2. Mixed Hodge Modules and Spectral Pairs

New York J. Math. 1(1995)149–177.

Spectral Pairs, Mixed Hodge Modules, and Series of Plane Curve Singularities

A. N´emethi and J.H.M. Steenbrink

Abstract. We consider a mixed Hodge moduleMon a normal surface sin- gularity (X, x) and a holomorphic function germf : (X, x)→(C,0). For the case that Mhas an abelian local monodromy group, we give a formula for the spectral pairs off with values inM. This result is applied to generalize the Sebastiani-Thom formula and to describe the behaviour of spectral pairs in series of singularities.

Contents

1. Introduction 150

2. Mixed Hodge Modules and Spectral Pairs 151

3. The General Setup 154

4. The Definition ofSpp_Γ 154

5. The Main Result 157

5.1. The proof of Theorem 5.1 158

6. Examples 162

6.1. Abelian coverings 162

6.2. The case of the trivial mixed Hodge module 164

7. Topological Series of Curve Singularities 165

7.1. Geometric meaning 166

7.2. Topologically trivial series 166

7.3. Intrinsic invariants 167

8. Topological Series of Plane Singularities with Coefficients in a Mixed

Hodge Module 167

8.1. Limit mixed Hodge structures 168

8.2. Intrinsic meaning 168

8.3. Further computations 168

9. The Spectral Pairs of Series of Plane Singularities 169

Received December 15, 1994.

Mathematics Subject Classification. 14B, 14C.

Key words and phrases. singularity spectrum, series of singularities.

A. N´emethi was supported by the Netherlands Organisation for the Advancement of Scientific Research N.W.O.

1995 State University of New Yorkc ISSN 1076-9803/95

149

9.1. General formula 169 9.2. The case of topologically trivial series 170 10. Spectral Pairs of Series of Composed Singularities 172 10.1. Some invariants of an ICIS with 2-dimensional base space 172 10.2. Topological series of composed singularities 173 11. A Generalized Sebastiani-Thom Type Result 174

References 176

1. Introduction

Spectral pairs were introduced first in [17] as discrete invariants of the mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singular- ity. The spectral pairs which are considered in this article are defined following a slightly different convention, as in [11]. This invariant encodes the dimensions of the eigenspaces of the semisimple partT_s of the monodromy acting on each sub- quotientGr_p+q^W Gr^p_F of the vanishing cohomology, and takes its values in the group ringZ[Q×Z].

Instead of vanishing cycles with constant coefficients one may consider vanishing cycles with coefficients in a mixed Hodge module [10]. We are led to consider these in the study of composed functions f = p◦φ where φ : (X, x) → (C²,0) is a 2-parameter smoothing of an isolated complete intersection singularity and p: (C²,0)→(C,0) is a holomorphic function germ. The main result of this article gives a formula for the spectral pairs for suchpat 0 with values in a mixed Hodge module on (C²,0) in terms of a decorated graph associated withp⁻¹(0)∪∆, where

∆ is the discriminant of the mixed Hodge module, under the assumption that the latter has an abelian local monodromy groupG. In fact, in the Main Theorem (5.1), (C²,0) has been replaced by an arbitrary normal surface singularity. The mixed Hodge module we consider gives rise to a limit mixed Hodge structure on which Gacts [9] and this action is used as input for the formula. The assumption about abelian monodromy is always fulfilled in case the complement of ∆ has abelian local fundamental group, e.g., when ∆ has normal crossings. We obtain generalizations of the Sebastiani-Thom formula (the case where φ=f×g with f and g isolated hypersurface singularities) in Section 11. We also obtain formulas describing the behaviour of the spectral pairs in certain series of singularities, which generalize [11], where the case of Yomdin’s series was treated.

A quick review of mixed Hodge modules and vanishing cycle functors is given in Section 2, which also contains the definition of spectral pairs and their basic proper- ties. The flavour of our result is described inSection 3by reformulating the case of a 1-dimensional base. The ingredients of the main formula are defined inSection 4, whereas its statement and proof form the content of Section 5. Some illustrative examples are treated inSection 6. Sections 7–10 deal with the application to series of singularities.

The authors thank the MSRI at Berkeley for its hospitality in May 1993, when part of this work was done. The first author thanks the University of Nijmegen for its hospitality in the academic year 1993-94, when this paper was finished.

2. Mixed Hodge Modules and Spectral Pairs

LetX be a (separated and reduced) complex analytic space. In [10] the category MHM(X) of mixed Hodge modules onX is associated with X. This category is stable under certain cohomological functors, for example underH^jf^∗andH^jf^! as- sociated with a morphismf of complex analytic spaces, and underH^jf∗associated with a projective (or proper K¨ahler) morphismf. Moreover, ifg is a holomorphic function onX andX₀=g⁻¹(0), then the vanishing and the nearby cycle functors ϕ_g, ψ_g : MHM(X)→MHM(X₀) are defined. All these functors are compatible with the corresponding perverse cohomological functors on the underlying perverse sheaves via the forgetful (exact) functor

rat:MHM(X)→P erv(QX)

which assigns to a mixed Hodge module the underlying perverse sheaf (with Q coefficients). (For the definition of the functors ϕg and ψg at the level of the constructible sheaves, see [2].)

