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GENERALIZED ZARISKI-VAN KAMPEN THEOREM AND ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES

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ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES

ICHIRO SHIMADA

Dedicated to the memory of Professor Nguyen Huu Duc

Abstract. We formulate and prove a generalization of Zariski-van Kam- pen theorem on the topological fundamental groups of smooth complex al- gebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz-Zariski-van Kampen type for the fundamental groups of the com- plements to the Grassmannian dual varieties.

1. Introduction

We work over the complex number field C. By a variety, we mean a reduced irreducible quasi-projective scheme. The fundamental group π1(V) of a variety V is the topological fundamental group of the analytic space underlying V. The conjunction of paths is read from left to right; that is, for pathsα:I:= [0,1]→V andβ:I→V, we defineαβ:I→V only whenα(1) =β(0).

For a subsetS of a groupG, we denote by 〈S〉the subgroup ofGgenerated by the elements ofS. Let a group Γ act onGfrom the right. Then the subgroup

NΓ:=〈 {g1gγ|g∈G, γ∈Γ} 〉

ofGis normal, becauseh1(g1gγ)h= ((gh)1(gh)γ)(h1hγ)1. We then put G//Γ :=G/NΓ,

and callG//Γ theZariski-van Kampen quotient ofGby Γ.

Letf :X →Y be a dominant morphism from a smooth varietyX to a smooth variety Y with a connected general fiber. There exists a non-empty Zariski open subsetY ⊂Y such thatf is locally trivial in theC-category overY. We put X:=f1(Y), and denote byf:X→Ythe restriction off toX. We choose a base pointb∈Y, putFb :=f1(b), and choose a base point ˜b∈Fb.

We investigate the kernel of the homomorphism ι : π1(Fb,˜b) π1(X,˜b)

induced by the inclusionι:Fb ,→X. The classical Zariski-van Kampen theorem, which started from [29], describes Ker(ι) in terms of the monodromy action of π1(Y, b) onπ1(Fb,˜b)under the assumption that a cross-section off passing through

˜b exists. The cross-section plays a double role; one is to define the monodromy action ofπ1(Y, b) onπ1(Fb,˜b), and the other is to preventπ2(Y) from contributing

2000Mathematics Subject Classification. 14F35, 14D05.

Partially supported by JSPS Core-to-Core Program 18005: “New Developments of Arithmetic Geometry, Motive, Galois Theory, and Their Practical Applications”.

1

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to Ker(ι). However, the cross-section rarely exists in applications. If we do not have any cross-section, then the monodromy of π1(Y, b) on π1(Fb) is not well- defined, and moreoverπ2(Y) may contribute to Ker(ι). (See Example 3.4.)

In this paper, we give a generalization of Zariski-van Kampen theorem (Theo- rem 3.20), which describes Ker(ι) under weaker conditions on the existence of the cross-section. Informally, our theorem states that, if there exists a cross-section on a subspace ofY whoseπ2surjects toπ2(Y), then, under additional assumptions on the singular fibers of f, Ker(ι) is generated by the monodromy relations arising from thelifted monodromy, which is defined as follows.

Sincef:X→Yis locally trivial, the groupsπ1(f1(f(x)), x) form a locally constant system onX whenxmoves onX, and henceπ1(X,˜b) acts onπ1(Fb,˜b) from the right in a natural way. We denote this action by

(1.1) µ : π1(X,˜b) Aut(π1(Fb,˜b)), and callµthelifted monodromy.

Combining our main result with Nori’s lemma [14] (see Proposition 3.1), we obtain the following:

Corollary 1.1. Suppose that the following three conditions are satisfied:

(C1) the locusSing(f)of critical points off is of codimension≥2 inX, (C2) there exists a Zariski closed subsetΞ0 ofY with codimension≥2such that

Fy:=f1(y) is non-empty and irreducible for anyy∈Y \Ξ0, and (Z) there exist a subspace Z ⊂Y containingb and a continuous cross-section

sZ :Z→f1(Z)off overZ satisfyingsZ(Z)Sing(f) = andsZ(b) = ˜b such that the inclusionZ ,→Y induces a surjectionπ2(Z, b)→→π2(Y, b).

Let iX : π1(X,˜b) π1(X,˜b) be the homomorphism induced by the inclusion iX :X,→X. ThenKer(ι) is equal to

(1.2) R:=〈 {g1gµ(γ)|g∈π1(Fb,˜b), γ∈Ker(iX)} 〉, and we have the exact sequence

1 −→ π1(Fb,˜b)//Ker(iX) −→ι π1(X,˜b) −→f π1(Y, b) −→ 1.

