{Y}}$ -coherent$¥mathcal{O}_{¥mathrm{Y}}$ -submodule of $¥Theta_{¥mathrm{Y}}¥mathrm{O}¥times(|)Y¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}$ . The inverse image of it under $¥Theta_{¥mathrm{Y}}¥rightarrow¥Theta_{¥mathrm{Y}}¥mathrm{O}¥times(|)_{Y}¥ovalbox{¥tt¥small REJECT}_{Y}$

is an $¥mathcal{O}_{¥mathrm{Y}}$ -coherent $¥mathcal{O}_{¥mathrm{Y}}$ -submodule of $¥Theta_{¥mathrm{Y}}$ , which we denote by $f_{¥star}E$ . If $E¥subset¥Theta_{X}$

is a coherent foliation on $X$, then $f_{¥star}E¥subset¥Theta_{¥mathrm{Y}}$ is a coherent foliation on $Y$.

Definition 1.2.3. For a coherent foliation $E¥subset¥Theta_{X}$ on $X$, we call thecoherent foliation

$f_{¥star}E¥subset¥Theta_{¥mathrm{Y}}$

on $Y$ the pushed-forward of $E$ by $f$.

Since $f$ is biholomorphic outside of thin analytic sets, we have

Theorem 1.2.4. Let $f:X¥rightarrow Y$ be a proper modification of a reducedcomplex space Y. Then the correspondences

$F¥subset¥Omega_{¥mathrm{Y}}¥rightarrow(f^{¥star}F)^{a}¥subset¥Theta_{X}$ and $E¥subset¥Theta_{X}¥rightarrow(f_{¥star}E)^{¥perp}¥subset¥Omega_{¥mathrm{Y}}$

are, restricted to reduced coherent foliations, the inverse of each other.

Thus, under proper modifications, reduced coherent foliations are“essentially invariant“. We can investigate reduced coherent foliations on areduced complex space $X$ by investigating the pulled-back of them by a


desingularization $¥tilde{X}¥rightarrow X$ of $X$ .

1.3. Closed embeddings and foliationsNext we consider the case that $X$ is a reduced complex space embedded

in a domain $D¥subset C^{m}$ as a closed subspace defined by a coherent $¥mathcal{O}_{¥mathrm{D}}$-ideal$¥mathcal{J}$ . Of course any coherent foliation $F$ $¥subset¥Omega_{D}$ on $D$ with $S(F)¥cap X$ thin in $X$

determines the pulledback foliation $¥iota^{¥star}F¥subset¥Omega_{X}$ , where $¥iota:X¥sigma$ $D$ is the embeddingmorphism.

Roughly speaking, vector fields on $D$ also define foliations on $X$ . First,we review logarithmic vector fields after Saito [Sai].

Definition 1.3.0. (Saito [Sai] pp. 267-268, (1.4) Definition.) A vector field$¥mathrm{v}¥in¥Theta_{D,x}$ at $x¥in D$ is logarithmic along $X$ if it satisfies

$¥mathrm{v}(¥mathcal{J}_{x})¥subset ¥mathcal{J}_{x}$ .

An $¥mathcal{O}_{¥mathrm{D}}$-coherent $¥mathcal{O}_{¥mathrm{Y}}$ -submodule $¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}}$ $(¥log X)$ of $¥Theta_{D}$ is defined by

$(¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}}(¥log X))_{x}:=$ Der$D,x$$(¥log X)$

$:=$ { $¥mathrm{v}¥in¥Theta_{D,x}|¥mathrm{v}$ is logarithmic along $X$ }.

Remark. It should be noted that $¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}}$ $(¥log X)$ is closed under the bracket$[¥cdot, ¥cdot]$ .

The sheaf $¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}}$ $(¥log X)$ of germs of logarithmic vector fields along $X$ isclosely related with the sheaf $¥Theta_{X}=¥Omega_{X}^{*}$ of germs of holomorphic vector fieldson $X$ as we see below. A coherent $¥mathcal{O}_{¥mathrm{D}}$-submodule $¥ovalbox{¥tt¥small REJECT}_{el_{C}}$ $(¥mathcal{O}_{D}, ¥mathcal{J})$ is defined by

$(¥ovalbox{¥tt¥small REJECT} e¥iota_{C}(¥mathcal{O}_{D}, ¥mathcal{J}))_{x}:=¥{¥mathrm{v}¥in¥Theta_{D,x}|¥mathrm{v}(¥mathcal{O}_{D,x})¥subset ¥mathcal{J}_{x}¥}$ .

We have

$¥ovalbox{¥tt¥small REJECT}_{ei_{C}(¥mathcal{O}_{D},¥mathcal{J})¥simeq¥ovalbox{¥tt¥small REJECT} am_{¥mathcal{O}_{D}}(¥Omega_{D},¥mathcal{J})}$

and, taking a coordinate system $(z_{1},¥ldots,z_{m})$ ,

$m$

$¥ovalbox{¥tt¥small REJECT}_{l}a_{C}(¥mathcal{O}_{D}, ¥mathcal{J})=¥bigoplus_{j=1}¥mathcal{J}(¥partial/¥partial z_{j})$.

It holds that

Der $D(¥log X)¥supset¥ovalbox{¥tt¥small REJECT} ei_{C}$ $(¥mathcal{O}_{D}, ¥mathcal{J})$

and that

(1.3.1) $¥Theta_{X}¥simeq$ $(¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}} (¥log X)/¥ovalbox{¥tt¥small REJECT} e¥iota_{C} (¥mathcal{O}_{D}, J))$ $|_{X}$ .

At $x¥in X$, for each $v¥in¥Theta_{X,x}$ , there exists a logarithmic vector field $¥mathrm{v}¥in¥Theta_{D,x}$

making the folowing diagram commute:

128 Akihiro SAEKI

(1.3.2)$¥mathcal{O}_{D,x}¥downarrow¥downarrow¥rightarrow^{¥mathrm{v}}¥mathcal{O}_{D,x}$

$¥mathcal{O}_{X,X¥vec{v}}¥mathcal{O}_{X,x}$

Conversely, any $¥mathrm{v}¥in¥Theta_{D,x}$ satisfying $¥mathrm{v}(¥mathcal{J}_{x})¥subset ¥mathcal{J}_{X}$ determines $v¥in¥Theta_{X,x}$ satisfying(1.3.2).

Now we see how foliations on $D$ defined by vector fields determinefoliations on $X$ . Any $¥mathcal{O}_{¥mathrm{D}}$-coherent $¥mathcal{O}_{¥mathrm{D}}$-submodule $E^{D}$ of Der$D(¥log X)$ definesan $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$ -submodule $E$ of $¥Theta_{X}$ :

$¥Theta_{X}=$ $((¥mathrm{D}¥mathrm{e}¥mathrm{r}_{D}(¥log X))/(¥ovalbox{¥tt¥small REJECT} e¥iota_{C}(¥mathcal{O}_{D}, ¥mathcal{J})))|_{X}$

$¥cup$$¥cup$

$E:=((E^{D}+¥ovalbox{¥tt¥small REJECT} ae¥iota_{¥mathrm{C}}(¥mathcal{O}_{D}, J))/(¥ovalbox{¥tt¥small REJECT} ei_{¥mathrm{C}}(¥mathcal{O}_{D}, ¥mathcal{J})))|_{X}$.

$¥simeq$$|$ ?

$(E^{D}/(E^{D}¥mathrm{n}¥ovalbox{¥tt¥small REJECT}_{ei_{C}} (¥mathcal{O}_{D}, ¥mathcal{J})))|_{X}$

If $E$ satisfies the integrability condition (1.0.2), we have a coherent foliationon $X$ . Note that $E$ does satisfies (1.0.2) on $X$ if $E^{D}$ satisfies on $D$ .

Proposition 1.3.3. Let $E^{D}$ be an $¥mathcal{O}_{D^{-}}$coherent $¥mathcal{O}_{D^{-}}$submodule of Der$D(¥log X)$

$¥subset¥Theta_{D}$ . Assume that

(1.3.4) $[E_{x}^{D}, E_{x}^{D}]¥subset(E^{D}+ Der_{C} (¥mathcal{O}_{D}, J))_{X}$

at $¥mathrm{x}¥in D-A$ , where $A$ is an analytic set in $D$ satisfying $¥iota^{-1}(A)$ is thin inX. Then $E=$ $((E^{D}+¥ovalbox{¥tt¥small REJECT}_{el_{C}} (¥mathcal{O}_{D}, ¥mathcal{J}))/¥ovalbox{¥tt¥small REJECT}_{el_{¥mathrm{C}}} (¥mathcal{O}_{D}, ¥mathcal{J}))|_{X}$ is, canonically identifiedwith the $¥mathcal{O}_{X^{-}}$submodule of $¥Theta_{X}$ , a coherent foliation on X. In particular, if$ff$ $¥subset¥Theta_{D}$ is a coherent foliation on $D$ , then $ E^{D}:=¥mathcal{B}¥cap$ Der$D(¥log X)$ satisfies theabove condition {with $ A=¥emptyset$) and we have a coherent foliation

(1.3.5) $¥iota^{¥star}¥ovalbox{¥tt¥small REJECT}:=((E^{D}+¥ovalbox{¥tt¥small REJECT}_{el_{C}}(¥mathcal{O}_{D}, l))/¥ovalbox{¥tt¥small REJECT}_{¥ell l_{C}}(¥mathcal{O}_{D}, ¥mathcal{J}))|_{X}¥subset¥Theta_{X}$

on $X$ .

More generally, an $¥mathcal{O}_{¥mathrm{D}}$-coherent $¥mathcal{O}_{D}$ -submodule $E^{D}$ of $¥Theta_{D}¥mathrm{O}¥times(|)D¥ovalbox{¥tt¥small REJECT}_{D}$ definesan $¥mathcal{O}X$ -submodule $E_{0}$ of $¥Theta_{X}¥mathrm{O}¥times_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ if $l^{-1}(S(E^{D}))$ is thin in $X$ and if, for any$x¥in D$ , there are an open neighbourhood $U$ of $X$ and $s¥in¥Gamma(U, ¥ovalbox{¥tt¥small REJECT}_{¥mathcal{L}_{D}})$ such that$sE^{D}|_{U}$ is a coherent $¥mathcal{O}_{¥mathrm{U}}$-submodule of Der$D(¥log X)|_{U}$ . By Proposition $¥mathrm{A}.0.9$,$E_{0}$ is coherent. Let $E:=¥mu_{¥Theta_{X}}^{-1}(E_{0})$ be the inverse image of $E_{0}$ under$¥mu_{¥Theta_{X}}$ : $¥Theta_{X}¥rightarrow¥Theta_{X}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ . $E$ is a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥theta_{X}$ . Thus, if $E$

satisfies the integrability condition, we have a coherent foliation $E$ derivedfrom $E^{D}$ .


From an $¥mathcal{O}_{¥mathrm{D}}$-coherent foliation $¥ovalbox{¥tt¥small REJECT}$

)

$¥in¥Theta_{D}$ on $D$ , two $¥mathcal{O}_{¥mathrm{X}}$-coherent foliations$l^{¥star}E¥subset¥Theta_{X}$ and $l^{¥star}(¥ovalbox{¥tt¥small REJECT}^{¥perp})¥subset¥Omega_{X}$ are induced, for which the following holds.

Proposition 1.3.6. Assume $l^{-1}(A)$ is thin in $X$ for$A:=$ Sing $(¥Omega_{D}/(¥mathcal{B}^{¥perp}+¥mathcal{O}_{D}d¥mathcal{J}))¥cup$ Sing $((d^{¥perp}+¥mathcal{O}_{D}dJ)/¥ovalbox{¥tt¥small REJECT}^{¥perp})$ .

Then

$((¥iota^{¥star}¥ovalbox{¥tt¥small REJECT})^{¥perp})^{a}=(¥iota^{¥star}(¥ovalbox{¥tt¥small REJECT}^{¥perp}))^{a}$ .

To prove this proposition, we need some definitions. For a coherent$¥mathcal{O}_{¥mathrm{X}}$-module $¥ovalbox{¥tt¥small REJECT}$ and a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{a^{-}}$ of $¥ovalbox{¥tt¥small REJECT}$ , two submodules $J_{F}^{¥varpi}$

and $J_{SF}^{F}$ of $¥Psi$ are defined as follows: For $U¥subset X$,

$J_{F}^{¥varpi}(U):=¥{¥xi¥in¥Psi(U)|¥xi|_{U-S(¥ovalbox{¥tt¥small REJECT})}¥in¥swarrow^{¥varpi}(U-S(¥Psi))¥}$ .

These $J_{F}^{¥varpi}(U)$ ’s with the canonical restriction define a complete presheaf, $¥mathrm{i}.¥mathrm{e}$ . asheaf. For $U¥subset X$,

$T_{F}(U):=$ { $¥xi¥in¥ovalbox{¥tt¥small REJECT}(U)|¥xi|_{UA}¥_¥in/^{¥varpi}(U-A)$ for some thin analytic set $A$ in $U.$ }.

These $T_{SF}(U)$ ’s with the canonical restriction define a presheaf. The associatedsheaf is defined to be $J_{SF}^{¥varpi}$ . If $¥ovalbox{¥tt¥small REJECT}=¥ovalbox{¥tt¥small REJECT}^{*}$ for a certain coherent $¥mathcal{O}_{¥mathrm{X}}$ -module $¥ovalbox{¥tt¥small REJECT}$ ,then an $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi¥perp a}=(J^{¥tau¥perp})^{a}$ is also defined as in §1.0. In §2 below,we will see that

$J_{F}^{¥varpi}=J_{SF}^{¥varpi}=J^{¥varpi¥perp a}$

Proof of Proposition 1.3.6. It should be noted that

$¥iota^{¥star}(¥ovalbox{¥tt¥small REJECT}^{¥perp})=((¥ovalbox{¥tt¥small REJECT}^{¥perp}+¥mathcal{O}_{D}d¥mathcal{J})/¥mathcal{O}_{D}d¥mathcal{J})|_{X}¥simeq(¥ovalbox{¥tt¥small REJECT}^{¥perp}/(¥ovalbox{¥tt¥small REJECT}^{¥perp}¥cap ¥mathcal{O}_{D}d¥mathcal{J}))|_{X}$.

Since $((¥iota^{¥star}ff)^{¥perp})^{a}=(¥iota^{¥star}ff)_{SF}$ , it is sufficient to prove that, at $x¥in D-A$ , $¥mathrm{v}¥in¥Theta_{D,x}$

satisfies

(1.3.7) $¥mathrm{v}((ff^{¥perp}+¥mathcal{O}_{D}df)_{X})¥subset ¥mathcal{J}_{x}$

if and only if $¥mathrm{v}¥in$ $(E^{D}+¥ovalbox{¥tt¥small REJECT}_{l}¥iota_{¥mathrm{C}} (¥mathcal{O}_{D}, ¥mathcal{J}))_{x}$ , where $ E^{D}:=ff_{F}¥cap$ Der$D(¥log X)$ . The ‘if’part directly follows from the definition of $¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}}$ $(¥log X)$ . Assume $¥mathrm{v}¥in¥Theta_{D,x}$

satisfies (1.3.7). Note that $¥mathrm{v}¥in$ Der$D,x$$(¥log X)$ . At $x¥in D-A$, we can take direct

sum decompositions

$(ff^{¥perp}+¥mathcal{O}_{D}d¥mathcal{J})_{x}=$ $(¥ovalbox{¥tt¥small REJECT}^{)¥perp})_{X}¥oplus S$ and $¥Omega_{D,x}=(ff^{¥perp}+¥mathcal{O}_{D}d¥mathcal{J})_{x}¥oplus T$.