The vanishing and nearby cycle functors have a functor automorphismT_sof finite order. It is provided by the Jordan decompositionT =T_s·T_u of the monodromy T.

One has the decompositions:

ψg=ψg,1⊕ψg,6=1 respectively ϕg=ϕg,1⊕ϕg,6=1

such thatTs is the identity onψg,1andϕg,1and has no 1-eigenspace onψg,6=1 and ϕg,6=1. One has the canonical morphisms:

can:ψ_g→ϕ_g and V ar:ϕ_g→ψ_g(−1), compatible with the action ofT_s, such that

can:ψ_g,6=1−→^∼ ϕ_g,6=1 (1)

is an isomorphism.

Let D^bMHM(X) be the derived category of MHM(X) (i.e., the category of bounded complexes whose cohomologies are mixed Hodge modules on X). Let i:Y →X be a closed immersion and j:U →X the inclusion of the complement ofY. Then the cohomological functors are lifted to functors i∗, i^!, i^∗, j^∗, j∗, j!; and we have the functorial distinguished triangles forM ∈D^bMHM(X):

→j_!j^∗M →M →i_∗i^∗M−−→⁺¹

→i_∗i^!M →M →j_∗j^∗M−−→⁺¹ . (2)

The connection between the two sets of functors is the following. SetX0=g⁻¹(0) and leti:X0→X be the corresponding immersion. Then forM ∈Ob MHM(X) one has:

0→ H⁻¹i^∗M →ψ_g,1M−−→^can ϕ_g,1M → H⁰i^∗M →0;

0→ H⁰i^!M →ϕg,1M−−→^{V ar} ψg,1M(−1)→ H¹i^!M →0;

(3)

andH^k−1i^∗M=H^ki^!M= 0 ifk6∈ {0,1}.

On the other hand, iff :X →Y is a proper morphism and gis a holomorphic function onY, then for any M ∈Ob MHM(X) one has:

ψ_gH^jf_∗M=H^jf_∗ψ_g◦fM (and similarly forϕ).

(4)

Example 2.1. Assume thatXis smooth. A mixed Hodge moduleM ∈MHM(X) is called smooth ifratMis a local system ([10]).

Example 2.2. The moduleMis called pure of weightn(or a polarizable Hodge module of weightn) ifGr^W_i M= 0 fori6=n.

The category of smooth polarizable mixed Hodge modules is equivalent to the category of variation of polarizable mixed Hodge structures which are admissible in the sense of [5].

Example 2.3. MHM(point) is the category of polarizableQ-mixed Hodge struc- tures ([10] (3.9)).

If g₁ and g₂ are two holomorphic functions such that g₁⁻¹(0) intersects g₂⁻¹(0) transversally alongX₀, then

ψ_g1ψ_g2=ψ_g2ψ_g1 :MHM(X)→MHM(X₀) (the same forϕ’s).

In this caseψg1ψg2Mhas two commuting monodromiesT1andT2 induced by the ψ-functors.

Moreover, consider the holomorphic functions g₁, . . . , g_s such that s = dimX and the intersection∩^s_i=1g⁻¹_i (0) is a regular pointx∈X, and the divisor∪^s_i=1g_i⁻¹(0) in a neighbourhood ofxhas normal crossings. Then onψ_g1· · ·ψ_gsM ∈MHM({x}) the commuting set of monodromies T₁, . . . , T_s acts. We make the set of this type of objects more explicit. For the definition of mixed Hodge structures, see [1].

Definition 2.4. For any abelian group Gwe letMHS(G) denote the category of representations

ρ:G→AutMHS(H)

forH a mixed Hodge structure. For suchρwe letρ^pqdenote the induced represen- tation ofGinAutC(H^pq), whereH^pq=Gr^W_p+qGr_F^pHC.

Example 2.5. LetM be a mixed Hodge module onX,g :X →Cholomorphic andx∈g⁻¹(0). Then for allj∈Z, we have the objectsH^ji^∗_xψgMand H^ji^∗_xϕgM ofMHS(Z), where the action of 1∈Zis the semisimple part of the monodromy.

By the monodromy theorem, this is an automorphism of finite order.

Definition 2.6. Forρ:Z→Aut(H) inMHS(Z) with finite order one defines:

h^pq_λ := multiplicity oft−λas a factor of the characteristic polynomial ofρ^pq(1) (forλ∈C);

and Spp(ρ) =X

α,w

h^{[α],w−[α]}_e2πiα (α, w)∈Z[Q×Z],

where [α] is the integral part of α. Moreover, for g : X → C holomorphic and x∈g⁻¹(0) one defines for a mixed Hodge moduleMonX:

Sppψ(M, g, x) :=X

j

(−1)^jSppH^ji^∗_xψgM;

Sppϕ(M, g, x) :=X

j

(−1)^jSppH^ji^∗_xϕgM.