Remark1.2. The condition (Z) is trivially satisfied ifπ2(Y) = 0; for example, when Y is an affine spaceAN, an abelian variety, or a Riemann surface of genus>0.

In our previous papers [17], [23] and [24], we have given three different proofs to a special case of Theorem 3.20, where Y is an affine space AN. Even this special case has yielded many applications ([16, 18, 19, 20, 21, 22, 25]). Thus we can expect more applications of the generalized Zariski-van Kampen theorem of this paper.

As an easy application, we obtain the following:

Corollary 1.3. Letf :X →Y be a morphism from a smooth varietyXto a smooth variety Y. Suppose that π2(Y) = 0, that f is projective with the general fiberFb

being connected, and thatSing(f) is of codimension≥3 inX. Letι:Fb,→X be the inclusion. Then the sequence

1 −→ π1(Fb) −→ι π1(X) −→f π1(Y) −→ 1 is exact.

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As the next application, we investigate the fundamental group of the comple- ment of theGrassmannian dual variety, and prove a hyperplane section theorem of Zariski-Lefschetz-van Kampen type.

A Zariski closed subset of a projective spacePN is said to benon-degenerate if it is not contained in any hyperplane of PN. We denote by Grc(PN) the Grass- mannian variety of (N −c)-dimensional linear subspaces of PN. For a point t∈(PN)= Gr1(PN) of the dual projective space, letHtPN denote the corre- sponding hyperplane.

LetW be a closed subscheme ofPN such that every irreducible component is of dimensionn. Forc≤n, theGrassmannian dual variety ofW inGrc(PN) is defined to be the locus of L∈ Grc(PN) such that the scheme-theoretic intersection ofW and the linear subspaceL⊂PN fails to be smooth of dimensionn−c. For a non- negative integerk, we denote byUk(W,PN) the complement of the Grassmannian dual variety of W in Grnk(PN); that is, Uk(W,PN) Grnk(PN) is the Zariski open subset of all L Grnk(PN) that intersect W along a smooth scheme of dimensionk.

LetX⊂PN be a smooth non-degenerate projective variety of dimensionn≥2.

The fundamental groupπ1((PN)\X) =π1(Un1(X,PN)) of the complement of the dual variety has been studied in several papers (for example, [3, 4]). However, there seem to be few studies on its generalization to Grassmannian varieties. We will investigate the fundamental groupsπ1(Uk(X,PN)) fork= 0, . . . , n−2.

We choose a general line Λ in (PN), and consider the corresponding pencil {Ht}tΛ of hyperplanes. LetA:=∩

Ht=PN2 denote the axis of the pencil. We put

Yt:=X∩Ht and ZΛ:=X∩A.

Let k be an integer such that 0 k n−2. Regarding Grc1(Ht) as a closed subvariety of Grc(PN), and Grc2(A) as a closed subvariety of Grc1(Ht), we have canonical inclusions

Uk(ZΛ, A) ,→ Uk(Yt, Ht) ,→ Uk(X,PN).

Since k≤n−2, the spaceUk(ZΛ, A) is non-empty. (Whenk=n−2, the space Un2(ZΛ, A) is equal to the one-point set Gr0(A) ={A}.) We choose a base point

Lo Uk(ZΛ, A),

which serves also as a base point of Uk(X,PN) and of Uk(Yt, Ht) by the natural inclusions above. Consider the space

Uk(X,PN,Λ) :={(L, t)∈Uk(X,PN)×Λ | L⊂Ht} with the projection

fΛ : Uk(X,PN,Λ) Λ.

The fiber offΛ overt Λ is canonically identified withUk(Yt, Ht), and the point Lo furnishes us with a holomorphic section

so : Λ → Uk(X,PN,Λ)

offΛ. There exists a proper Zariski closed subset ΣΛ of Λ such that fΛ is locally trivial over Λ\ΣΛ in theC-category. We choose a base point 0Λ\ΣΛ. By the sectionso, the fundamental group π1\ΣΛ,0) acts onπ1(Uk(Y0, H0), Lo) in the classical (not lifted) monodromy.

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Using the fact that Λ,→(PN)induces an isomorphismπ2(Λ)=π2((PN)), we derive from Theorem 3.20 the following:

Theorem 1.4. Consider the homomorphism

ι : π1(Uk(Y0, H0), Lo) π1(Uk(X,PN), Lo) induced by the inclusionι:Uk(Y0, H0),→Uk(X,PN).

(1)If k≤n−2, thenι is surjective and induces an isomorphism π1(Uk(Y0, H0), Lo)// π1\ΣΛ,0) →∼ π1(Uk(X,PN), Lo).