$¥mathrm{T}¥mathrm{u}¥in(¥ovalbox{¥tt¥small REJECT}_{¥ell}a_{¥mathrm{C}}(¥mathcal{O}_{D},¥mathcal{J}))_{x}¥mathrm{a}¥mathrm{n}¥mathrm{d}¥mathrm{v}-¥mathrm{u}¥in(¥ovalbox{¥tt¥small REJECT}_{F}¥cap ¥mathrm{D}¥mathrm{e}¥mathrm{r}_{D}(1¥mathrm{o}¥mathrm{g}X))_{x}¥mathrm{h}¥mathrm{e}¥mathrm{r}¥mathrm{e}¥mathrm{i}¥mathrm{s}¥mathrm{a}¥mathrm{v}¥mathrm{e}¥mathrm{c}¥mathrm{t}¥mathrm{o}¥mathrm{r}¥mathrm{f}¥mathrm{i}¥mathrm{e}¥mathrm{l}¥mathrm{d}¥mathrm{u}¥in¥Theta_{D,x}¥mathrm{s}¥mathrm{a}¥mathrm{t}¥mathrm{i}¥mathrm{s}¥mathrm{f}¥mathrm{y}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥mathrm{u}|_{d_{¥mathrm{X}}^{¥perp}}=¥mathrm{v}|_{¥theta_{¥mathrm{x}}^{¥perp}}.¥mathrm{a}.¥mathrm{n}¥mathrm{d}¥mathrm{u}|_{S¥oplus T}=¥mathrm{v}|_{S¥oplus T}$

. Thus

130 Akihiro SAEKI

1.4. Quotient singularitiesLet $W$ be a complex space, $G$ a finite subgroup of the group of

isomorphisms of $W$ and $X:=G¥backslash W$, the quotient complex space. Note that$X$ is normal if $W$ is. We denote by $¥pi:W¥rightarrow X$ the canonical surjection, whichis a finite morphism, moreover, an analytic covering. For any $g¥in G$ , thefollowing diagram commutes:

$W¥rightarrow gW$

(1.4.0) $¥backslash _{¥pi}¥downarrow¥pi$ .

$X$

Let $B_{X}$ be the branching locus of $¥pi$ , $¥mathrm{i}.¥mathrm{e}$ . $ B_{X}:=¥{¥mathrm{x}¥in X|¥pi$ is not locallybiholomorphic in a neighbourhood of $¥pi^{-1}(x)¥}$ and $B_{W}:=¥pi^{-1}(B_{X})$ . These arethin analytic sets in $X$ and $W$, respectively. Each $g¥in G$ induces isomorphismsof $¥mathcal{O}_{¥mathrm{W}}$ -modules $g^{*}:$ $¥Omega_{W}¥rightarrow¥Omega_{W}$ and $g_{*}:$ $¥Theta_{W}¥rightarrow¥Theta_{W}$ in a natural manner. Set$X_{0}:=X-B_{X}$ and $W_{0}:=¥pi^{-1}(X_{0})=W-B_{W}$ . Note that the restriction of $¥pi$

to $W_{0}$ is locally biholomorphic and that any $g¥in G$ induces a coveringtransformation of $¥pi:W_{0}¥rightarrow X_{0}$ . (See (1.4.0).)

Let $E$ be a $¥mathrm{G}$-invariant coherent $¥mathcal{O}_{¥mathrm{W}_{¥mathrm{O}}}$-submodule of $¥Theta_{W_{¥mathrm{O}}}$ . $E$ induces acoherent $¥mathcal{O}_{X_{0}}-$ submodule $E_{X_{0}}$ of $¥Theta_{X¥mathrm{o}}$ .

Let $x¥in B_{X}$ and $¥pi^{-1}(¥mathrm{x})=¥{u_{1},¥cdots, w_{n}¥}$ , where $w_{i}¥neq w_{j}$ if $i¥neq j$ . We can takeopen neighbourhoods $V¥subset X$ and $U_{j}¥subset W$ of $x$ and $w_{j}$ , respectively, such that

$n$

$U:=¥pi^{-1}(V)=¥bigcup_{j=1}U_{j}$ ,

$gU_{j}=U_{j}$ for $g¥in G_{j}:=¥{g¥in G|gw_{j}=w_{j}¥}$ ,(1.4. 1)

$ U_{i}¥cap U_{j}=¥emptyset$ if $i¥neq j$ and

$gU_{i}=U_{j}$ if $gw_{i}=w_{j}$ .

Such $F$ ’s form a fundamental system of neighbourhoods of $x$ . Note thata vector field $¥mathrm{v}¥in¥Theta_{W}(U)$ defines a (uniquely determined) vector field $v¥in¥Theta_{X}(V)$

if and only if $¥mathrm{v}$ is $¥mathrm{G}$-invariant. Let $E$ be a $¥mathrm{G}$-invariant coherent $¥mathcal{O}_{¥mathrm{W}}$ -submoduleof $¥Theta_{W}$ . Assume that, for every $x¥in B_{X}$ , there exist neighbourhoods $V$ of $¥mathrm{x}$ and$U_{j}$ of each $w_{j}¥in¥pi^{-1}(x)$ satisfying (1.4.1) and $¥mathrm{v}_{j,1},¥cdots$ , $¥mathrm{v}_{j,k}¥in¥Theta_{W}(U_{j})$ with thefollowing properties:

$¥mathrm{v}_{j,1},¥cdots,¥mathrm{v}_{j,k}¥in¥Theta_{W}(U_{j})$ generatecoherent $¥mathcal{O}$ -module $E$ on $U_{j}$ and(1.4.2)

$g_{*}¥mathrm{v}_{j,¥mathrm{v}}=¥mathrm{v}_{j,v}$ for $g¥in G;v$ $=1,¥cdots$ , $ k;j=1,¥cdots$ , $n$ .

Then $E$ defines a well defined coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $E_{X}$ of $¥Theta_{X}$ . If $E¥subset¥Theta_{W}$

is a coherent foliation then so is $E_{X}¥subset¥Theta_{X}$ . Thus we have


Proposition and Definition 1.4.3.Let $E¥subset¥Theta_{W}$ be a $G$-invariant coherent foliation on W. Assume there is $a$

coherent foliation $E_{1}¥subset¥Theta_{W}$ on $W$ satisfying the condition (1.4.2) and$ E|_{WA}¥_=E_{1}|_{WA}¥_$ for some thin analytic set $A$ in W. Then $E_{1}$ defines $a$

well-defined coherent foliation $E_{1X}¥subset¥Theta_{X}$ on X. We call the coherent foliation$¥pi¥star E:=(E_{1X})_{F}¥subset¥Theta_{X}$ on $X$ the foliation induced from $E$ by $¥pi$ , which is independentof the choice of $E_{1}$ described as above.

This construction tells us that it is worth while to consider non-reducedfoliations. See examples below in §1.6.

1.5. Foliations of dimension one or codimension oneIn this section, we investigate foliations of dimension one or of

codimension one and show that, in one dimensional case, our definition is anextension of Gomez-Mont’s definition of “foliations by curves” in [GM1, 2].

Let $X$ be a reduced irreducible complex space. For any coherent$¥mathcal{O}_{¥mathrm{X}}$-module $¥ovalbox{¥tt¥small REJECT}$ , Sing $¥Psi$ is a thin analytic set in $X$ and $X-$ Sing $¥ovalbox{¥tt¥small REJECT}$ isconnected. Thus the rank of locally free sheaf $¥ovalbox{¥tt¥small REJECT}|_{X-¥mathrm{S}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥ovalbox{¥tt¥small REJECT}}$ on $X-$ Sing $l$ iswell defined, which we call the rank of the coherent sheaf $¥ovalbox{¥tt¥small REJECT}$ .

Definition 1.5.0.A coherent foliation $E¥subset¥Theta_{X}$ is called of dimension $p$ if the coherent

$¥mathcal{O}_{¥mathrm{X}}$-module $E$ is of rank $p$ .A coherent foliation $F¥subset¥Omega_{X}$ is called of codimension $q$ if the coherent

$¥mathcal{O}_{¥mathrm{X}}$-module $F$ is of rank $q$ .

A coherent foliation $E¥subset¥Theta_{X}$ is said of codimension $q$ if $E^{¥perp}$ is ofcodimention $q$ .

A coherent foliation $F¥subset¥Omega_{X}$ is said of dimension $p$ if $F^{a}$ is of dimension $p$ .

Remark. A coherent foliation $E¥subset¥Theta_{X}$ of dimension $p$ defines anon-singular foliation of dimension $p$ on the (connected) complex manifold$X-$ ($ S(E)¥cup$ Sing $X$ ). A coherent foliation $F$ $¥subset¥Omega_{X}$ of codimension $q$ defines anon-singular foliation of codimension $q$ on the connected) complex manifold$X-S(F)$ .

We use the following terminologies.

Definition 1.5.1.A complex space $X$ is said to be of pure dimension $n$ at $x¥in X$ if every

prime component of $X$ at $¥mathrm{x}$ is of dimension $n$ at $x$ .$X$ is called of pure dimension $n$ if $X$ is of pure dimension $n$ at each $x¥in X$ .$X$ is non-singular in codimension one if the singular locus Sing $X$ of $X$ is

of codimension strictly greater than one in $X$ .

132 Akihiro SAEKI

Remark. In the case that $X$ is locally irreducible, a coherent sheaf $l$ isof rank $r$ if the restriction of $¥ovalbox{¥tt¥small REJECT}$ to each component of $X$ is coherent of rank $r$ .

A coherent foliation $E¥subset¥Theta_{X}$ is of dimension $p$ if the coherent $¥mathcal{O}_{¥mathrm{X}}$-module$E$ is of rank $p$ and a coherent foliation $E¥in¥Omega_{X}$ is of codimension $q$ if thecoherent $¥mathcal{O}_{¥mathrm{X}}$-module $F$ is of rank $q$ , as above. The ranks of $E^{¥perp}$ and of $F^{a}$

are not always well-defined. If $X$ is of pure dimension, the codimension of$E¥subset¥Theta_{X}$ and the dimension of $F¥subset¥Omega_{X}$ are defined as in Definition 1.5.0.

Proposition 1.5.2.0) On a complex manifold $M$, every reduced coherent foliation $E¥subset¥Theta_{M}$ of

dimension one is an invertible subsheaf of $¥Theta_{M}$ .

1) Let $(X, ¥mathcal{O}_{X})$ be a reduced locally irreducible complex space and $A$ ananalytic set in $X$ thin of order two and containing Sing X. For anycoherent foliation $ E_{0}¥subset¥Theta_{XA}¥_$ on $X-A$ of dimension one, there existsa uniquely determined reduced foliation $E¥subset¥Theta_{X}$ on $X$ of dimention onesatisfying $E|_{X(A¥cup S(E_{0}))}¥_=E_{0}|_{X-(A¥cup S(E¥mathrm{o}))}$ .

2) Let $(X, ¥mathcal{O}_{X})$ be a reduced locally irreducible complex space and $Ea$

coherent foliation on X. Then $E¥otimes_{(¥emptyset ¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}$ is a locally free $¥ovalbox{¥tt¥small REJECT}_{X^{-}}$submodule ofrank one of $¥Theta_{X}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ . Thus, on $X$ , reduced coherent foliations of rank oneare in one-to-one correspondence with locally free $¥ovalbox{¥tt¥small REJECT}_{X^{-}}$submodules of rank oneof rank one of $¥Theta_{X}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ .

Proof. These are the direct consequences of Proposition A.2.2 and ofProposition A.2.1, respectively. $¥blacksquare$

In [GM1, 2], Gomez-Mont defined foliations by curves on complex spacesas follows:

Definition 1.5.3. (Gomez-Mont) A foliation $E$ by curves on a complexmanifold $M$ is defined in the following two ways, which are equivalent toeach other.

a) An invertible $¥mathcal{O}_{¥mathrm{M}}$ -submodule of $¥Theta_{M}$ .

b) A (natural equivalence class of a) morphism $¥Phi:L¥rightarrow T$ of holomorphicvector bundle on $M$ not identically zero on each component of $M$,where $L$ is a holomorphic line bundle on $M$ and $T:=TM$ is theholomorphic tangent bundle of $M$ . Two such morphisms $¥Phi_{1}$ : $L_{1}¥rightarrow T$

and $¥Phi_{2}$ : $L_{2}¥rightarrow T$ are equivalent if there are an isomorphism $¥psi:L_{1}¥rightarrow L_{2}$

and a never vanishing holomorphic function $¥lambda¥in ¥mathcal{O}_{M}(M)$ such that thefollowing diagram commutes:

$¥psi L_{1^{¥rightarrow}}^{¥Phi_{1}}¥downarrow¥downarrow¥lambda T$

.

$L_{2_{¥vec{¥Phi_{2}}}}T$


Let $(X, ¥mathcal{O}_{X})$ be a reduced complex space of pure dimension andnon-singular in codimension one. A foliation $F$ by curves on $X$ is

c) A (natural equivalence class of a) pair $(F_{A}, A)$ , where $A$ is an analyticset in $X$ thin of order two and containing Sing $X$ and $F_{A}$ is a non-singularfoliation on the complex manifold $X$ A. Two such pairs $(F_{A}, A)$ and $(F_{B}, B)$

are equivalent if$F_{A}|_{X-(A¥cup B)}=F_{B}|_{X-(A¥cup B)}$ .

Remark.0) This definition of foliations on manifolds may seem rather strict. In

the case of dimension one, Baum-Bott [B-B] and Suwa [Sul, 2] definedthem to be coherent subsheaves of rank one of $¥Theta_{M}$ , not neccesarilyinvertible. Proposition 1.5.2 tells us that these definitions are essentiallyequivalent with one another.

1) Gomez-Mont showed in 1) of Theorem 5 of [GM1] (pp. 131-132), thatfoliations by curves in his sense on a reduced locally irreducible complexspace $X$, which is nonsingular in codimension one, correspond one-to-one with locally free $¥ovalbox{¥tt¥small REJECT}_{¥mathrm{X}}$-submodules of rank one of $¥Theta_{X}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$. Thushis definition is equivalent with ours. He proved this result consideringan embedding into a domain in some complex number space. Ourmethod is applicable to the case of foliations of codimensionone as wesee below.

2) 2) of Theorem 5 of [GM1] is written in our language as follows:

Proposition 1.5.4. {Gomez-Mont) Let $E¥subset¥Theta_{X}$ be a coherent foliation ofdimension one on a normal complex space X. Assume the germ of the zerolocus of $v¥in E_{x}$ at $x¥in X$ is of codimension strictly greater than one. Then $E$ isreduced at $x$ , invertible at $x$ and written $E_{¥mathrm{X}}=¥mathcal{O}_{X,x}v$.

Definition 1.5.5. (Suwa [Sul] p. 184 (1.10) Definition) A foliation $F¥subset¥Omega_{M}$

on a complex manifold $M$ is of complete intersection type if $F$ is a locallyfree $¥mathcal{O}_{¥mathrm{M}}$ -module.

As Proposition 1.5.2, we have

Proposition 1.5.6.0) On a complex manifold $M$, every reduced coherent foliation $F$ $¥subset¥Omega_{M}$ of

codimension one is of complete intersection type.

1) Let $(X, ¥mathcal{O}_{X})$ be a locally irreducible complex space and $A$ an analytic setin $X$ thin of order two and containing Sing X. For any coherent foliation$F_{0}¥subset¥Omega_{X-A}$ on $X-A$ of codimension one, there exists a uniquelydetermined reduced foliation $F¥subset¥Omega_{X}$ on $X$ of codimension one satisfying$F|_{X-(A¥cup S(F_{0}))}=F_{0}|_{X-(A¥cup S(F_{¥mathrm{O}}))}$ .

134 Akihiro SAEKI

Proof. 0) is obvious.1) In the case that $¥Omega_{X}$ is torsion-free, it follows directly from Proposition

A.2.1. Otherwise, we consider the canonical $¥mathcal{O}_{¥mathrm{X}}$ -morphism $¥tau_{¥Omega_{X}}$ : $¥Omega_{X}¥rightarrow¥Omega_{X}^{**}$ .