(2) If k < n−2, the monodromy action ofπ1\ΣΛ,0) onπ1(Uk(Y0, H0), Lo) is trivial. In particular, the homomorphismι is an isomorphism fork < n−2.

Remark that this theorem resembles the classical Lefschetz hyperplane section theorem on the homotopy groups of smooth projective varieties: namely, the inclu- sion Y0 ,→X induces surjective homomorphismsπk(Y0)→→πk(X) for k≤n−1, and isomorphismsπk(Y0) →∼ πk(X) fork < n−1.

The isomorphism in the assertion (2) of Theorem 1.4 seems to fail to hold for k=n−2, as can be seen from the argument in§6 of this paper.

As the third application, we studyπ1(Uk(X,PN), Lo) fork= 0. By Theorem 1.4, it is enough to investigate the case where dimX = 2, and to study the monodromy action of π1\ ΣΛ,0) on π1(U0(Y0, H0), Lo), where Y0 = X ∩H0 is a smooth compact Riemann surface.

First we define the simple braid groupSBdg ofdstrings on a compact Riemann surfaceC of genusg >0. We denote by Divd(C) the variety of effective divisors of degreed onC, and by rDivd(C)Divd(C) the Zariski open subset consisting of reduced divisors. We fix a base point

D0=p1+· · ·+pd

of rDivd(C). The braid group Bgd = B(C, D0) is defined to be the fundamental groupπ1(rDivd(C), D0). (See [2].)

Definition 1.5. Thesimple braid group SBdg = SB(C, D0) is defined to be the kernel of the homomorphismB(C, D0)→π1(Divd(C), D0) induced by the inclusion rDivd(C),→Divd(C).

LetMdg =M(C, D0) be the topological group of orientation-preserving diffeo- morphismsγ ofC acting from the right that satisfypiγ =pi for each pointpi of D0. We denote by

Γgd=Γ(C, D0) :=π0(M(C, D0))

the group of isotopy classes of diffeomorphisms inMdg=M(C, D0), which acts on SBdg=SB(C, D0) from the right in a natural way.

LetC⊂PM be a smooth non-degenerate projective curve of degreedand genus g >0, and letD0rDivd(C) be a general hyperplane section. We will investigate π1(U0(C,PM), D0); that is, the fundamental group of the complement of the dual hypersurface ofC.

In [8] and [23], we studied this group under conditions thatd≥2g+ 2 and that the invertible sheaf OC(D0) corresponds to a general point of the Picard variety Picd(C) of isomorphism classes of line bundles of degreed.

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Using the fact that π2(Picd(C)) = 0, we derive from our main theorem (Theo- rem 3.20) the following result, which states the same result as in [8] and [23] under weaker conditions.

Definition 1.6. We say thatC⊂PM isPl¨ucker general if the dual curveρ(C) (P2) of the image ρ(C) P2 of the general projection ρ : C P2 has only ordinary nodes and ordinary cusps as its singularities.

Theorem 1.7. Suppose thatd≥g+ 4and thatCis Pl¨ucker general inPM. Then π1(U0(C,PM), D0)is isomorphic to SB(C, D0).

LetX⊂PN be a smooth non-degenerate projective surface of degree d, and let {Yt}tΛ be a pencil of hyperplane sections of X parameterized by a general line Λ (PN) with the base locus ZΛ :=X ∩A, where A=∩

Ht is the axis of the pencil{Ht}tΛ of hyperplanes. Let

ϕ : Y:={(x, t)∈X×Λ | x∈Ht} → Λ

be the fibration of the pencil. Then ϕ is locally trivial over Λ\ΣΛ in the C- category, where ΣΛ is the set of critical values ofϕ. Let 0 be a general point of Λ.

The corresponding memberY0is a compact Riemann surface of genus g:= (d+H0·KX)/2 + 1.

Note thatU0(ZΛ, A) ={A}, and that each point ofZΛyields a holomorphic section ofϕ:Y →Λ. By the classical monodromy, we obtain a homomorphism

(1.3) π1\ΣΛ,0) Γgd =Γ(Y0, ZΛ),

and hence π1\ΣΛ,0) acts on the simple braid group SBdg =SB(Y0, ZΛ) from the right. We denote by

ΓΛ Γgd =Γ(Y0, ZΛ)

the image of the monodromy homomorphism (1.3). Combining Theorems 1.4 and 1.7, we obtain the following:

Corollary 1.8. Let X, {Yt}tΛ, ZΛ =X ∩A and ΓΛ be as above. Suppose that g > 0, d g + 4, and that a general hyperplane section of X is Pl¨ucker gen- eral. Then π1(U0(X,PN), A) is isomorphic to the Zariski-van Kampen quotient SB(Y0, ZΛ)//ΓΛ.