Note that $¥Omega_{X}^{**}$ is torsion-free in this situation. $¥tau_{¥Omega ¥mathrm{x}}(F_{0})¥subset¥Omega_{X}^{**}|_{XA}¥_$ is aninvertible subsheaf and determines a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $F_{1}$ of $¥Omega_{X}^{**}$ ofrank one by Proposition A.2.1. The inverse image $F_{2}:=¥tau_{¥Omega ¥mathrm{x}}^{-1}(F_{1})¥subset¥Omega_{X}$ iscoherent of rank one and $F:=(F_{2})_{F}$ is the required one. $¥blacksquare$

We consider foliations in neighbourhoods of quotient singularitiesagain. We use the notations in 1.4. The following is induced fromProposition 1.5.2 and Proposition 1.5.5.

Proposition 1.5.7. Let $G$, $¥pi:W¥rightarrow X$ , $B_{W}$ and $B_{X}$ be as in 1.4. Assumethat $X$ is a locally irreducible complex space and that $B_{X}$ is thin of order two

in $X$ and containing Sing X. If a coherent foliation $E¥subset¥Theta_{W}$ of dimension oneis $G$ -invariant then there is a uniquely determined reduced foliation $E_{X}¥subset¥Theta_{X}$

satisfying $E_{X}|_{X-B¥mathrm{x}}=(¥pi¥star(E|_{WB_{W}}¥_))_{F}$ .

A similar statement holds for foliations of codimension one.Especially, if $W$ is a complex manifold and if $B_{W}$ is thin of order two in

$W$ then the above conditions are satisfied.1.6. Examples. The complex space X defined by $z^{2}$ ?xy $=0$

Let $D$ be $C^{3}$ with a natural coordinate system $(x, y, z)$ , $¥mathcal{J}:=(z^{2}-xy)¥mathcal{O}_{D}$

and $X$ the closed complex subspace of $D$ defined by the coherent $¥mathcal{O}_{¥mathrm{D}}$ -ideal$¥mathcal{J}$ . We denote the closed embedding by $l:X¥mathrm{s}$ D. $X$ is a normal complexspace with the singular locus Sing $X=¥{¥mathrm{X}_{0}¥}$ , where $¥iota(¥mathrm{x}_{0})=0$ . The complexmanifold $U:=X-$ Sing $X$ satisfies

$U=U_{¥alpha}¥cup U_{¥beta}$ ,with

$U_{¥alpha}¥simeq C^{¥times}¥times C$ and

$U_{a}¥simeq C^{¥times}¥times X$,

where holomorphic maps

$ C^{¥times}¥times C¥rightarrow$ $D=C^{3}$

$(¥mathrm{x}, z)$ $¥mapsto(¥mathrm{x}, x^{-1}z, z)$

and

$ C^{¥times}¥times C¥rightarrow$ $D=C^{3}$

$(y, z)$ $¥mapsto(y^{-1}z, y, z)$


define the above isomorphisms, respectively. Here $C^{¥times}:=C-¥{0¥}$ .

A desingularization $¥sigma:¥tilde{X}¥rightarrow X$ is defined as follows:Let $¥tilde{X}_{a}$ and $¥tilde{X}_{¥beta}$ be $C^{2}$ ’s with coordinate systems $(s_{a}, t_{¥alpha})$ and $(s_{¥beta}, t_{¥beta})$ ,

trdprvyibrly. We define a complex manifold $¥tilde{X}$ by

$¥tilde{X}:=¥tilde{X}_{a}¥cup¥tilde{X}_{¥beta}$

with

$¥tilde{X}_{¥alpha}¥cap¥tilde{X}_{¥beta}:=¥{(s_{¥alpha}, t_{a})|t_{¥alpha}¥neq 0¥}¥subset¥tilde{X}_{a}$ and$¥tilde{X}_{a}¥cap¥tilde{X}_{¥beta}:=¥{(s_{¥beta}, t_{¥beta})|t_{¥beta}¥neq 0¥}¥subset¥tilde{X}_{¥beta}$,

patched by

$s_{¥alpha}=s_{¥beta}t_{¥beta}^{2}$ and $s_{¥beta}=s_{a}t_{a}^{2}$

$t_{a}=t_{¥beta}^{-1}$ $t_{¥beta}=t_{a}^{-1}$ .

Holomorphic maps$¥tilde{X}_{¥alpha}$ $¥rightarrow D=C^{3}$ $¥tilde{X}_{¥beta}$ $¥rightarrow D=C^{3}$

$(s_{a}, t_{¥alpha})¥mapsto(¥mathrm{x}, y, z)$ $(s_{¥beta}, t_{¥beta})¥mapsto(x, y, z)$

$=$ $s_{a}$ and $=$ $s_{¥beta}t_{¥beta}^{2}$

$y$ $=$ $s_{¥alpha}t_{a}^{2}$$y$ $=$

$s_{¥beta}$

$=$ $s_{a}t_{a}$ $=$ $s_{¥beta}t_{¥beta}$

are patched together to define a proper modification $¥sigma:¥tilde{X}¥rightarrow X$ .

Note that $X$ is isomorphic to the quotient complex space $G¥backslash W$, where$W=C^{2}$ with a coordinate system $(w_{1}, w_{2})$ and $G=¥{id_{W}, J¥}$ . Here $J$ is theinvolution on $W=C^{2}$ . A holomorphic map $¥Pi:W¥rightarrow D$ is defined by

$¥Pi$ $x$ $=$ $w_{1}^{2}$

$W=C^{2}¥rightarrow D=C^{3}$ ; $y=w_{2}^{2}$ ,

$(w_{1}, w_{2})¥mapsto(x, y, z)$ $Z=w_{1}w_{2}$

which induces the surjection $¥pi:W¥rightarrow G¥backslash W¥simeq X$ so that the following diagramcommutes:

$¥pi W¥downarrow¥backslash ¥Pi$

$X=D$ .

136 Akihiro SAEKI

. Foliations on $X$

Example 1.6.0. Let $¥mathrm{v}_{0}:=¥mathrm{x}(¥partial/¥partial x)+y(¥partial/¥partial y)+z$ $(¥partial/¥partial z)¥subset¥Theta_{D}(D)$ . Since $¥mathrm{v}_{0}¥in$

(Der$D(¥log X)$ ) $(D)$ , $¥mathrm{v}_{0}$ defines a well-defined vector field $v_{0}¥in¥Theta_{X}(X)$ . Set$E_{0}:=¥mathcal{O}_{X}v_{0}¥subset¥Theta_{X}$ . $E_{0}$ is a coherent foliation on $X$ of dimension one. It isreduced by Proposition 1.5.2.

We define two holomorphic 1-forms on $D$ by

$--.0:=zdx-xdz¥in¥Omega_{D}(D)$ and

$¥mathrm{H}_{0}:=zdy-ydz¥in¥Omega_{D}(D)$ .

They define $¥xi_{0}¥in¥Omega_{X}(X)$ and $¥eta_{0}¥in¥Omega_{X}(X)$ , respectively, which are as follows:On $U_{a}$ ,

$¥xi_{0}|_{U_{¥alpha}}=zd¥mathrm{x}-xdz$ and

$¥eta_{0}|_{U_{¥alpha}}=-x^{-2}z^{2}(zdx-¥mathrm{x}dz)=-x^{-2}z^{2}¥xi_{0}|_{U_{¥alpha}}$ .

On $U_{¥beta}$ ,

$¥eta_{0}|_{U_{¥beta}}=zd¥mathrm{x}-xdz$ and

$¥xi_{0}|_{U_{¥beta}}=-x^{-2}z^{2}(zdx-xdz)=-X^{-2}z^{2}¥eta_{0}|_{U_{¥beta}}$ .

The coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $F_{0}:=¥mathcal{O}_{X}¥xi_{0}+¥mathcal{O}_{X}¥eta_{0}$ of $¥Omega_{X}$ is a coherent foliationon $X$ of codimension one and $E_{0}=(F_{0})^{a}$ holds.

Let $¥ovalbox{¥tt¥small REJECT}_{0}:=¥mathcal{O}_{D}¥mathrm{v}_{0}¥subset¥Theta_{D}$ and $¥swarrow_{0}^{¥varpi}:=(¥mathcal{O}_{D}^{-}-.0+¥mathcal{O}_{D}¥mathrm{H}_{0})¥subset¥Omega_{D}$ . They are coherentfoliations on $D$ and $e_{0}=(A_{0}^{¥Gamma})^{a}$ and $¥swarrow_{0}^{¥varpi}=$ $(¥ovalbox{¥tt¥small REJECT}_{0})^{¥perp}$ holds.

Since $¥mathrm{v}_{0}¥in$$(¥mathrm{D}¥mathrm{e}¥mathrm{r}_{¥mathrm{D}} (¥log X))$ $(D)$ , $¥ovalbox{¥tt¥small REJECT}_{0}¥subset$ Der$D(¥log X))$ and, by Proposition 1.3.4,

the coherent foliation $¥ovalbox{¥tt¥small REJECT}_{0}¥subset¥Theta_{D}$ defines a coherent foliation $l^{*e_{0}¥subset¥Theta_{X}}$ on$X$ . Since both of them are reduced, $E_{0}=l^{¥star¥ovalbox{¥tt¥small REJECT}_{0}}$ holds.

The coherent foliation $F_{0}¥subset¥Omega_{X}$ on $X$ defines the pull-back foliation$¥sigma^{¥star}F_{0}¥subset¥Omega_{¥tilde{X}}$ on $¥tilde{X}$ , which is as follows. The forms $¥xi_{0}¥in¥Omega_{X}(X)$ and $¥eta_{0}¥in¥Omega_{X}(X)$

define forms $¥sigma^{*}¥xi_{0}¥in¥Omega_{¥tilde{X}}(¥tilde{X})$ and $¥sigma^{*}¥eta_{0}¥in¥Omega_{¥tilde{X}}(¥tilde{X})$ , respectively, which are on $¥tilde{X}_{¥alpha}$ ,

$¥sigma^{*}¥xi_{0}|_{¥tilde{X}_{¥alpha}}=-s_{a}^{2}dt_{a}$ and $¥sigma^{*}¥eta_{0}|_{¥tilde{X}_{a}}=s_{a}^{2}t_{a}^{2}dt_{a}$

and on $¥tilde{X}_{¥beta}$ ,

$¥sigma^{*}¥eta_{0}|_{¥tilde{X}_{¥beta}}=-s_{¥beta}^{2}dt_{¥beta}$ and $¥sigma^{*}¥xi_{0}|_{¥tilde{X}_{¥beta}}=s_{¥beta}^{2}t_{¥beta}^{2}dt_{¥beta}$ .

Thus $¥sigma^{¥star}F_{0}=¥mathcal{O}_{¥tilde{X}}¥xi_{0}+¥mathcal{O}_{¥tilde{X}}¥eta_{0}¥subset¥Omega_{¥tilde{X}}$ .

Since $¥sigma:¥tilde{X}¥rightarrow X$ is a proper modification, the coherent foliation $(¥sigma^{¥star}F_{0})^{a}¥subset$

$¥Theta_{¥tilde{X}}$ on $¥tilde{X}$ defines a uniquely determined reduced foliation $F_{¥acute{0}}¥subset¥Omega_{X}$ asfollows. The direct image $¥sigma_{*}((¥sigma^{¥star}F_{0})^{a})$ is considered as a coherent $¥mathcal{O}_{X^{-}}$

submodule of $¥Theta_{X}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ . $E_{¥acute{0}}:=¥sigma¥star((¥sigma^{¥star}F_{0})^{a}):=¥mu_{¥Theta_{X}}^{-1}$ $(¥sigma_{*}((¥sigma^{¥star}F_{0})^{a}))¥subset¥Theta_{X}$ is $¥mathrm{a}$


coherent foliation on $X$ and $F_{¥acute{0}}:=(E_{¥acute{0}})^{¥perp}¥subset¥Omega_{X}$ is the desired one. Moreexplicitly, we define a coherent foliation $E_{0¥tilde{X}}¥subset _{¥tilde{X}}$ on $¥tilde{X}$ by $E_{0¥tilde{X}}:=¥mathcal{O}_{¥tilde{X}}v_{¥tilde{X}}$ with$v_{¥tilde{X}}¥in¥Theta_{¥tilde{X}}$ such as, $v_{¥tilde{X}}|_{¥tilde{X}_{¥alpha}}=s_{a}(¥partial/¥partial s_{¥alpha})$ on $¥tilde{X}_{¥alpha}$ and $v_{¥tilde{X}}|_{¥tilde{X}_{¥beta}}=s_{¥beta}(¥partial/¥partial s_{¥beta})$ on $¥tilde{X}_{¥beta}$ . Wehave $¥sigma_{*}v_{¥tilde{X}}=v_{0}¥in _{X}(X)$ . Thus $E_{0}=¥mathcal{O}_{X}v_{0}¥subset¥sigma_{¥star}E_{0¥tilde{X}}:=¥mu_{¥Theta_{X}}^{-1}$ $(¥sigma_{*}E_{0¥tilde{X}})¥subset¥Theta_{X}$ and$E¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}=(¥sigma_{¥star}E_{0¥tilde{X}})¥mathrm{O}¥times_{¥mathit{0}¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}=$ $(¥mathcal{O}_{X}v_{0})¥otimes_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}=¥ovalbox{¥tt¥small REJECT}_{X}v_{0}¥subset _{X}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ . Since$E_{0}$ is reduced, $E_{0}=¥sigma_{¥star}E_{0¥tilde{X}}$ . Note that $E_{0¥tilde{X}}¥subset(¥sigma*F_{0})^{a}$ and $(E_{0¥tilde{X}})_{F}=$ $(¥sigma¥star F_{0})^{a}$ . Itfollows that $E_{¥acute{0}}=¥sigma¥star E_{0¥tilde{X}}=E_{0}$ and that $F_{¥acute{0}}=(¥sigma*E_{0¥tilde{X}})^{¥perp}=(E_{0})^{¥perp}=(F)_{F}=F_{0}$ .

Example 1.6.1. Let $u_{1}:=w_{1}(¥partial/¥partial w_{1})+w_{2}(¥partial/¥partial w_{2})¥in¥Theta_{W}(W)$ and $E_{W_{1}}:=$

$¥mathcal{O}_{W}u_{1}¥subset¥Theta_{W}$ . Since $u_{1}$ is $¥mathrm{G}$-invariant vector field, $E_{W_{1}}$ is a $¥mathrm{G}$-invariant coherentfoliation on $W$ satisfying the condition 1.4.3. By Proposition 1.4.3, $E_{W_{1}}$ definesa uniquely determined reduced coherent foliation $¥pi*E_{W_{1}}¥subset _{X}$ on $X$ . Namelythe $¥mathrm{G}$ -invariant vector field $u_{1}$ defines a holomorphic vector field $¥pi_{¥star}u_{1}$ on $X$,which is actually the above vector field $v_{0}¥in¥Theta_{X}(X)$ , and

$¥pi¥star E_{W_{1}}=(¥mathcal{O}_{X}(¥pi_{¥star}u_{1}))_{F}=E_{0}$ .