A motivation of the study of the fundamental groupπ1(U0(X,PN)) for a surface X PN is the conjecture of Auroux, Donaldson, Katzarkov and Yotov [1] about the fundamental groupπ1(P2\B) of the complement of the branch curveB⊂P2of the general projection X→P2, which had been intensively studied by Moishezon, Teicher, Robb. The weakening of the conditions from our previous works ([8], [23]) to the present result (Theorem 1.7) is important with respect to this application.

See Remark 6.4.

The plan of this paper is as follows. In§2, we state some elementary facts about Zariski-van Kampen quotients. In§3, we prove the generalized Zariski-van Kampen theorem (Theorem 3.20). We then prove its variant (Theorem 3.33), and deduce Corollaries 1.1 and 1.3. The main ingredient of the proof is the notion of free loop pairs of monodromy relation type (Definitions 3.23 and 3.24), and Proposition 3.29.

Using these results, we prove Theorem 1.4 in §4, and Theorem 1.7 in §5. In the

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last section, we explain the relation betweenπ1(U0(X,PN)) and the conjecture of Auroux, Donaldson, Katzarkov, Yotov.

Conventions and Notation

(1) The constant map to a pointP is denoted by 1P.

(2) We denote by I Rthe interval [0,1], by ∆C the open unit disc, and by ¯∆Cthe closed unit disc.

(3) For a continuous mapδ: ¯∆→T to a topological spaceT, we denote by

εδ : I T the loop given byt7→δ(exp(2π√

1t)).

2. Zariski-van Kampen quotient

Definition 2.1. LetGbe a group, and letS be a subset ofG. We denote by〈S〉G

or simply by〈S〉the smallest subgroup ofGcontainingS, and by〈〈S〉〉G or simply by〈〈S〉〉the smallest normal subgroup ofGcontainingS.

We let a group Γ act on a groupGfrom the right. The following are easy:

Lemma 2.2. For any γ Γ, the subgroup 〈{g1gγ|g∈G}〉G of G is normal.

Hence, for any subsetΣΓ, the subgroup〈{g1gσ|g∈G, σ∈Σ}〉G is normal.

Lemma 2.3. Let S be a subset ofG, and letΣbe a subset ofΓ. IfG=〈S〉G and Γ =ΣΓ, then we have

〈〈{s1sσ|s∈S, σ∈Σ}〉〉G=〈{g1gσ|g∈G, σ∈Σ}〉G=〈{g1gγ|g∈G, γ∈Γ}〉G. Definition 2.4. We define GoΓ to be the group with the underlying setΓ and with the product defined by

(g, γ)(h, δ) := (( h(γ1)

) , γδ).

We then define homomorphismsi:G→GoΓ, p:GΓ ands: Γ→GoΓ byi(g) := (g,1),p(g, γ) :=γands(γ) := (1, γ). Then we obtain an exact sequence

(2.1) 1 −→ G −→i G−→p Γ −→ 1

with the cross-sections ofp, and the actiong7→gγ of γ∈Γ onGcoincides with the inner-automorphismg7→s(γ)1gs(γ) bys(γ)∈GoΓ on the normal subgroup G=i(G) ofGoΓ.

The following two lemmas are elementary:

Lemma 2.5. Let G be a group. Suppose that we are given an exact sequence

(2.2) 1 −→ G −→ Gi −→p Γ −→ 1

with a cross-section s : Γ→ G of p that is a homomorphism of groups. Suppose also that the action ofγ∈Γ ong∈Gis equal to the inner-automorphism bys(γ);

that is, we have i(gγ) = s(γ)1i(g)s(γ)for any g ∈G and γ Γ. Then there exists an isomorphism G ∼= Gsuch that the exact sequences (2.1) and (2.2) coincide and the cross-section scorresponds tos by this isomorphism.

Lemma 2.6. The composite homomorphism

G −→i G−→ (GoΓ)/〈〈s(Γ)〉〉GoΓ

is surjective, and its kernel is equal to〈{g1gγ|g∈G, γ∈Γ}〉; that is, the Zariski- van Kampen quotient G//Γis isomorphic to (GoΓ)/〈〈s(Γ)〉〉.

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3. Fundamental groups of algebraic fiber spaces

LetX andY be smooth varieties, and letf :X →Y be a dominant morphism.

We denote by Sing(f)⊂X the Zariski closed subset of the critical points off. For a pointy ∈Y, we put

Fy:=f1(y).

Letα:T →Y be a continuous map from a topological spaceT. Then a continuous map ˜α:T →X is said to be alift ofαiff◦α˜ =α.