Example 1.6.2. Let $u_{2}:=(¥partial/¥partial w_{2})¥in _{W}(W)$ and $E_{W_{2}}:=¥mathcal{O}_{W}u_{2}¥subset¥Theta_{W}$ . Notethat $E_{W_{2}}$ is a $¥mathrm{G}$ -invariant coherent foliation on $W$, though $u_{2}$ is not a$¥mathrm{G}$ -invariant vector field. $E_{W_{2}}$ defines a uniquely determined reduced coherentfoliation $¥pi¥star E_{W_{2}}¥subset¥Theta_{X}$ on $X$ , since the surjection $¥pi:W¥rightarrow X$ satisfies the conditionof Proposition 1.5.7. Namely, $w_{2}u_{2}=w_{2}(¥partial/¥partial w_{2})¥subset¥Theta_{W}(W)$ is a $¥mathrm{G}$ -invariantvector field and $E_{W_{2}}^{¥prime}:=¥mathcal{O}_{W}(w_{2}u_{2})$ is a $¥mathrm{G}$ -invariant coherent foliation on $W$ $E_{W_{2}}^{¥prime}$

defines a coherent foliation on $X$ as in Example 1.6.1 and $¥pi¥star E_{W_{2}}=¥pi*(E_{W_{2}}^{¥prime})$

since $E_{W_{2}}=(E_{W_{2}}^{¥prime})_{F}$ . Explicitly, $w_{2}u_{2}$ defines a vector field $u_{x_{2}}¥in _{X}(X)$ such as$u_{X_{2}}|_{U_{¥alpha}}=z(¥partial/¥partial z)$ on $U_{a}$ and $u_{X_{2}}|_{U¥rho}=2y(¥partial/¥partial y)+z(¥partial/¥partial z)$ on $U_{¥beta}$ . $¥pi¥star E_{W_{2}}=$ $(¥mathcal{O}_{X}u_{X_{2}})_{F}$ .

§2. Coherent modules, their duals and their submodules

2.0. Preliminaries.First, we recall the following definitions.

Definition 2.0.0.0) A complex space $(X, ¥mathcal{O}_{X})$ is reduced at $x¥in X$ if $¥mathcal{O}_{X,x}$ is reduced, $¥mathrm{i}.¥mathrm{e}$ . $¥mathcal{O}_{X,x}$

.has no non-zero nilpotent element. $(X, ¥mathcal{O}_{X})$ is said to be reduced if it is

reduced at each $x¥in X$ .

1) A complex space $(X, ¥mathcal{O}_{X})$ is (locally) irreducible at $x¥in X$ if $¥mathcal{O}_{X,x}$ is anintegral domain. $(X, ¥mathcal{O}_{X})$ is said to be locally irreducible if it is locallyirreducible at each $x¥in X$.

2) A complex space $(X, ¥mathcal{O}_{X})$ is normal at $x¥in X$ if $¥mathcal{O}_{X,x}$ is an integrally closeddomain. $(X, ¥mathcal{O}_{X})$ is said to be normal if it is normal at each $x¥in X$ .

3) A complex space $(X, ¥mathcal{O}_{X})$ is locally factorial at $x¥in X$ if $¥mathcal{O}_{X,x}$ is a U.F.D..

138 Akihiro SAEKI

$(X, ¥mathcal{O}_{X})$ is said to be loclly factorial if it is locally factorial at each $¥mathrm{x}¥in X$ .

4) A closed subset $N$ of a complex space $X$ is called thin at $x¥in X$ if thereexist an open neighbourhood $U¥subset X$ of $x$ and a nowhere dense analyticset $A$ in $U$ satisfying $N¥cap U¥subset A$ . $N$ is said to be thin of order $k$ at$x$ if we can take the above $U$ and $A$ so that $¥dim_{x}X-¥dim_{X}A¥geq k$. $N$

is thin if $N$ is thin at each $x¥in X$ .

Let $(X, ¥mathcal{O}_{X})$ be a complex space, $l$ a coherent $¥mathcal{O}_{¥mathrm{X}}$-module, $J^{¥varpi}$ a coherent$¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥Psi$ and $¥mathit{9}:=¥Psi/J^{¥varpi}$ the quotient:

$0¥rightarrow J^{¥varpi}¥rightarrow¥ovalbox{¥tt¥small REJECT}¥rightarrow ¥mathit{9}-0$ (exact).

In this section, we define for $J^{¥varpi}$ four $¥mathcal{O}_{¥mathrm{X}}$-submodules $J_{F}^{¥varpi}$ , $J_{SF}^{¥varpi}$ , $J^{¥varpi a¥perp}$ and $J_{¥ovalbox{¥tt¥small REJECT}}^{¥varpi}$

of $¥Psi$ and investigate the relations among them. In particular, we prove thatthey coincide with one another if the complex space $X$ satisfies someconditions. (Proposition 2.5.1 and Theorem 2.6.1.)

We use the following notations.

Notation 2.0.1.0) For an $¥mathcal{O}_{¥mathrm{X}}$-module $¥ovalbox{¥tt¥small REJECT}$ , we set

Sing $¥ovalbox{¥tt¥small REJECT}:=$ { $x¥in X|¥Psi_{x}$ is not $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -free}

and call it the singular locus of $¥Psi$ .

1) For an $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi}$ of $¥Psi$ , we write

$S(J^{¥varpi}):=$ Sing $¥Psi¥cup$ Sing $l/J^{¥varpi}$

If $¥Psi$ is coherent, then Sing $¥Psi$ is an analytic set in $X$ . If, in addition,the complex space $X$ is reduced, then Sing $¥Psi$ is thin.

For a coherent $¥mathcal{O}_{¥mathrm{X}}$-module $¥ovalbox{¥tt¥small REJECT}$ and a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi}$ of $l$ , thequotient module $¥mathit{9}=¥Psi/J^{¥varpi}$ is also coherent and $S(J^{¥varpi})$ is an analytic set in$X$ . It is thin in $X$ if $X$ is reduced. At $¥mathrm{x}¥in X$ , we consider the exact sequenceof $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -modules:

$0¥rightarrow J_{x}^{¥varpi}¥rightarrow l_{x}¥rightarrow ¥mathit{9}_{x}¥rightarrow 0$ .

If $x¥in X-S(J^{¥varpi})$ , then 9$X$

is a free $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -module and the above sequencesplits. This implies that $J_{X}^{¥varpi}$ is a direct summund of a free $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$.-module $¥ovalbox{¥tt¥small REJECT}_{x}$ ,$¥mathrm{i}.¥mathrm{e}$ . a projective $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -module. Since $¥mathcal{O}_{X,x}$ is a local ring, $J_{¥mathrm{X}}^{¥varpi}$ is also a free$¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -module. Thus we have $ S(J^{¥varpi})¥supset$ Sing $J^{¥varpi}$ . Summarizing:

Proposition 2.0.2. For a coherent $¥mathcal{O}_{X^{-}}$module $¥Psi$ and a coherent $¥mathcal{O}_{X^{-}}$

submodule $J^{¥Gamma}$ of $¥Psi$ , $S(J^{¥varpi})$ is an analytic set in $X$ satisfying

$ S(J^{¥Gamma})¥supset$ Sing $J^{¥varpi}$ .


On $X-S(J^{¥varpi}),$ $¥swarrow¥varpi$ is locally a direct summand of $¥Psi$ .

2.1. The $¥mathcal{O}_{¥mathrm{X}}$ -submodule $¥mathscr{T}_{F}$

Definition 2.1.0. For a coherent $¥mathcal{O}_{¥mathrm{X}}$-module $¥Psi$ and a coherent $¥mathcal{O}_{X^{-}}$

submodule $¥mathscr{T}^{¥varpi}$ of $¥ovalbox{¥tt¥small REJECT}$ , an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}_{F}^{¥varpi}$ of $¥Psi$ is defined by a completepresheaf

$¥mathscr{T}_{F}^{¥varpi}(U):=$ $¥{¥xi¥in¥Psi(U)|¥xi|_{US(¥ovalbox{¥tt¥small REJECT})}¥_¥in ¥mathscr{T}^{¥varpi}(U-S(¥Psi))¥}$ for $U¥subset X$ .

The coherence of $¥mathscr{T}_{F}^{¥varpi}$ is shown in 2.5.

2.2. The $¥mathcal{O}_{¥mathrm{X}}$ -submodule $¥mathscr{T}_{SF}$

Definition 2.2.0. For a coherent $¥mathcal{O}_{¥mathrm{X}}$-module $¥Psi$ and a coherent $¥mathcal{O}_{X^{-}}$

submodule $¥mathscr{T}^{¥Gamma}$ of $¥Psi$ , an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}_{SF}^{¥varpi}$ of $¥Psi$ is defined as the associatiedsheaf of the presheaf $¥{T_{F}(U)¥}$ :

$T_{F}(U):=$ { $¥xi¥in¥ovalbox{¥tt¥small REJECT}(U)|¥xi|_{UA}¥_¥in/^{¥tau}(U-A)$ for some thin analytic set $A$ in $U.$ }.

The coherence of $¥mathscr{T}_{SF}^{¥Gamma}$ is shown in 2.5.

2.3. The $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}^{a¥neg l1}$

Let $l$ be a coherent $¥mathcal{O}_{¥mathrm{X}}$-module and $¥ovalbox{¥tt¥small REJECT}*$ the dual $¥mathcal{O}_{¥mathrm{X}}$-module of $¥Psi$ . First,we define an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}^{¥varpi a}$ of $¥ovalbox{¥tt¥small REJECT}*$ for $¥mathscr{T}^{¥varpi}$ and $¥mathcal{O}_{¥mathrm{X}}$-submodule of $l$ andthen, using $¥mathscr{T}^{¥varpi a}$ , an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}^{¥varpi a1}$ of $¥Psi$ is defined.

Definition 2.3.0. For an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}^{¥Gamma}$ of $¥Psi$ , $¥mathscr{T}^{¥varpi a}$ is an $¥mathcal{O}_{¥mathrm{X}}$-submoduleof $¥Psi^{*}$ whose stalk on $¥mathrm{x}¥in X$ is

$¥mathscr{T}_{x}^{¥varpi a}:=$ { $v¥in¥Psi_{x}^{*}|¥langle v$, $¥xi¥rangle=0$ for all $¥xi¥in ¥mathscr{T}_{X}^{¥varpi}$ }.

Remark. If $¥mathscr{T}^{¥varpi}$ is coherent, then, as $¥mathcal{O}_{¥mathrm{X}}$-modules, $Y^{a}¥simeq ¥mathit{9}^{*}$ , where 9* isthe dual of the quotient module $¥mathit{9}=¥ovalbox{¥tt¥small REJECT}/¥mathscr{T}^{¥varpi}$. Thus $¥mathscr{T}^{¥varpi a}$ is coherent if $¥mathscr{T}^{¥Gamma}$ is. Ifit is the case,

$S(¥mathscr{T}^{¥varpi a})¥subset S(¥mathscr{T}^{¥varpi})$

hold.

Using this $¥mathscr{T}^{¥varpi a}$ , we define an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mathscr{T}^{¥Gamma a¥perp}$ of $¥Psi$ .

Definition 2.3.1. For an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥ovalbox{¥tt¥small REJECT}$ of $¥ovalbox{¥tt¥small REJECT}*$ , we define an$¥mathcal{O}_{¥mathrm{X}}$-submodule $¥ovalbox{¥tt¥small REJECT}¥perp$ of $¥ovalbox{¥tt¥small REJECT}$ as follows:

$¥ovalbox{¥tt¥small REJECT}_{x}^{¥perp}:=$ { $¥xi¥in l_{x}|¥langle v$, $¥xi¥rangle=0$ for all $v¥in¥ovalbox{¥tt¥small REJECT}_{x}$ }.Remark. $¥ovalbox{¥tt¥small REJECT}¥perp$ is not always isomorphic to the $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥ovalbox{¥tt¥small REJECT}^{a}$ of the

bidual $¥ovalbox{¥tt¥small REJECT}**$ of $¥ovalbox{¥tt¥small REJECT}$ . It is the inverse image of $¥ovalbox{¥tt¥small REJECT}^{a}$ by the canonical $¥mathcal{O}_{¥mathrm{X}}$-morphism

140 Akihiro SAEKI

$¥Psi¥rightarrow¥varphi**$ . Thus $¥ovalbox{¥tt¥small REJECT}^{¥perp}$ is the kernel of the composite $¥mathcal{O}_{¥mathrm{X}}$-morphism

$¥Psi¥rightarrow¥varphi**¥rightarrow¥Psi^{**}/¥ovalbox{¥tt¥small REJECT}^{a}$

and the coherence of $¥ovalbox{¥tt¥small REJECT}¥perp$ follows from that of $¥varphi**/¥ovalbox{¥tt¥small REJECT}^{a}$ . Note that $¥varphi**/¥ovalbox{¥tt¥small REJECT}^{a}$

is coherent if and only if $¥ovalbox{¥tt¥small REJECT}^{a}$ is, and that the coherence of $¥ovalbox{¥tt¥small REJECT}^{a}$ is induced fromthat of $¥ovalbox{¥tt¥small REJECT}$ .

Definition 2.3.2. For an $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi}$ of $¥Psi$ , we define an$¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi a1_{-}}$submodule $¥swarrow¥varpi a¥perp$ of $¥ovalbox{¥tt¥small REJECT}$ by

$J^{¥Gamma a1}:=(J^{¥varpi a})^{¥perp}$ .

We have the following:

Proposition 2.3.3. If an $¥mathcal{O}_{X^{-}}$submodule $J^{¥varpi}$ of $l$ is coherent, so are $J^{¥tau a}$

and $J^{¥varpi a¥perp}$ .

2.4. The $¥mathcal{O}_{¥mathrm{X}}$ -submodule $¥mathscr{T}_{¥mathscr{M}}$

First, we review the sheaf $¥ovalbox{¥tt¥small REJECT}_{X}$ of germs of meromorphic functions on $X$ .

Definition 2.4.0.0) At $x¥in X$, $f¥in ¥mathcal{O}_{X,x}$ is said to be active if the image of $f$ by

$¥mathcal{O}_{X,x}¥rightarrow ¥mathcal{O}_{X,x}/¥Lambda_{X}^{¥prime}$

is a non-zero divisor in $¥mathcal{O}_{X,x}/¥Lambda_{X}^{¥prime}$ , where $¥Lambda_{x}^{¥prime}$ is the nilradical of $¥mathcal{O}_{X,x}$ .

1) Let $¥ovalbox{¥tt¥small REJECT} c$ be a subsheaf (of sets) of $¥mathcal{O}_{X}$ defined, at $x¥in X$, by

$¥ovalbox{¥tt¥small REJECT} c_{X}:=$ { $f¥in ¥mathcal{O}_{X,x}|f$ is active at $x$ }.

This is a multiplicative system in $¥mathcal{O}_{X}$ . We define the sheaf $¥ovalbox{¥tt¥small REJECT}_{X}$ ofgerms of meromorphic functions on $X$ by

$¥ovalbox{¥tt¥small REJECT}_{X}:=¥ovalbox{¥tt¥small REJECT} c^{-1}¥mathcal{O}_{X}$ ,

whose stalk on $x¥in X$ satisfies$¥ovalbox{¥tt¥small REJECT}_{X,x}=¥ovalbox{¥tt¥small REJECT} c_{x}^{-1}¥mathcal{O}_{X,x}$ : the localization of $¥mathcal{O}_{X,x}$ by the multiplicative system$¥ovalbox{¥tt¥small REJECT} c_{¥mathrm{x}}$ .

Remark.0) For $U¥subset X$ ,

$¥ovalbox{¥tt¥small REJECT}_{X}(U)¥supset¥ovalbox{¥tt¥small REJECT} c(U)^{-1}¥mathcal{O}_{X}(U)$ but $¥ovalbox{¥tt¥small REJECT}_{X}(U)¥neq¥ovalbox{¥tt¥small REJECT} c(U)^{-1}¥mathcal{O}_{X}(U)$,

in general.1) If the complex space $X$ is reduced at $x$ , then

$¥ovalbox{¥tt¥small REJECT} c_{x}=$ { $f¥in ¥mathcal{O}_{X,x}|f$ is a non-zero divisor in $¥mathcal{O}_{X.x}$ }


and the canonical $¥mathrm{C}$-algebra homomorphism

$¥mu:¥mathcal{O}_{X,x}¥rightarrow¥ovalbox{¥tt¥small REJECT}_{X,x}$

is injective.