We fix, once and for all, a proper Zariski closed subset Σ⊂Y

such thatf:X→Y is locally trivial in theC-category, where Y:=Y \Σ, X:=f1(Y) and f:=f|X :X→Y.

(In particular, Sing(f) is contained inf1(Σ).) It follows from Hironaka’s resolution of singularities that such a proper Zariski closed subset Σ⊂Y exists. We then fix base points

b∈Y and ˜b∈Fb⊂X, and consider the homomorphisms

ι : π1(Fb,˜b) π1(X,˜b) and f : π1(X,˜b) π1(Y, b)

induced by the inclusion ι: Fb ,→X and the morphism f :X →Y, respectively.

The aim of Zariski-van Kampen theorem is to describe Ker(ι).

The following result of Nori [14] will be used throughout this paper:

Proposition 3.1. Suppose that Fb is connected, and that there exists a Zariski closed subset Ξ⊂Y of codimension≥2 such thatFy\(FySing(f))̸= for any y ∈Y \Ξ. Then f : π1(X,˜b) π1(Y, b) is surjective, and its kernel is equal to the image ofι:π1(Fb,˜b)→π1(X,˜b).

Proof. See Nori [14, Lemma 1.5] and [23, Proposition 3.1]. ¤ Let ˜α : I X be a path, and we put α:= f◦α. Then ˜˜ α induces an iso- morphismπ1(Fα(0)(0))˜ →∼π1(Fα(1)(1)), which depends only on the homotopy˜ class (relative to∂I) of the path ˜α. Hence we can write this isomorphism as

[ ˜α] : π1(Fα(0)(0))˜ →∼ π1(Fα(1)(1)).˜ Thelifted monodromy

µ : π1(X,˜b) Aut(π1(Fb,˜b))

introduced in§1 (see (1.1)) is obtained by applying this construction to the loops inX with the base point ˜b. By the definition, we have the following:

Proposition 3.2. For any [ ˜α]∈π1(X,˜b) andg∈π1(Fb,˜b), we have ι(gµ([ ˜α])) = [ ˜α]1·ι(g)·[ ˜α]

inπ1(X,˜b), whereι:π1(Fb,˜b)→π1(X,˜b)is the homomorphism induced by the inclusion ι:Fb,→X.

First we prove the following:

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>

γ

α˜

>

γ

˜

α φ −→pr2

α

Figure 3.1. The extensionφ

Proposition 3.3. Suppose that a loop α˜ : (I, ∂I)(X,˜b)is null-homotopic in (X,˜b). Then g1gµ([ ˜α])Ker(ι) for anyg∈π1(Fb,˜b).

Proof. We put α:= f◦α, and˜ := (I× {0})(∂I ×I). Let g ∈π1(Fb,˜b) be represented by a loopγ: (I, ∂I)(Fb,˜b). We defineφ:⊔ →X by

φ(s,0) :=γ(s), φ(0, t) := ˜α(t), and φ(1, t) := ˜α(t).

Then we havef◦φ= (α◦pr2)|, where pr2:I×I→Iis the second projection.

Sinceis a strong deformation retract ofI×Iandfis locally trivial, the extension of (α◦pr2)|:⊔ →Ytoα◦pr2:I×I→Ylifts to an extension fromφ:⊔ →X to a continuous mapφ:I×I →X that satisfiesφ|=φ andf◦φ=α◦pr2. (See Figure 3.1.) Then the loop

γ :=φ|I×{1} : (I, ∂I) (Fb,˜b) representsgµ([ ˜α]). Sinceφ|{0I = ˜αandφ|{1I = ˜α, we have

[γ]1[ ˜α][γ][ ˜α]1= 1

inπ1(X,˜b). Since [ ˜α] = 1 inπ1(X,˜b) by the assumption, we have [γ]1[γ] = 1 in

π1(X,˜b). ¤

By Proposition 3.3, the normal subgroup R defined by (1.2) is contained in Ker(ι). HoweverRis not equal to Ker(ι) in general. We give two examples.

Example 3.4. LetL P1 be a line bundle of degree d >0, and let L× ⊂L be the complement of the zero-section. Since the projectionf :X =L× →Y =P1 is locally trivial, we can put Σ =, and henceR={1}. However, the kernel of

ι : π1(Fb) =π1(C×)=Z π1(L×)=Z/dZ

is non-trivial. Indeed, Ker(ι) is equal to the image of the boundary homomorphism π2(P1)→π1(C×) in the homotopy exact sequence.