For a morphism $(f,¥tilde{f}):(X, ¥mathcal{O}_{X})¥rightarrow(Y, ¥mathcal{O}_{¥mathrm{Y}})$ , the morphism of structuresheaves $¥tilde{f}:¥mathcal{O}_{¥mathrm{Y}}¥rightarrow f_{*}¥mathcal{O}_{X}$ does not always induce the morphism of sheaves ofmeromorphic functions $¥tilde{f}:¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}¥rightarrow f_{*}.¥ovalbox{¥tt¥small REJECT}_{X}$ .

Proposition 2.4.1. Suppose that a morphism $(f,¥tilde{f}):(X, ¥mathcal{O}_{X})¥rightarrow(Y, ¥mathcal{O}_{¥mathrm{Y}})$ ofreduced complex spaces satisfies the thinness condition: For any open $U¥subset ¥mathrm{Y}$

and any thin set $N$ in $U$ , the inverse image

(2.4.2.) $f^{-1}(N)$ is thin in $f^{-1}(U)$ .

Then the morphism of structure sheaves $¥tilde{f}:¥mathcal{O}_{¥mathrm{Y}}¥rightarrow f_{*}¥mathcal{O}_{X}$ uniquely extends to themorphism of sheaves of meromorphic functions $¥tilde{f}:¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}¥rightarrow f_{*}¥ovalbox{¥tt¥small REJECT}_{X}$ . Furthermore,if the underlying continuous map $f:¥mathrm{s}¥mathrm{p}$ $X¥rightarrow ¥mathrm{s}¥mathrm{p}Y$ is surjective, then $¥tilde{f}:¥ovalbox{¥tt¥small REJECT}_{¥mathrm{Y}}¥rightarrow f_{*}¥ovalbox{¥tt¥small REJECT}_{X}$

is injective.

Note that the functor $¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ is exact in the category of $¥mathcal{O}_{¥mathrm{X}}$-modules,$¥mathrm{i}.¥mathrm{e}$ . for any exact sequence of (not necessalily coherent) $¥mathcal{O}_{¥mathrm{X}}$-modules

$0¥rightarrow¥Psi_{0}¥rightarrow l_{1}¥rightarrow l_{2}¥rightarrow 0$,

the induced sequence

$0¥rightarrow¥ovalbox{¥tt¥small REJECT}_{0}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow¥ovalbox{¥tt¥small REJECT}_{1}¥otimes_{¥mathcal{O}¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow¥ovalbox{¥tt¥small REJECT}_{2}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow 0$

is also exact.For any $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥Psi$ , let

$¥mu_{¥ovalbox{¥tt¥small REJECT}}$ : $¥ovalbox{¥tt¥small REJECT}¥rightarrow¥ovalbox{¥tt¥small REJECT} ¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$

be the canonical $¥mathcal{O}_{¥mathrm{X}}$-morphism.Now we define the $¥mathcal{O}_{¥mathrm{X}}$-submodule $J_{¥ovalbox{¥tt¥small REJECT}}^{¥varpi}$ . For an $¥mathcal{O}_{¥mathrm{X}}$-module $l$ and an

$¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi}$ of $¥Psi$ , we have the following commutative diagram, whoserows are exact:

$ 0¥rightarrow$

$¥mu_{J^{¥varpi}}J^{¥Gamma}¥downarrow$

$¥rightarrow$

$¥mu¥ovalbox{¥tt¥small REJECT}¥Psi¥downarrow$

$¥rightarrow$

$¥mu p¥ovalbox{¥tt¥small REJECT}¥downarrow$

$¥rightarrow 0$

$0¥rightarrow J^{¥varpi}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow¥Psi¥otimes_{¥mathcal{O}¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow ¥mathit{9}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow 0$

where $¥mathit{9}=¥ovalbox{¥tt¥small REJECT}/J^{¥varpi}$ is the quotient $¥mathcal{O}_{¥mathrm{X}}$-module. By this diagram, we consider$J^{¥varpi}¥otimes_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ as an $¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥ovalbox{¥tt¥small REJECT}¥otimes_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ .

142 Akihiro SAEKI

Definition 2.4.3. For an $¥mathcal{O}_{¥mathrm{X}}$-submodule $J^{¥varpi}$ of $¥ovalbox{¥tt¥small REJECT}$ , $J_{¥ovalbox{¥tt¥small REJECT}}^{¥varpi}$ is the $¥mathcal{O}_{¥mathrm{X}}$-submoduleof $¥Psi$ defined by

$¥swarrow_{¥ovalbox{¥tt¥small REJECT}}^{¥varpi}:=¥mu_{¥ovalbox{¥tt¥small REJECT}}^{-1}(J^{¥varpi}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X})$ .

2.5. Relations of $¥mathscr{T}_{F}$ , $¥mathscr{T}_{SF}$ and $¥mathscr{T}^{a¥perp}$

Let $(X, ¥mathcal{O}_{X})$ be, at first, an arbitrary complex space. Let $¥Psi$ be a coherent$¥mathcal{O}_{¥mathrm{X}}$-module and $¥swarrow¥varpi$ a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥ovalbox{¥tt¥small REJECT}$ . Let $¥mathscr{T}_{F}^{¥varpi}$ , $¥mathscr{T}_{SF}^{¥varpi}$ and $¥mathscr{T}^{¥Gamma a¥perp}$ be$¥mathcal{O}_{¥mathrm{X}}$-submodules of $¥Psi$ defined in 2.1, 2.2 and 2.3, respectively. From thedefinitions of $¥mathscr{T}_{F}^{¥varpi}$ , $¥mathscr{T}_{SF}^{¥varpi}$ and $¥mathscr{T}^{¥varpi a¥perp}$ and Proposition 2.0.2, we have the following.

Lemma 2.5.0. As $¥mathcal{O}_{X^{-}}$submodules of $¥Psi$ , the following holds:0) $(¥mathscr{T}^{¥varpi a¥perp})^{a¥perp}=¥mathscr{T}^{¥varpi a¥perp}$ .

1) $¥mathscr{T}^{¥varpi}¥subset ¥mathscr{T}_{F}^{¥varpi}$ , $¥mathscr{T}^{¥varpi}¥subset ¥mathscr{T}_{SF}^{¥Gamma}$ and $¥mathscr{T}^{¥varpi}¥subset ¥mathscr{T}^{¥varpi a¥perp}$ .

2) $/^{¥varpi}|_{X-S(J^{¥varpi})}=¥mathscr{T}_{F}^{¥varpi}|_{X-S(J^{¥varpi})}=¥mathscr{T}_{SF}^{¥varpi}|_{X-S(J^{¥varpi})}=¥mathscr{T}^{¥varpi a¥perp}|_{X-S(J^{¥varpi})}$ .

3) If $S(¥mathscr{T}^{¥varpi})$ is thin in $X$, then

$¥mathscr{T}_{F}^{¥varpi}¥subset ¥mathscr{T}_{SF}^{¥Gamma}$ .

Note that, if the complex space $X$ is reduced, the hypothesis of 3) ofLemma 2.5.0 is satisfied. In what follows, we always assume complex spacesare reduced unless explicitly stated otherwise.

Proposition 2.5.1. Let $(X, ¥mathcal{O}_{X})$ be a reduced complex space. For a coherent$¥mathcal{O}_{X^{-}}$module $¥ovalbox{¥tt¥small REJECT}$ and a coherent $¥mathcal{O}_{X^{-}}$submodule $¥mathscr{T}^{¥varpi}$ of $¥ovalbox{¥tt¥small REJECT}$ , it holds that

$¥mathscr{T}_{F}^{¥varpi}=¥mathscr{T}_{SF}^{¥varpi}=¥mathscr{T}^{¥tau a¥perp}$

and all of them are coherent.

Proof. We show $¥mathscr{T}_{SF}^{¥varpi}¥subset ¥mathscr{T}^{¥varpi a¥perp}$ and $¥mathscr{T}^{¥Gamma a¥perp}¥subset ¥mathscr{T}_{F}^{¥Gamma}$ .. $¥mathscr{T}_{SF}^{¥Gamma}¥subset ¥mathscr{T}^{¥Gamma a¥perp}$

Let $x¥in X$ , $¥xi¥in(¥mathscr{T}_{SF}^{¥varpi})_{¥mathrm{x}}$ and $v¥in(¥mathscr{T}^{¥varpi a})_{X}$ . Assume that $¥xi$ and $v$ have representatives$¥xi¥in¥swarrow_{SF}^{¥varpi}(U)$ and $v¥in¥swarrow^{¥varpi a}(U)$ on a connected open neighbourhood $U$ of $x$ ,respectively, and that there is a thin analytic set $A$ in $U$ such that$¥xi|_{UA}¥_¥in ¥mathscr{T}^{¥varpi}(U-A)$ . Since, by definition, $¥langle¥xi, v¥rangle¥in ¥mathcal{O}_{X}(U)$ satisfies

$¥langle¥xi, v¥rangle|_{U-A}=¥langle¥xi|_{U-A}, v|_{U-A}¥rangle=0$ ,

it follows from the thinness of $A$ in $U$ and the connected of $U$ that$¥langle¥xi, v¥rangle=0$ .

This implies $¥xi¥in(¥mathscr{T}^{¥varpi a¥perp})_{¥mathrm{x}}$ .. $¥mathscr{T}^{¥varpi a¥perp}¥subset ¥mathscr{T}_{F}^{¥varpi}$

Since, in this case, $S$ ($/¥gamma$ is thin, this follows immediately from 2) of Lemma2.5.0. $¥blacksquare$


We use the following terminology.

Definition 2.5.2. Let $¥Psi$ be a coherent $¥mathcal{O}_{¥mathrm{X}}$-module. A coherent $¥mathcal{O}_{X^{-}}$

submodule $J^{¥varpi}$ of $¥Psi$ is said to be full in $¥Psi$ if it satisfies

$J^{¥varpi}=J_{F}^{¥varpi}$ .

Corollary 2.5.3.a) The following statements are equivalent:

0) $¥swarrow¥varpi$ is full in $¥Psi$ .

1) $¥swarrow¥varpi=J_{SF}^{¥Gamma}$ .

2) $J^{¥varpi}=J^{¥Gamma^{a¥perp}}$ .

b) If two coherent $¥mathcal{O}_{X^{-}}$submodules $J_{0}^{¥varpi}$ and $J_{1}^{¥varpi}$ satisfy

$J_{0}^{¥varpi}|_{X-A}=J_{1}^{¥varpi}|_{X-A}$

for a thin analytic set $A$ in $X$, then

$(J_{0}^{¥varpi})_{F}=(J_{1}^{¥varpi})_{F}$

From the proof of Proposition 2.5.1, we have the following

Corollary 2.5.4.For a coherent $¥mathcal{O}_{X^{-}}$ submodule $J^{¥varpi}$ of a coherent $¥mathcal{O}_{X^{-}}$ module $¥ovalbox{¥tt¥small REJECT}$ , we have

$J^{¥varpi a}=(J_{F}^{¥varpi})^{a}=(J_{SF}^{¥varpi})^{a}=(J^{¥varpi a¥perp})^{a}$ .

It is a coherent $¥mathcal{O}_{X^{-}}$ submodule of $¥Psi^{*}$ .

Now we investigate submodules of $¥varphi*$ (cf. Definition 2.3.1).

Proposition 2.5.5. For a coherent $¥mathcal{O}_{X^{-}}$ submodule $¥ovalbox{¥tt¥small REJECT}$ of $¥varphi*$ , the followingholds.

0) $¥ovalbox{¥tt¥small REJECT}¥subset¥ovalbox{¥tt¥small REJECT}^{¥perp a}$ and $¥ovalbox{¥tt¥small REJECT}|_{X-S(¥ovalbox{¥tt¥small REJECT})}=¥ovalbox{¥tt¥small REJECT}^{¥perp a}|_{X-S(¥ovalbox{¥tt¥small REJECT})}$ .

1) $(¥ovalbox{¥tt¥small REJECT}^{¥perp a})^{¥perp}=¥ovalbox{¥tt¥small REJECT}^{¥perp}and$ $(¥ovalbox{¥tt¥small REJECT}^{¥perp a})^{¥perp a}=¥ovalbox{¥tt¥small REJECT}¥perp a$ .

2) $¥ovalbox{¥tt¥small REJECT}^{¥perp a}=¥ovalbox{¥tt¥small REJECT}_{F}$ .

3) $¥ovalbox{¥tt¥small REJECT}$ is full in $¥varphi*$ if and only if $¥ovalbox{¥tt¥small REJECT}^{¥perp a}=¥ovalbox{¥tt¥small REJECT}$ .

Proof. 0) obvious.1) $(¥ovalbox{¥tt¥small REJECT}^{¥perp a})^{¥perp a}=¥ovalbox{¥tt¥small REJECT}^{¥perp a}$ follows from Corolary 2.5.4, setting $J^{¥varpi}=¥ovalbox{¥tt¥small REJECT}^{¥perp}$ . $(¥ovalbox{¥tt¥small REJECT}^{¥perp a})^{¥perp}¥subset$

$¥ovalbox{¥tt¥small REJECT}¥perp$ is derived from $¥ovalbox{¥tt¥small REJECT}¥subset¥ovalbox{¥tt¥small REJECT}^{¥perp a}$ and $(¥ovalbox{¥tt¥small REJECT}^{¥perp a})^{¥perp}¥supset¥ovalbox{¥tt¥small REJECT}^{¥perp}$ is deduced from 1) ofLemma 2.5.0.

2) Proved in a similar manner to Proposition 2.5.1.3) directly follows from Definition 2.5.2. $¥blacksquare$

Let us summarize the attributes of the correspondences . $a$ and $.¥perp$

144 Akihiro SAEKI

Theorem 2.5.6.a) By the correspondence $¥mathscr{T}^{c7^{-}}¥subset ¥mathscr{S}¥rightarrow ¥mathscr{T}^{¥varpi a}¥subset ¥mathscr{S}^{*}$ , a coherent $¥mathcal{O}_{X^{-}}$ submodule $¥mathscr{T}^{¥varpi}$ of

$¥ovalbox{¥tt¥small REJECT}$ is corresponded to the coherent $¥mathcal{O}_{X^{-}}$submodule $¥mathscr{T}^{¥varpi a}$ of $¥varphi*$ , which is full in$¥ovalbox{¥tt¥small REJECT}*$ . This correspondence inverts the including relations.

b) By the correspondence $¥ovalbox{¥tt¥small REJECT}¥subset ¥mathscr{S}^{*}¥rightarrow¥ovalbox{¥tt¥small REJECT}^{¥perp}¥subset ¥mathscr{S}$, a coherent $¥mathcal{O}_{X^{-}}$submodule $¥ovalbox{¥tt¥small REJECT}$ of$¥ovalbox{¥tt¥small REJECT}*$ is corresponded to the coherent $¥mathcal{O}_{X^{-}}$submodule $¥ovalbox{¥tt¥small REJECT}¥perp of$ $¥mathscr{S}$ , which is full in$¥epsilon¥varphi$ . This correspondence inverts the including relations.

c) These correspondeces, restricted to full submodules, are the inverse of eachother.

2.6. Relations with $¥mathscr{T}_{¥mathscr{M}}$

Assume, as above, $(X, ¥mathcal{O}_{X})$ is a reduced complex space. Let $¥epsilon¥varphi$ be acoherent $¥mathcal{O}_{¥mathrm{X}}$-module, $¥mathscr{T}^{¥varpi}$ a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule of $l$ and $¥mathscr{T}_{¥mathscr{M}}^{¥varpi}$ the$¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥ovalbox{¥tt¥small REJECT}$ defined in 2.4.

Lemma 2.6.0. As $¥mathcal{O}_{X^{-}}$ submodules of $¥ovalbox{¥tt¥small REJECT}$ , the following holds:0) $(¥mathscr{T}_{¥mathscr{M}}^{¥varpi})_{¥mathscr{M}}=¥mathscr{T}_{¥mathscr{M}}^{¥varpi}$

1) $¥mathscr{T}^{¥varpi}¥subset ¥mathscr{T}_{¥mathscr{M}}^{¥varpi}$

2) $¥mathscr{T}^{¥varpi}|_{X-S(J^{¥varpi})}=¥mathscr{T}_{¥ovalbox{¥tt¥small REJECT}}^{¥Gamma}|_{X-S(J^{¥varpi})}$

3) $¥mathscr{T}^{¥varpi}¥subset ¥mathscr{T}_{¥ovalbox{¥tt¥small REJECT}}^{¥Gamma}¥subset ¥mathscr{T}^{¥varpi a1}$

Proof. This follows from Proposition 2.0.2 and Remark followingDefinition 2.4.0. $¥blacksquare$

On locally irreducible complex spacs, a stronger assertion holds:

Theorem 2.6.1. Let $(X, ¥mathcal{O}_{X})$ be a locally irreducible complex space, $¥mathscr{S}$$a$

coherent $¥mathcal{O}_{X^{-}}$module and $¥mathscr{T}^{¥varpi}$ a coherent $¥mathcal{O}_{X^{-}}$submodule of $¥ovalbox{¥tt¥small REJECT}$ . Let $¥mathrm{e}/_{F}^{¥varpi}$ , $¥mathscr{T}_{SF}^{¥varpi}$ , $¥mathscr{T}^{¥varpi a1}$

and $¥mathscr{T}_{¥mathscr{M}}^{¥Gamma}$ be the $¥mathcal{O}_{X^{-}}$submodules of $l$ defined in 2.1, 2.2, 2.3 and 2.4,respectively. Then

$¥mathscr{T}_{F}^{¥varpi}=¥mathscr{T}_{SF}^{¥varpi}=¥mathscr{T}^{¥varpi a1}=¥mathscr{T}_{¥ovalbox{¥tt¥small REJECT}}^{¥varpi}$ .

This follows from the observations in Appendices A.0 and A.1.

Appendices

A.0. Algebraic observations on coherent sheaves on reduced complex spaces.Let $A$ be a commutative ring with 1. In what follows, we always assume

that $S$ is a multiplicative system in $A$ and that every $s¥in S$ is a non-zero-divisorin $A$ . Then the localization

$¥mu:A¥rightarrow S^{-1}A=A_{S}$

is injective, by which we consider $A$ as a subring of $B=S^{-1}A=A_{S}$ . Since$B$ is A-flat, for any A-module $M$ and any A-submodule $L$ of $M$, the induced


$¥mathrm{B}$-linear map

$L¥otimes_{A}B¥rightarrow M¥otimes_{A}B$

is injective, by which we regard $L¥otimes_{A}B$ as 5-submodule of $M¥mathrm{O}¥times_{A}B$. Thuswe have a commutative diagram:

(A.O.O)$¥downarrow L$

$¥subset$

$ M¥downarrow$

$L¥mathrm{O}¥times_{A}B¥subset M¥mathrm{O}¥times_{A}B$

We use the following notations.

Notation A.O.1.For an A-module $M$,

$M^{*}:=¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathrm{A}}(¥mathrm{M}, A)$ : the 4-dual of $M$.

For an 5-submodule $L$ of $M$ ,

$L^{a}:=$ { $¥varphi¥in M^{*}|¥varphi(u)=0$ for all $u¥in L$ },$L^{a¥perp}:=$ { $u¥in M|¥varphi(u)=0$ for all $¥varphi¥in L^{a}$}

and

$L_{B}:=$ the inverse image of $L¥otimes_{A}B¥subset M¥otimes_{A}B$ under

$M¥rightarrow M¥mathrm{O}¥times_{A}B$ in (A.O.O) (an 5-submodule of $M$).

For a 5-submodule $N$,

$N^{¥star}:=¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathrm{B}}(¥mathrm{N}, B)$ : the $¥mathrm{B}$-dual of $N$ .

For a 5-submodule $T$ of $N$ ,

$T^{a}:=$ { $¥psi¥in N^{¥star}|¥psi(u)=0$ for all $u¥in T$ }

and

$T^{¥alpha¥perp}:=$ { $u¥in N|¥psi(u)=0$ for all $¥psi¥in T^{a}$ }.

It follows from the definitions that

Lemma A.0.2.

$L^{a¥perp}¥subset L$ and $(L^{a¥perp})^{a}¥subset L^{2¥perp}$ .

Lemma A.0.3. For a finitely generated $A$ -module $M$,

$(M^{*})¥mathrm{O}¥times_{A}B¥simeq(M¥otimes_{A}B)^{¥star}$ .

146 Akihiro SAEKI

Proof. Since there always exists the canonical $¥mathrm{B}$-linear isomorphism

$¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathrm{A}}(¥mathrm{M}, B)¥simeq(M¥otimes_{A}B)^{¥star}=Hom_{B}(MO¥times_{A}B, B)$

;$¥varphi¥mapsto¥varphi_{B}$

$¥varphi_{B}(u¥otimes b)=¥varphi(u)b$ for $u¥in M$ and $b¥in B$ ,

we have only to show that the $¥mathrm{B}$-linear map$(M^{*})O¥times_{A}BHom_{A}(M, B)$

;$¥psi¥otimes b$ $¥mapsto$ $b$ . $¥psi$

$(b. ¥psi)(u)=b$ . $¥psi(u)$ in $B$ for $u¥in M$

is an isomorphism. Since every element of $(M^{*})¥otimes_{A}B$ is written in the form

$¥psi ¥mathrm{O}¥times s^{-1}$ $¥psi¥in M^{*}$ and $s¥in S$ ,

the injectivity follows from that of $A¥rightarrow B$ . Surjectivity is induced from that$M$ is finitely generated over $A$ . Take a generating system $u_{1},¥cdots,u_{n}¥in M$ . Forany $¥varphi¥in ¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathrm{A}}(¥mathrm{M}, B)$ , $¥varphi(u_{1}),¥cdots$ , $¥varphi(u_{n})¥in B$ can be written as

$¥varphi(u_{j})=s_{j}^{-1}a_{j}$ $s_{j}¥in S$ and $a_{j}¥in A(j =1,¥cdots,n)$ .

Let $s=s_{1}¥cdots s_{n}¥in S$ . $ s¥varphi$ define a well-defined element $¥psi¥in M^{*}$ and $¥varphi$ is the imageof $¥psi¥otimes s^{-1}$ . $¥blacksquare$

We identify $(M^{*})¥otimes_{A}B$ and $(M ¥mathrm{O}¥times_{A}B)^{¥star}$ with each other by the compositeisomorphism

$(M^{*})¥otimes_{A}B¥simeq Hom_{A}(M, B)¥simeq(M¥otimes_{A}B)^{¥star}$ .

The following lemma is derived from (A.O.O), Notation A.O.1 and LemmataA.0.2 and A.O.3.

Lemma A.0.4. For an $A$-submodule $L$ of a finitely generated $A$-module $M$,

$(L^{a})¥mathrm{O}¥times_{A}B$ $=$ $(L¥otimes_{A}B)^{a}$

$||$

$((L^{a¥perp})^{a})¥otimes_{A}B=((L^{a¥perp})¥mathrm{O}¥times_{A}B)^{¥alpha}$

as $B$-submodules of $(M^{*})¥otimes_{A}B=(M¥otimes_{A}B)^{¥star}$ .

Let $(X, ¥mathcal{O}_{X})$ be a reduced complex space, $¥ovalbox{¥tt¥small REJECT}_{X}$ the sheaf of germs ofmeromorphic functions on $X$ and $¥ovalbox{¥tt¥small REJECT}$ a coherent $¥mathcal{O}_{¥mathrm{X}}$-module. It follows fromthe coherence of $T$ that, for any (not necessarily $¥mathcal{O}_{¥mathrm{X}}$ -coherent) $¥mathcal{O}_{¥mathrm{X}}$-module $¥ovalbox{¥tt¥small REJECT}$ ,

the natural $¥mathcal{O}_{¥mathrm{X}}$-homomorphism

$(¥ovalbox{¥tt¥small REJECT}_{o¥ovalbox{¥tt¥small REJECT}_{x}}(¥ovalbox{¥tt¥small REJECT}, ¥ovalbox{¥tt¥small REJECT}))_{x}¥rightarrow ¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathit{0}_{X.¥mathrm{x}}}(_{¥epsilon}¥varphi_{X}, ¥ovalbox{¥tt¥small REJECT}_{x})$


is, at each $x¥in X$ , an isomorphism. Especially

$(l^{*})_{X}=(¥ovalbox{¥tt¥small REJECT}_{¥theta d¥ovalbox{¥tt¥small REJECT} ¥mathrm{p}_{¥mathrm{X}}}(l, ¥mathcal{O}_{X}))_{X}¥simeq ¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathcal{O}_{X,¥mathrm{x}}}(¥ovalbox{¥tt¥small REJECT}_{x}, ¥mathcal{O}_{X,¥mathrm{x}})=(¥Psi_{¥mathrm{X}})^{*}$

holds. We identify these $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -modules with each other and denote themby $l_{x}^{*}$ . We consider the natural $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -homomorphism $¥ovalbox{¥tt¥small REJECT}_{x}^{*}¥otimes_{o_{X,¥mathrm{x}}}¥ovalbox{¥tt¥small REJECT}_{¥mathrm{x}}¥rightarrow$

$¥mathrm{H}¥mathrm{o}¥mathrm{m}_{¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}}(¥Psi_{¥mathrm{x}}, ¥ovalbox{¥tt¥small REJECT}_{x})$ . Since the canonical $¥mathcal{O}_{¥mathrm{X},¥mathrm{x}}$ -homomorphism $(¥ovalbox{¥tt¥small REJECT}^{*}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT})_{X}¥rightarrow$

$¥ovalbox{¥tt¥small REJECT}_{x}^{*}¥otimes_{o_{X,x}}¥ovalbox{¥tt¥small REJECT}_{x}$ is always an isomorphism, for a coherent $¥mathcal{O}_{¥mathrm{X}}$-module $¥Psi$ , amorphism of $¥mathcal{O}_{¥mathrm{X}}$-modules $y*¥otimes_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥rightarrow¥ovalbox{¥tt¥small REJECT}_{a¥ovalbox{¥tt¥small REJECT}_{¥mathrm{x}}}(¥Psi, ¥ovalbox{¥tt¥small REJECT}_{X})$ is well-defined. FromLemma A.O.3, the following is induced.

Proposition A.0.5. On a reduced complex space $X$ , for any coherent$¥mathcal{O}_{X^{-}}$module $¥ovalbox{¥tt¥small REJECT}$ ,

$¥epsilon¥varphi*¥otimes_{¥mathcal{O}x}¥ovalbox{¥tt¥small REJECT}_{X}¥simeq¥ovalbox{¥tt¥small REJECT}_{¥rho¥ovalbox{¥tt¥small REJECT}}4q_{X}(l, ¥ovalbox{¥tt¥small REJECT}_{X})$ .

Let $A$ , $S$ and $B$ be as above and $M$ an A-module. If every $s¥in S$ satisfies

(A.0.6) $u¥in Msu$ $=0=u=0$then the canonical A-hommorphism $¥mu_{M}$ : $M¥rightarrow M¥mathrm{O}¥times_{A}B$ is injective. Especially,if $M$ is torsion free, it is fulfilled. If $s¥in S$ satisfies the condition (A.O.6), themthe induced A-linear map

$M¥otimes_{A}B¥rightarrow M¥mathrm{O}¥times_{A}B$

$¥mapsto$ $sv$

is an isomorphism. Let $L_{0}$ be a finitely generated A-submodule of$M¥mathrm{O}¥times_{A}B$. Take a generating system $u_{1}¥otimes s_{1}^{-1},¥cdots,u_{k}¥otimes s_{k}^{-1}$ of the A-module$L_{0}$ , where $u_{1},¥cdots,u_{k}¥in M$ and $s_{1},¥cdots,s_{k}¥in S$ . Assume $s=s_{1}¥cdots s_{k}¥in S$ satisfies (A.O.6).Then $s$ induces an A-isomorphism and the image $L_{1}$ of $L_{0}$ by this isomorphismis contained in $¥mu_{M}(M)$ . If $¥mu_{M}$ is injective, $L:=¥mu_{M}^{-1}(L_{1})¥subset M$ is isomorphic to$L_{1}$ . If each $s¥in S$ satisfies (A.O.6), it is the case. We have the following.

Lemma A.0.7. Let $M$ be an $A$-module for which every $s¥in S$ satisfies(A.0.6). Then the canonical $A$ -linear map

$¥mu_{M}$ : $M¥rightarrow M¥otimes_{A}B$

is injective. For any finitely generated $A$-submodule $L_{0}$ of $M¥otimes_{A}B$ , there existsan $A$-submodule $L$ of $M$ which is isomorphic to $L_{0}$ . If $M$ is torsion free, thecondition (A.0.6) is satisfied.

Let $l$ be an $¥mathcal{O}_{¥mathrm{X}}$-module on a reduced complex space $X$ . Assume that$¥ovalbox{¥tt¥small REJECT}$ satisfies the following condition.

At each $x¥in X$, for every $s¥in¥ovalbox{¥tt¥small REJECT} c_{x}$ ,

148 Akihiro SAEKI

(A.O.8) $u¥in l_{x}su=0=u=0$

holds.

By the above observation, any $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥swarrow¥varpi$ of $¥Psi¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ which is of$¥mathcal{O}_{¥mathrm{X}}$ -finite type is locally $-isomorphic to an $¥mathcal{O}$ -submodule of $¥ovalbox{¥tt¥small REJECT}$ . Since an$¥mathcal{O}_{¥mathrm{X}}$-submodule of an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-module is $¥mathcal{O}_{¥mathrm{X}}$-coherent if and only if it isof $¥mathcal{O}_{¥mathrm{X}}$-finite type, we have the following.

Proposition A.0.9. Let $¥Psi$ be a coherent $¥mathcal{O}_{X^{-}}$module on a reduced complexspace X. Assume that $¥Psi$ satisfies the condition $(¥mathrm{A},0.8)$ . Then any $¥mathcal{O}_{X^{-}}$

submodule $J^{¥varpi}$ of $¥ovalbox{¥tt¥small REJECT}¥otimes_{¥mathcal{O}¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}$ of $¥mathcal{O}_{X}$ -finite type is $¥mathcal{O}_{X^{-}}$ coherent. If $¥Psi$ is torsionfree, (A.0.8) is satisfied.

We review the following definitions.

Definition A.O.10. Let $(X, ¥mathcal{O}_{X})$ be an arbitrary complex space (or, moregenerally, a ringed space whose structure sheaf $¥mathcal{O}_{X}$ is coherent). For an$¥mathcal{O}_{¥mathrm{X}}$-module $¥ovalbox{¥tt¥small REJECT}$ , we define

Supp $¥ovalbox{¥tt¥small REJECT}:=¥{x¥in X|l_{x}¥neq 0¥}$ .

At each $x¥in X$, we define an ideal $(¥ovalbox{¥tt¥small REJECT}_{n_{¥epsilon}}¥varphi)_{¥mathrm{x}}$ of $¥mathcal{O}_{X,x}$ by

$(¥ovalbox{¥tt¥small REJECT}_{n}y)_{X}:=¥{a¥in ¥mathcal{O}_{X,¥mathrm{x}}|a¥cdot l_{¥mathrm{X}}=0¥}$.

These $(¥ovalbox{¥tt¥small REJECT}_{n}¥varphi)_{x}$ ’s define an $¥mathcal{O}_{¥mathrm{X}}$-ideal $¥ovalbox{¥tt¥small REJECT}_{p¥iota}¥varphi$, which is called the annihilator of $¥Psi$ .For an $¥mathcal{O}_{¥mathrm{X}}$-ideal $¥mathcal{J}$ , we define the zero set $N(¥Psi)$ by

$N(¥Psi):=$ Supp $¥mathcal{O}_{X}/¥mathcal{J}$ .