Example 3.5. Consider the morphism

f : X =C2 Y =C

given by f(x, y) := xy. We can put Σ = {0}, and hence the fundamental group ofX =C2\ {xy = 0} is isomorphic toZ2. The general fiberFb is isomorphic to P1 minus two points, and the lifted monodromy action ofπ1(X) onπ1(Fb)=Zis trivial. Therefore we haveR={1}, while we have Ker(ι) =π1(Fb)=Z.

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Our ultimate goal is to show that the three conditions in Corollary 1.1 is sufficient forR= Ker(ι) to hold.

From now on, we suppose that f : X Y satisfies the first two of the three conditions in Corollary 1.1; namely, we assume the following:

(C1) Sing(f) is of codimension2 inX, and

(C2) there exists a Zariski closed subset Ξ0 ⊂Y of codimension2 such that Fy is non-empty and irreducible for anyy∈Y \Ξ0.

Remark3.6. By the conditions (C1) and (C2), the following hold:

(C0) fory∈Y, the fiberFy is connected, and

(C3) there exists a Zariski closed subset Ξ1 ⊂Y of codimension2 such that Fy\(FySing(f)) is non-empty and connected for everyy∈Y \Ξ1. In particular, we see that f is surjective and Im(ι) = Ker(f) holds by Nori’s lemma (Proposition 3.1).

Let Σ1, . . . , ΣN be the irreducible components of Σ with codimension 1 inY. There exists a proper Zariski closed subset ΞΣ with the following properties. We put

Y:=Y \Ξ, Σi := Σi\iΞ) = Σi∩Y, Σ:= Σ\Ξ = Σ∩Y. (Ξ0) The codimension of Ξ inY is2.

(Ξ1) The Zariski closed subsets Ξ0 ⊂Y in the condition (C2) and Ξ1 Y in the condition (C3) are contained in Ξ.

(Ξ2) Each Σi is a smooth hypersurface of Y, and Σ is a disjoint union of Σ1, . . . ,ΣN; that is, Ξ contains all the irreducible components of Σ with codimension2 inY and the singular locus of Σ.

(Ξ3) For each y∈Σi, there exist an open neighborhoodU ⊂Y ofy inY and an analytic isomorphism

φ: (U, U∩Σ) −→m1×(∆,0), wherem= dimY,

with the following properties. Let ψ : U m1 be the composite of φ:U = ∆m1×∆ and the projection ∆m1×m1. Then

Ψ :=ψ◦f : f1(U) m1 is smooth, and the commutative diagram

f1(U) −→f U

Ψ ψ

m1

is a trivial family ofC-maps over ∆m1 in theC-category.

Because of the choice of Ξ, forany pointy∈Σi, there exists an open disc ∆⊂Y with the following properties:

(∆1) ∆Σ ={y}, and ∆ intersects Σi transversely aty, (∆2) f1(∆) is a complex manifold,

(∆3) f|f1(∆) : f1(∆) ∆ is a one-dimensional family of complex analytic spaces that is locally trivial in theC-category over ∆\ {y}, and

(∆4) the central fiber Fy := f1(y) is an irreducible hypersurface of f1(∆), andFy\(FySing(f)) is non-empty and connected.

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Moreover the diffeomorphism type off|f1(∆):f1(∆)∆ depends only on the indexiof Σi.

We put

X:=f1(Y), f:=f|X :X→Y, Θi:= (f)1i) and Θ:= (f)1).

Then each Θi is an irreducible hypersurface of X, and Θ is a disjoint union of Θ1, . . . ,ΘN. Note that we haveX=X\Θ.

Remark 3.7. By the condition (C1), the Zariski closed subsetf1(Ξ) of X is also of codimension 2, and hence the inclusions induce isomorphisms π1(X,˜b) = π1(X,˜b) andπ1(Y, b)=π1(Y, b).

We introduce notions oftransversal discs, leashed discs andlassos.

Definition 3.8. LetH ⊂M be a reduced hypersurface of a complex manifoldM of dimensionm, and let H1, . . . , Hl be the irreducible components ofH. We fix a base pointbM ∈M \H .

(1) LetN be a realk-dimensionalC-manifold with 2≤k≤2m(possibly with boundaries and corners), and letφ:N→M be a continuous map. Letpbe a point ofN that is not in the corner ofN. Ifk= 2, we further assume that p /∈∂N. We say that φ:N →M intersects H atptransversely if the following hold:

(φ1) φ(p)∈H\Sing(H), and

(φ2) there exist local coordinates (u1, . . . , uk) of N at pand local coordinates (v1, . . . , v2m) of theC-manifold underlying M atφ(p) such that

p= (0, . . . ,0), φ(p) = (0, . . . ,0),

ifp∈∂N, then N is given byuk0 locally atp,

H is locally defined byv1=v2= 0 inM, and

φis given by (u1, . . . , uk)7→(v1, . . . , v2m) = (u1, . . . , uk,0, . . . ,0).

We say that φ: N M intersects H transversely if φ1(H) is disjoint from the corner of N (whenk = 2, we assume thatφ1(H)∩∂N =) and φ intersectsH transversely at every point ofφ1(H).

If φ intersects H transversely, then φ1(H) is a real (k−2)-dimensional sub- manifold of N. If k >2, then the boundary ofφ1(H) is equal toφ1(H)∩∂N, while ifk= 2, then φ1(H) is a finite set of points in the interior ofN.

(2) A continuous map δ : ¯∆ M is called a transversal disc around Hi if δ1(H) = {0}, δ(0) ∈Hi and δ intersects H transversely at 0. In this case, the sign ofδis the local intersection number (+1 or1) ofδwithHi atδ(0).

(3) Anisotopybetween transversal discsδandδaroundHiis a continuous map h : ¯∆×I M

such that, for eacht∈I, the restrictionδt:=h|¯×{t}: ¯∆→M ofhto ¯∆× {t} is a transversal disc aroundHi, and such that δ0=δandδ1=δ hold.

(4) A leashed disc around Hi with the base point bM is a pair ρ= (δ, η) of a transversal disc δ : ¯∆ M around Hi and a path η : I M \H from δ(1) =

εδ(0) =εδ(1) to bM. (Recall thatεδis the loop given byt7→δ(exp(2π√

1t)).

See Convention (3).) Thesign of a leashed discρ= (δ, η) is the sign of δ.

(5) Thelassoλ(ρ) associated with a leashed discρ= (δ, η) is the loopη1·(εδ)·η inM \H with the base pointbM.

(11)

(6) Anisotopy of leashed discs aroundHi with the base point bM is the pair of continuous maps

(h¯, hI) : ( ¯∆, I)×I (M, M\H)

such that, for eacht∈I, the restriction of (h¯, hI) to ( ¯∆, I)× {t}is a leashed disc aroundHi with the base pointbM.

Remark 3.9. The isotopy class of a leashed discρ is denoted by [ρ]. If [ρ] = [ρ], then [λ(ρ)] = [λ(ρ)] holds inπ1(M \H, bM).

The following is obvious:

Proposition 3.10. (1) Any two transversal discs around Hi with the same sign are isotopic.

(2) The homotopy classes of lassos associated with all the leashed discs around Hi with a fixed sign form a conjugacy class inπ1(M\H, bM).

(3) The kernel of the homomorphism π1(M \H, bM) →π1(M, bM) induced by the inclusion is generated by the homotopy classes of all lassos aroundH1, . . . , Hl.

We apply these notions to the hypersurfaces

Σ= Σ1∪ · · · ∪ΣN ofY, and Θ= Θ1∪ · · · ∪ΘN ofX.

Definition 3.11. (1) Atransversal lift of a transversal discδ: ¯∆→Yaround Σi is a lift ˜δ: ¯∆→X ofδwith ˜δ(0)∈/ Sing(f) such that ˜δ intersects the irreducible hypersurface Θi transversely at 0.

(2) Letρ= (δ, η) be a leashed disc around Σiwith the base pointb. Atransversal lift ofρis a pair ˜ρ= (˜δ,η) such that ˜˜ δ: ¯∆→Xis a transversal lift ofδ: ¯∆→Y and ˜η:I→X is a lift ofη:I→Y such that ˜η(0) = ˜δ(1) and ˜η(1) = ˜b.

Remark3.12. Any transversal lift of a transversal disc (resp. a leashed disc) around Σi is a transversal disc (resp. a leashed disc) around Θi. Moreover the lifting does not change the sign.

Definition 3.13. (1) Letδ0andδ1be two transversal discs onYaround Σi, and leth: ¯∆×I →Y be an isotopy of transversal discs from δ0 to δ1. A lift of the isotopyhis a continuous map

˜h : ¯∆×I X

such that, for each t I, the restriction ˜δt := ˜h|¯×{t} is a transversal lift of the transversal disc δt := h|¯×{t} on Y. In particular, we have f ˜h = h and

˜h( ¯∆×I)Sing(f) =. Moreover ˜his an isotopy of transversal discs around Θi from ˜δ0 to ˜δ1. By abuse of notation, we sometimes say that the isotopy ˜δtis the transversal lift of the isotopyδt, understanding thatt is the homotopy parameter.