Remark. If $¥Psi$ is $¥mathcal{O}_{¥mathrm{X}}$-coherent, so is $¥ovalbox{¥tt¥small REJECT}_{n}¥varphi$ . For this, the coherence of$¥mathcal{O}_{X}$ is essential.

Supp $¥Psi=N(¥ovalbox{¥tt¥small REJECT}_{n}y)$

holds.

Let $(X, ¥mathcal{O}_{X})$ is a reduced complex space and $¥ovalbox{¥tt¥small REJECT}$ is an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-modulefor which $¥ovalbox{¥tt¥small REJECT} c$ satisfies (A.O.8). The natural $¥mathcal{O}_{¥mathrm{X}}$-morphism $¥mu_{¥ovalbox{¥tt¥small REJECT}}$ : $¥ovalbox{¥tt¥small REJECT}¥rightarrow l¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$

is injective, by which we identify $¥ovalbox{¥tt¥small REJECT}$ with the $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥mu_{¥ovalbox{¥tt¥small REJECT}}(¥ovalbox{¥tt¥small REJECT})$ of$¥ovalbox{¥tt¥small REJECT}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ . Let $J^{¥varpi}$ be an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule of $l¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ . Theinverse image $¥mu_{¥ovalbox{¥tt¥small REJECT}}^{-1}(J^{¥varpi})$ , which we often denote by $J^{¥varpi}¥cap l$ , is an $¥mathcal{O}_{¥mathrm{X}}$-coherent$¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥Psi$ . $¥mu_{¥ovalbox{¥tt¥small REJECT}}(J^{¥Gamma}¥cap l)$ , which is isomorphic to $J^{¥varpi}¥cap l$ and alsodenoted by $J^{¥varpi}¥cap¥ovalbox{¥tt¥small REJECT}$ , is an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule of $l$ $¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ containedin $J^{¥varpi}$ . Thus an $¥mathcal{O}_{¥mathrm{X}}$-module

$J^{¥varpi}/(J^{¥varpi}¥cap¥Psi)$


is well-defined and $¥mathcal{O}_{¥mathrm{X}}$-coherent.

Definition A.O. $¥mathrm{I}¥mathrm{I}$ . We define the sheaf $¥ovalbox{¥tt¥small REJECT}=¥ovalbox{¥tt¥small REJECT}_{¥varpi}$, of denominators of $J^{¥Gamma}$ by

$¥ovalbox{¥tt¥small REJECT}_{¥varpi},:=¥ovalbox{¥tt¥small REJECT} n(J^{¥varpi}/(J^{¥varpi}¥cap¥Psi))$ ,

which is an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-ideal.The polar set $P(J^{¥varpi})$ of $J^{¥varpi}$ is defined by

$P(J^{¥varpi}):=N(¥ovalbox{¥tt¥small REJECT}_{¥varpi},)=$ Supp $(J^{¥varpi}/(J^{¥Gamma}¥cap l))$ ,

which is an analytic set in $X$ .

A.1. Algebraic observations on coherent sheaves on locally irreducible complexspaces

Let $A$ be an integral domain, $S=A-0$ and $K=S^{-1}$ $A$ the quotient fieldof $A$ . By the canonical A-algebra homomorphism

(A. 1.0) $¥mu:A¥rightarrow K$,

we always consider $A$ as an A-subalgebra of $K$ .

If $M$ is a finitely generated A-module, then $M¥otimes_{A}K$ is a finite dimensionalvector space over $K$ . Since, for two $¥mathrm{K}$ -vector subspaces $T_{1}$ and $T_{2}$ of $M¥mathrm{O}¥times_{A}K$,

$T_{1}=T_{2}=(T_{1})^{¥alpha}=(T_{2})^{a}$

holds, Lemma A.0.4 induces the following.

Lemma A.1.1. $A$ and $K$ are as above. For any finitely generated $A$ -module$M$ and any $A$-submodule $L$ of $M$,

$L¥otimes_{A}K=(L^{a¥perp})¥mathrm{O}¥times_{A}K$ (as $K$ -vector subspaces of $M¥otimes_{A}K$)

and

$L_{K}=$$L^{a¥perp}$ (as $A$-submodules of $M$)

holds.

Thus we have

Theorem 2.6.1. On a locally irreducible complex space $X$, for any coherent$¥mathcal{O}_{X^{-}}$ module $¥Psi$ ,

$¥Psi_{¥ovalbox{¥tt¥small REJECT}}=¥Psi^{a1}$

holds. (cf. Definition 2.3.2 and Definition 2.4.3.)

Let $M$ be an A-module. Since $A$ is an integral domain,

150 Akihiro SAEKI

(A.1.3) $M_{r}:=$ { $u¥in M|su$ $=0$ for some $0¥neq a¥in A$ }

is an A-submodule of $M$ . We denote the quotient module $M/M_{t}$ by$M_{Q}$ . Thus we have an exact sequence of A-modules:

(A. 1.2) $0¥rightarrow M_{¥mathrm{r}}¥rightarrow M¥rightarrow M_{Q}¥rightarrow 0$

Since $K$ is A-fiat, the exact sequence (A.1.3) induces the following commutativediagram whose rows $¥mathrm{r}¥mathrm{e}$ exact.

(A.1.4)

$ 0¥rightarrow$

$ M_{t}¥downarrow$

$¥rightarrow$

$ M¥downarrow$

$¥rightarrow$

$ M_{Q}¥downarrow$

$¥rightarrow 0$

0 $M_{t}¥otimes_{A}KM¥otimes_{A}KM_{Q}¥otimes_{A}K¥mathit{0}$

From the definition, we have the following.

Lemma A.1.5. The $A$-module $M_{t}$ is the kernel of the canonical$A$-homomorphism $¥mu_{M}$ : $M¥rightarrow M¥mathrm{O}¥times_{A}K$ and we have a commutative diagram:

$M_{t}=¥mu_{M}^{-1}(0)$

0$¥downarrow$

$M¥otimes_{A}M¥downarrow 1^{Q}¥rightarrow MK¥rightarrow M_{Q}¥mathrm{O}¥times_{A}K¥rightarrow 0$

We denote, as usual, by $M^{*}$ and $M^{**}$ the dual of $M$ and the bidual of$M$, respectively:

$M^{*}:=¥mathrm{H}¥mathrm{o}¥mathrm{m}_{A}(M, A)$

and

$M^{**}:=¥mathrm{H}¥mathrm{o}¥mathrm{m}_{A}(M^{*}, A)$

There exists a natural A-homomorphism

$¥tau_{M}$ : $M¥rightarrow M^{**}$ ,

whose kernel we denote by $M_{Z}$ :

$M_{Z}:=¥tau_{M}^{-1}(0)$

Lemma A.1.5.

0) $M_{t}¥subset M_{Z}$

1) $M_{t}^{*}=0$


Proof. 0) Let $u¥in M_{t}$ . Assume that $0¥neq a¥in A$ and that $au=0$. For any$¥varphi¥in M^{*}$ , $a¥langle u, ¥varphi¥rangle=0$ holds. Since $A$ is an integral domain and $a¥neq 0$, wehave $¥langle u, ¥varphi¥rangle=0$ . Thus $u¥in M_{Z}$ . 1) is proved similarly. $¥blacksquare$

The exact sequence (A.I.3) induces the following exact sequence.

$M_{t}^{*}-M^{*}-M_{Q}^{*}-0$

The following is induced from Lemma A.1.6.

Lemma A.1.7. The canonical surjection $M¥rightarrow M_{Q}$ induces $(M_{Q})^{*}¥simeq M^{*}$ .

We have the following commutative diagram of the canonical $A$ -homomorphism.

$M^{**}M¥downarrow¥downarrow¥rightarrow M_{Q}¥simeq(M_{Q})^{**}¥rightarrow 0$

Actually, we have

Lemma A.1.8. If $M$ is finitely generated over $A$ , then

$M_{t}=M_{Z}$

and

$M_{Q}¥rightarrow(M_{Q})^{**}$

is injective.

To prove these assertions, the following lemma is useful.

Lemma A.1.9. Let $M$ be a finitely generated $A$ -module. If $M$ is torsionfree, then $M$ is isomorphic to an $A$-submodule of some free $A¥_ module$

$A^{n}$ . Moreover, we can take $n$ so that the following commutative diagram ofthe canonical $A$-homomorphisms exists:

0 0

$ 0¥rightarrow$ $¥mu MM¥downarrow¥downarrow$ $¥rightarrow A^{n}=¥bigoplus_{¥downarrow}^{¥downarrow}-jn=1Ae_{j}$

$M¥otimes_{A}K¥simeq K^{n}=¥oplus_{j=1}^{n}Ke_{j}$

Proof. Take a generating system $u_{1},¥cdots,u_{n}$ , $ v_{1},¥cdots$ , $v_{k}$ of $M$ so that $u_{1},¥cdots,u_{n}$

are linearly independent over $A$ and that, for any $i=1,¥cdots,k$, $u_{1},¥cdots,u_{n}$ , $v_{i}$ arelinearly dependent. There are non-trivial linear relations over $A$ :

152 Akihiro SAEKI

$¥alpha_{i}v_{i}=¥sum_{j=1}^{n}¥gamma_{i}^{j}u_{j}$ $(i =1,¥cdots,k)$ ,

where $¥alpha_{i}¥neq 0$ . Since $A$ is an integral domain, $¥alpha:=¥prod_{i=1}^{k}¥alpha_{i}¥neq 0$ and an $¥mathrm{K}$ -linearmap $¥varphi:M¥rightarrow A^{n}=¥oplus_{j=1}^{n}Ae_{j}$ is defined as follows.

$¥varphi$

$M¥rightarrow A^{n}=¥oplus_{j=1}^{n}Ae_{j}$

$ u_{j}¥mapsto$ $¥alpha e_{j}$ $(j =1,¥cdots,n)$ ,

$ v_{i}¥mapsto$ $¥sum_{j=1}^{n}¥beta_{i}^{j}e_{j}$ $(i =1,¥cdots,k)$

where $¥beta_{l}^{j}:=¥alpha_{i}^{-1}¥alpha¥gamma_{i}^{j}$ for $i=1,¥cdots,k$ and $j=1$ , $¥cdots,n$ . Since $u_{1},¥cdots,u_{n}$ are linearlyindependent over $A$ , $¥varphi:M¥rightarrow A^{n}$ is injective. $¥blacksquare$

Proof of Lemma A.I.8. Since $M_{Q}$ is torsion free we have an injective$¥mathrm{A}$ -homomorphism $M_{Q}¥rightarrow A^{n}¥rightarrow K^{n}$ . For any $0¥neq v¥in M_{Q}¥subset K^{n}$ , there is a$¥mathrm{K}$ -linear map $¥Phi¥in(K^{n})^{¥star}=(A^{n})^{*}¥mathrm{O}¥times_{A}K¥supset(A^{n})^{*}$ satisfying $¥Phi(v)¥neq 0$ and $¥Phi(A^{n})¥subset$

$A¥subset K$ . Restricting $¥Phi$ to $M_{Q}$ , we have $¥varphi¥in(M_{Q})^{*}$ such that $¥varphi(v)¥neq 0$ . $¥blacksquare$

Thus we have

Proposition A.I.IO.Let $A$ be an integral domain, $K$ the quotient field of $A$ and $M$ finitely

generated $A$-module. We have

$M_{t}=¥mu_{M}^{-1}(0)=¥tau_{M}^{-1}(0)$

and the following commutative diagrams of the canonical $A$-homomorphisms.

$¥downarrow 0$$¥downarrow 0$

$M¥otimes_{A}M¥downarrow¥downarrow^{Q}¥rightarrow MK¥simeq M_{Q}¥otimes_{A}K¥rightarrow 0$ $M^{**}M¥downarrow¥rightarrow¥simeq(M_{Q}M_{Q}¥downarrow)^{**}¥rightarrow 0$

Let $(X, ¥mathcal{O}_{X})$ be a locally irreducible complex space.

Definition A.l.ll.For an $¥mathcal{O}_{¥mathrm{X}}$-module $¥Psi$ , we define an $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥Psi_{t}$ of $l$ by

At $¥mathrm{x}¥in X$ ,

$(¥Psi_{t})_{x}:=$ { $u¥in¥Psi_{x}|su=0$ for some $0¥neq a¥in ¥mathcal{O}_{X,x}$}.

We define an $¥mathcal{O}_{¥mathrm{X}}$-module $l_{Q}$ by

$l_{Q}:=¥Psi/¥Psi_{t}$ .


By Proposition A.1.10, we have

Proposition A.1.12. For an $¥mathcal{O}_{X^{-}}$coherent $¥mathcal{O}_{X^{-}}$module $l$ on a loclly irreduciblecomplex space $X$,

$¥Psi_{t}=¥mu_{¥ovalbox{¥tt¥small REJECT}}^{-1}(0)=¥tau_{¥ovalbox{¥tt¥small REJECT}}^{-1}(0)$ ,

where $¥mu_{¥ovalbox{¥tt¥small REJECT}}$ : $¥ovalbox{¥tt¥small REJECT}¥rightarrow¥ovalbox{¥tt¥small REJECT}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ and $¥tau_{¥ovalbox{¥tt¥small REJECT}}$ : $¥Psi¥rightarrow¥ovalbox{¥tt¥small REJECT}^{**}and$ the canonical $¥mathcal{O}_{¥mathrm{X}}$-morphisms.Thus $¥ovalbox{¥tt¥small REJECT}_{t}$ and $¥ovalbox{¥tt¥small REJECT}_{Q}$ are also $¥mathcal{O}_{X^{-}}$coherent and we have the following commutative

diagrams of the canonical $¥mathcal{O}_{X^{-}}$morphisms.

$¥downarrow 0$ $¥downarrow 0$

$¥Psi¥downarrow$

$¥rightarrow$

$¥Psi_{Q}¥downarrow$

$¥rightarrow 0$

$¥ovalbox{¥tt¥small REJECT}¥downarrow¥rightarrow$ $¥epsilon_{¥mathrm{I}}r_{Q}$

$¥rightarrow 0$

$¥ovalbox{¥tt¥small REJECT}¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥simeq¥nu_{Q}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ $l^{**}¥simeq(¥ovalbox{¥tt¥small REJECT}_{Q})^{**}$

From this and Proposition A.0.9, the following is induced.

Proposition A.1.13. Let $(X, ¥mathcal{O}_{X})$ be a locally irreducible complex space and$¥Psi$ an $¥mathcal{O}_{X^{-}}$coherent $¥mathcal{O}_{X^{-}}$ module. An $¥mathcal{O}_{X^{-}}$submodule $J^{¥varpi}ofl$ $¥mathrm{O}¥times_{¥mathit{0}¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}$ is $¥mathcal{O}_{X^{-}}$ coherentif and only if $J^{¥varpi}$ is of $¥mathcal{O}_{X}$ -finite type. If it is the case, $¥mu_{¥ovalbox{¥tt¥small REJECT}}^{-1}(J^{¥varpi})$ is an $¥mathcal{O}_{X^{-}}$ coherent$¥mathcal{O}_{X^{-}}$submodule of $¥ovalbox{¥tt¥small REJECT}$ .