(2) Letρ0 and ρ1 be two leashed discs on Y around to Σi, and let (h¯, hI) : ( ¯∆, I)×I (Y, Y) be an isotopy of leashed discs from ρ0 to ρ1. A lift of the isotopy (h¯, hI) is a pair of continuous maps

h¯,h˜I) : ( ¯∆, I)×I (X, X)

such that, for eacht∈I, the restriction ˜ρt:= (˜h¯,˜hI)|( ¯,I)×{t} is a transversal lift of the leashed discρt:= (h¯, hI)|( ¯,I)×{t} onY.

The following are obvious from the condition (∆4):

(12)

Proposition 3.14. Every transversal disc around Σi has a transversal lift onX. Moreover, every isotopy δt of transversal discs around Σi from δ0 toδ1 lifts to an isotopyδ˜tfrom a given transversal lift ˜δ0 of δ0 to a given transversal liftδ˜1 of δ1. Remark 3.15. Every leashed disc on Y around Σi has a transversal lift on X. Moreover, every isotopyρt of leashed discs onY has a lift ˜ρt onX from a given transversal lift ˜ρ0 ofρ0, but the ending lift ˜ρ1 cannot be arbitrarily given.

Definition 3.16. Letρbe a leashed disc onYaround Σi, and let ˜ρbe a transversal lift ofρ. Then we have the lassoλ( ˜ρ), which is a loop inXwith the base point ˜b.

Recall thatµis the lifted monodromy. We put

N( ˜ρ) :=〈 {g1gµ([λ( ˜ρ)])|g∈π1(Fb,˜b)} 〉π1(Fb,˜b).

Proposition-Definition 3.17. Let ρ be a leashed disc on Y isotopic to ρ, and letρ˜ be a transversal lift of ρ. Then we have

N( ˜ρ) = N( ˜ρ).

Therefore, for an isotopy class [ρ]of leashed discs on Y, we can define a normal subgroup N[ρ] of π1(Fb,˜b) by choosing a transversal lift ρ˜of a representative ρ of [ρ], and putting

N[ρ]:=N( ˜ρ).

Proof. By Remarks 3.9 and 3.15, the isotopy fromρtoρ lifts to an isotopy from ˜ρ to some lift ˜ρ1ofρ, and we have [λ( ˜ρ)] = [λ( ˜ρ1)] inπ1(X,˜b). (However [λ( ˜ρ1)] and [λ( ˜ρ)] may be distinct in general.) Therefore it is enough to show thatN( ˜ρ(1)) = N( ˜ρ(2)) holds for any two transversal lifts ˜ρ(1) = (˜δ(1)˜(1)) and ˜ρ(2) = (˜δ(2)˜(2)) of a single leashed discρ= (δ, η) onY. We can assume that the transversal disc δ: ¯∆→Yaround Σi is an embedding of a complex manifold. We denote by ¯∆ρ the image of δ, and by ∆ρ the interior of ¯∆ρ. We can further assume that ¯∆ρ is sufficiently small, and that

Eρ:=f1(∆ρ)

is a smooth complex manifold by the condition (∆2). We then put Eρ =f1( ¯∆ρ), E×ρ =f1( ¯∆×ρ),

where ¯∆×ρ := ¯∆ρ\ {δ(0)}= ¯∆ρ∩Y. We also putq:=δ(1) =η(0)∈∂∆¯ρ and

˜

q(1):= ˜δ(1)(1) = ˜η(1)(0)∈Fq, q˜(2):= ˜δ(2)(1) = ˜η(2)(0)∈Fq.

Since f is locally trivial overη(I)⊂Y and= (∂I×I)(I× {1}) is a strong deformation retract ofI×I, there exists a continuous map Ω :I×I →X such that the following hold for anys, t∈I:

f(Ω(s, t)) =η(t), Ω(s,1) = ˜b, Ω(0, t) = ˜η(1)(t), Ω(1, t) = ˜η(2)(t).

(See Figure 3.2.) Then, for eacht∈I, the maps7→Ω(s, t) is a path inFη(t) from

˜

η(1)(t) to ˜η(2)(t). We denote byω:I→Fq the path inFq from ˜q(1)to ˜q(2)defined byω(s) := Ω(s,0). Then we have the following commutative diagram:

π1(Fb,˜b) ←−

[ ˜η(1)]

π1(Fq,q˜(1)) −→iq π1(Eρ,q˜(1))

[ω] [ω] π1(Fb,˜b) ←−

[ ˜η(2)]

π1(Fq,q˜(2)) −→i

Figure 3.1. The extension φ
Figure 3.2. The map Ω
Figure 3.3. (h | T ) ∼ and h L
Figure 3.4. An orientation of ∂I 2
+7

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