Let $¥ovalbox{¥tt¥small REJECT}$ be a coherent-module on an arbitrary complex space $(X, ¥mathcal{O}_{X})$

and $¥xi_{1},¥cdots$ , $¥xi_{n}¥in¥ovalbox{¥tt¥small REJECT}_{¥mathrm{x}}$ at $x¥in X$ . There is an open neighbourhood $U$ of $x$ , where$¥xi_{j}$’s have their representatives $¥xi_{jU}¥in¥ovalbox{¥tt¥small REJECT}(U)$ , $j=1,¥cdots,n$ . We define a morphismof $¥mathcal{O}_{¥mathrm{U}}$-morphisms $¥Phi:¥oplus_{j=1}^{n}¥mathcal{O}_{U}e_{j}=¥mathcal{O}_{U}^{n}¥rightarrow¥ovalbox{¥tt¥small REJECT}|_{U}$ by $¥Phi(e_{j}):=¥xi_{jU}$ . Let $¥swarrow¥varpi$ be thekernel of $¥Phi:¥mathcal{O}_{U}^{n}¥rightarrow¥ovalbox{¥tt¥small REJECT}|_{U}$. From the coherence of $l$ , the $¥mathcal{O}_{¥mathrm{U}}$-coherence of $J^{¥Gamma}$

follows. Thus we have

Lemma A.1.14.

Supp $J^{¥Gamma}$

$=$ { $ y¥in U|(¥xi_{1U})_{y},¥cdots$ , $(¥xi_{nU})_{y}¥in l_{y}$ are linearly dependent over $¥mathcal{O}_{X,y}$ }

is an analytic set in U. If $¥xi_{1},¥cdots$ , $¥xi_{n}¥in¥Psi_{x}$ are linearly independent over $¥mathcal{O}_{X,x}$ ,then there is an open neighbourhood $V¥subset U$ of $¥mathrm{x}$ such that $(¥xi_{1U})_{y},¥cdots$ , $(¥xi_{nU})_{y}¥in¥Psi_{y}$

are linearly independent over $¥mathcal{O}_{X,y}$ at any $y¥in V$.

Assume $¥xi_{1},¥cdots$ , $¥xi_{n}¥in¥Psi_{x}$ generate the $¥mathcal{O}_{X,x^{-}}$module $¥epsilon¥varphi_{x}$ at $x¥in X$ . Since $¥Psi$ is$¥mathcal{O}_{X^{-}}$ coherent, we can take an open neighbourhood $U$ of $x$ so that $¥Phi$ induces anexact sequence of $¥mathcal{O}_{U^{-}}$modules

$¥otimes_{j=1}^{n}¥mathcal{O}_{U}e_{j}=¥mathcal{O}_{U^{¥rightarrow}}^{n¥Phi}¥ovalbox{¥tt¥small REJECT}|_{U}¥rightarrow 0$ .

154 Akihiro SAEKI

From Lemma A.1.9, the following is induced.

Proposition A.1.15. Let $¥ovalbox{¥tt¥small REJECT}$ be a coherent torsion free $¥mathcal{O}_{X^{-}}$module on $a$

locally irreducible complex space $(X, ¥mathcal{O}_{X})$ . At each $x¥in X$ , there exist an openneighbourhood $U$ of $¥mathrm{x}$ and the following commutative diagram of $¥mathcal{O}_{U^{-}}$moduleswhose columns and rows are exact.

$¥downarrow 0$ $¥downarrow 0$

$ 0¥rightarrow$

$l¥downarrow|_{U}$ $¥rightarrow ¥mathcal{O}_{U}^{n}¥downarrow$

$0¥rightarrow¥ovalbox{¥tt¥small REJECT} ¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}|_{U}¥rightarrow ¥mathcal{O}_{U}^{n}$

A.2. Coherent sheaves of rank 1Let $(X, ¥mathcal{O}_{X})$ be a locally irreducible complex space and $¥ovalbox{¥tt¥small REJECT}$ an $¥mathcal{O}_{¥mathrm{X}}$-coherent

$¥mathcal{O}_{¥mathrm{X}}$-module.

Definition A.2.0.For each $x¥in X$, the rank of $¥Psi$ at $x$ is defined as follows: Take a connected

open neighbourhood $U$ of $¥mathrm{x}$ . Then $ U¥cap$ Sing $¥ovalbox{¥tt¥small REJECT}=$ Sing $(¥ovalbox{¥tt¥small REJECT}|_{U})$ is a thin analyticset in $U$ and $U-$ Sing $¥Psi$ is connected. Thus the rank of locally free $¥mathcal{O}$ -module$¥Psi|_{U-¥mathrm{S}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥ovalbox{¥tt¥small REJECT}}$ on the connected complex space $U-$ Sing $¥Psi$ is well-defined andindependent of the choice of the connected open neighbourhood $U$ of $x$ . Thisis the rank of $¥Psi$ at $x$ . It is a constant on each connected component of$X$ . If it is a constant $n$ on $X$ , we call $l$ of rank $n$ .

Proposition A.2.1.Let $¥Psi$ be a torsion free coherent $¥mathcal{O}_{X^{-}}$ module on a locally irreducible complex

space X. Assume $A$ is an analytic set in $X$ thin of order 2 containingSing X. For any invertible subsheaf $J_{0}^{¥varpi}$ of $¥epsilon¥Psi|_{X-A}$ , there exists a coherent$¥mathcal{O}_{X^{-}}$ submodule $J^{¥varpi}$ of $¥ovalbox{¥tt¥small REJECT}$ (on $X$ and of rank 1) such that

$J^{¥varpi}|_{X-(A¥cup S(J_{0}^{¥varpi}))}=J_{0}^{¥Gamma}|_{X-(A¥cup S(J_{¥mathrm{o}}^{¥varpi}))}$

and that, for every $x¥in X$, there is an open neighbourhood $U$ of $¥mathrm{X}$ where

$J^{¥varpi}¥otimes_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}|_{U}¥simeq¥ovalbox{¥tt¥small REJECT}_{U}$

holds.

Proof. Let $x¥in A$ . It follows from Proposition A.1.5 that $¥Psi|_{U}$ isconsidered as a coherent $¥mathcal{O}_{¥mathrm{U}}$-submodule of $¥mathcal{O}_{U}^{n}$ on an open neighbourhood$U¥subset X$ of $x$ . Note that $¥ovalbox{¥tt¥small REJECT}_{X}$ is $¥mathcal{O}_{¥mathrm{X}}$ -flat and $¥Psi¥rightarrow¥ovalbox{¥tt¥small REJECT} ¥mathrm{O}¥times_{¥mathcal{O}¥mathrm{x}}¥ovalbox{¥tt¥small REJECT}_{X}$ is injective. Weconsider


$¥Psi|_{U}$ $¥subset ¥mathcal{O}_{U}^{n}$

$¥cap$ $¥cap$

$(¥ovalbox{¥tt¥small REJECT} ¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X})|_{U}¥subset¥ovalbox{¥tt¥small REJECT}_{U}^{n}$

If a coherent $¥mathcal{O}_{¥mathrm{U}}$-submodule $J_{1}^{¥Gamma}$ of $¥mathcal{O}_{U}^{n}$ satisfies $J_{1}^{¥varpi}|_{U-(A¥cup S(J_{0}^{¥varpi}))}=J_{0}^{¥Gamma}|_{U-(A¥cup S(J_{0}^{¥varpi}))}$ ,

then $J_{2}^{¥varpi}:=(J_{1}^{¥Gamma}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X})¥cap(l|_{U})$ is a coherent $¥mathcal{O}_{¥mathrm{U}}$-submodule of $¥ovalbox{¥tt¥small REJECT}|_{U}$ satisfying$J_{2}^{¥varpi}|_{U-(A¥cup S(J_{¥mathrm{O}}^{¥varpi}))}=J_{0}^{¥varpi}|_{U-(A¥cup S(J^{¥Gamma}¥mathrm{o}))}$ . Such $J_{2}^{¥varpi}$ ’s define a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule $¥swarrow¥varpi$

of $¥Psi$ satisfiying the conditions of Proposition A.2.1. Thus we may assumethat $U=X$ is connected and that $¥Psi=¥mathcal{O}_{X}^{n}$ . There is an open covering$X-A=¥bigcup_{¥alpha}X_{¥alpha}$ with $J_{0}^{¥varpi}|_{X_{¥alpha}}=¥mathcal{O}_{X_{¥alpha}}¥eta_{a}$ where

$¥eta_{a}=(¥eta_{¥alpha 1},¥cdots,¥eta_{¥alpha n})¥in J_{0}^{¥varpi}(X_{¥alpha})¥subset ¥mathcal{O}_{X}^{n}(X_{a})=¥mathcal{O}_{X}(X_{¥alpha})^{n}$ .

Since that $J_{0}^{¥varpi}$ is invertible and that $X-A$ is connected, $¥eta_{¥alpha j}=0$ in $¥mathcal{O}_{X}(X_{a})$ forall $¥alpha$ if $¥eta_{aj}=0$ in $¥mathcal{O}_{X}(X_{a})$ for an $¥alpha$ . We may assume $¥eta_{a1}¥neq 0$ in $¥mathcal{O}_{X}(X_{¥alpha})$ forall $¥alpha$ . For all $¥alpha$ ,

$¥zeta_{a}:=¥eta_{a1}^{-1}¥eta_{a}=(1,¥cdots,¥eta_{a1}^{-1}¥eta_{aj},¥cdots,¥eta_{¥alpha 1}^{-1}¥eta_{an})¥in/_{0}^{¥varpi}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}(X_{a})¥subset¥ovalbox{¥tt¥small REJECT}_{X}(X_{¥alpha})^{n}$

is well-defined and, if $ X_{a}¥cap X_{¥beta}¥neq¥emptyset$ , $¥zeta_{¥alpha}|_{X_{¥alpha}¥mathrm{n}X_{¥beta}}=¥zeta_{¥beta}|_{X_{¥alpha}¥mathrm{n}X_{¥beta}}$ . Thus we have

$¥zeta_{X-A}=(1,¥cdots, ¥zeta_{X-A,j},¥cdots, ¥zeta_{X-A,n})¥in J_{0}^{¥varpi}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}(X-A)¥subset¥ovalbox{¥tt¥small REJECT}_{X}(X-A)$ .

Since $X$ is irreducible and $A$ is thin of order 2, $¥zeta_{X-A}$ extends (uniquely) to

$¥zeta=(1,¥cdots,¥zeta_{j},¥cdots,¥zeta_{n})¥in¥ovalbox{¥tt¥small REJECT}_{X}(X)^{n}$ .

It follows from Proposition A.1.13 that $¥mathcal{O}_{X}¥zeta$ is an $¥mathcal{O}_{¥mathrm{X}}$-coherent $¥mathcal{O}_{¥mathrm{X}}$-submoduleof $¥ovalbox{¥tt¥small REJECT}_{X}^{n}=¥Psi¥otimes_{o_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$ and that

$ J^{¥varpi}:=(q¥zeta)¥cap¥Psi$

is a coherent $¥mathcal{O}_{¥mathrm{X}}$-submodule of $¥ovalbox{¥tt¥small REJECT}$ with required properties. $¥blacksquare$

Proposition A.2.2. Let $(X, ¥mathcal{O}_{X})$ be a localy factorial complex space and $¥Psi$

a locally free $¥mathcal{O}_{X^{-}}$ module of finite rank $n$ . If an $¥mathcal{O}_{X^{-}}$ submodule $J^{¥varpi}$ of $¥ovalbox{¥tt¥small REJECT}$ iscoherent of rank 1 and full in $¥Psi$ $ then¥swarrow¥varpi$ is invertible.

Proof. This is of local property. Let $x¥in X$ and take an openneighbourhood $U$ of $x$ . We may assume that $X=U$ is connected, that$¥Psi=¥mathcal{O}_{X}^{n}$ and that $J^{¥varpi}=¥sum_{i=1}^{m}¥mathcal{O}_{X}¥eta_{i}$ with $0¥neq¥eta_{i}¥in J^{¥varpi}(X)$ . We consider that

$J^{¥varpi}$$¥subset ¥mathcal{O}_{X}^{n}=$ $¥ovalbox{¥tt¥small REJECT}$

$¥cap$$¥cap$

$J^{¥varpi}¥mathrm{O}¥times_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}¥subset¥ovalbox{¥tt¥small REJECT}_{X}^{n}=¥Psi ¥mathrm{O}¥times_{¥mathcal{O}_{X}}¥ovalbox{¥tt¥small REJECT}_{X}$

and that

156 Akihiro SAEKI

$¥eta_{i}=(¥eta_{i1},¥cdots,¥eta_{in})¥in J^{¥varpi}(X)¥subset l(X)=¥mathcal{O}_{X}^{n}(X)¥subset ¥mathcal{O}_{X}(X)^{n}$ .

Note that $A=$ Sing $¥swarrow¥varpi$ is a thin analytic set in $X$ , which we prove to beempty. Since $J^{¥Gamma}|_{X-A}$ is invertible on the connected complex space $X-A$,$¥eta_{ij}=0$ in $¥mathcal{O}_{X}(X)$ for an $i$ if and only if $¥eta_{ij}=0$ in $¥mathcal{O}_{X}(X)$ for all $i$ . We mayassume that $¥eta_{ij}¥neq 0$ in $¥mathcal{O}_{X}(X)$ for $i=1,¥cdots,m$ and $ j=1,¥cdots$ , $k$ and that $¥eta_{ij}=0$

in $¥mathcal{O}_{X}(X)$ for $i=1,¥cdots,m$ and $ j=k+1,¥cdots$ , $n$ . We consider the germs of $¥eta_{i}$ ’sat $x$ , which we also denote by $¥eta_{i}$ ’s:

$¥eta=(¥eta_{i1},¥cdots,¥eta_{in})¥in J_{x}^{¥varpi}¥subset¥Psi_{x}=¥mathcal{O}_{X,x}^{n}$ $i=1,¥cdots,m$

$¥eta_{ij}¥neq 0$ in $¥mathcal{O}_{X,x}$ for $i=1,¥cdots,m$ and $j=1,¥cdots,k$ and

$¥eta_{ij}=0$ in $¥mathcal{O}_{X,x}$ for $i=1,¥cdots,m$ and $j=k+1,¥cdots,n$ .

Since $X$ is locally factorial, $¥mathcal{O}_{X,x}$ is a U.F.D. and we have the G.C.M. $d_{i}$ of$¥eta_{i1},¥cdots,¥eta_{in}¥in ¥mathcal{O}_{X,x}$ for $i=1,¥cdots,m$. Note that $d_{i}¥in¥ovalbox{¥tt¥small REJECT} c_{¥mathrm{x}}$ . $¥zeta_{i}:=d_{i}^{-1}¥eta_{i}¥in((/^{¥varpi}¥mathrm{O}¥times_{¥mathit{0}_{X}}¥ovalbox{¥tt¥small REJECT}_{X})$

$¥cap¥ovalbox{¥tt¥small REJECT})_{X}$ is welldefined for $i=1,¥cdots,m$. Since $A=$ Sing $J^{¥varpi}$ is thin, all $¥zeta_{i}$ ’s coincidewith one another, which we denote by $¥zeta$ . Thus $¥eta_{i}=d_{i}¥zeta$ and $J^{¥varpi}|_{V}¥subset¥Psi|_{V}$ as$¥mathcal{O}_{¥mathrm{V}}$-submodule of $¥Psi|_{V}$ on an open neighbourhood $V¥subset U$ of $x$ . The polar set$P:=P(d_{i}^{-1})$ of the meromorphic function $d_{i}^{-1}¥in¥ovalbox{¥tt¥small REJECT}_{X}(V)$ is a thin analytic setin $V$ and $¥zeta|_{V-P}¥in/^{¥varpi}(V-P)$ . Since $J^{¥varpi}$ is full in $¥ovalbox{¥tt¥small REJECT}$ , $¥zeta¥in/^{¥varpi}(V)$ holds. Thus$ J^{¥varpi}|_{V}=¥mathcal{O}_{V}¥zeta$ . $¥blacksquare$

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nuna adreso:Department of MathematicsNagoya Institute of TechnologyShowa-ku, Nagoya 466Japan

(Ricevita la 10-an de marto, 1993)

mathfe.math.kobe-u.ac.jp/fe/fe_pdf_with_bookmark/fe35-40-en...funkcialajekvacioj, 38 (1995) 121-157...

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