These take their values inZ[Q×Z].

Remark 2.7. In [17] the invariant SppSt(g,0) of spectral pairs was defined for an isolated hypersurface singularity g : (Cⁿ⁺¹,0) → (C,0). Its relation with the invariants above is as follows:

if SppSt(g,0) =X

nα,w(α, w), then Sppϕ(Q^H_Cn+1[n+ 1], g,0) = X

α6∈Z

nα,w(n−α, w) +X

α∈Z

nα,w(n−α, w+ 1).

Example 2.8. Let X be a smooth space-germ, Y ⊂ X a reduced divisor, and x∈Y. LetV be a polarized variation of Hodge structure on X\Y such that its underlying representation is abelian and quasi-unipotent. Then one obtains a limit mixed Hodge structureLVatxequipped with a semi-simple action ofH₁(X\Y), cf. [9], i.e., an object ofMHS(H₁(X\Y)).

If Y has irreducible components Y₁, . . . , Y_s, thenH₁(X \Y) is free abelian on generatorsM₁, . . . , M_s, whereM_jis represented by an oriented circle in a transverse slice toYj.

Lemma 2.9. a) There is an unique way to extend the definition ofSpp_ψ(M, g, x) toM ∈Ob D^bMHM(X)in such a way that for any distinguished triangle

M⁰→ M → M⁰⁰−−→⁺¹ one has

Sppψ(M, g, x) =Sppψ(M⁰, g, x) +Sppψ(M⁰⁰, g, x).

b) Foru∈ O^∗_X,x one has

Spp_ψ(M, ug, x) =Spp_ψ(M, g, x).

c) Sppψ(M, g, x) =P

lSppψ(Gr_l^WM, g, x)forM ∈Ob MHM(X).

d) Let T(p, q) : Z[Q×Z] →Z[Q×Z] be the automorphism mapping (α, w)to (α+p, w+p+q). Then

Spp_ψ(M ⊗Q^H_X(k), g, x) =T(−k,−k)(Spp_ψ(M, g, x)).

e) We letH^ji^∗_xM ∈MHS(Z)with trivial representation. Then Sppϕ(M, g, x) =Sppψ(M, g, x) +X

j

(−1)^jSpp(H^ji^∗_xM).

f) Letcn:Z[Q×Z]→Z[Q×Z] (n∈N^∗)be the unique map which sends(α, w) to P_n−1

k=0([α] + ^{α}+k_n , w), (here [β] (resp. {β} =β−[β]) is the integer part (resp. the fractional part)ofβ). Then:

Spp_ψ(M, fⁿ, x) =c_nSpp_ψ(M, f, x).

The propertiesa)–d) also hold withϕinstead ofψ.

The proof is left to the reader.

During the paper the notation Q^H_X means a^∗_XQ^H_pt, where a_X : X → pt is the constant function (see [10], p. 324).

3. The General Setup

Assume a complex analytic spaceX and a pointx∈X are given. The invariant Sppψ(M, f, x) depends onMand onf in a complicated way. We want to decom- pose it into one step depending only onM, and a combinatorial step depending mainly onf. To illustrate this, we first treat the case where dim(X) = 1.

Example 3.1. Let X be one-dimensional, x ∈ X and f : X → C non-constant holomorphic withf(x) = 0. Assume thatX is irreducible atx. LetMbe a mixed Hodge module onX. We will indicate how to computeSppψ(M, f, x).

Letσ: ˜X →X be the normalization ofX, and lettbe a uniformizing parameter at ˜x = σ⁻¹(x). Then f ◦σ = u·tⁿ for some germ u ∈ O_X,˜^∗_˜x and n ∈ N^∗. If N =H⁰σ^∗M, then by(4)andLemma 2.9 bandf, one has:

Sppψ(M, f, x) =cn Sppψ(N, t,x).˜

Moreover,Sppψ(N, t,x) =˜ Spp(LM, Ts) withLMthe limit mixed Hodge structure ofN at ˜x(observe that the restriction ofN to a punctured neighbourhood of ˜xis an admissible variation of mixed Hodge structure) andTsis the semi-simple part of the monodromyT. HenceSpp_ψ(M, f, x) =c_n Spp(LM). HereLMdepends only on M, andndepends only onf. This means that the computation ofSpp_ψ(M, f, x) goes in two steps. The first one, the computation ofLMas an object ofMHM(Z), does not involvef. In the second step, only the multiplicitynof ˜xas a zero off◦σ matters.

We are going to generalize the previous example to the two-dimensional case.

The first step, passage fromMtoLM, is possible ifMhas an abelian monodromy group, which satisfies the condition of Example 2.8, and gives rise to an object LM ofMHM(G), where G=H₁(X \Y) andY is the critical locus of M. The second step involves identification of the relevant discrete invariants of the function f :X →Catx. We will use the decorated resolution graph Γ off with respect to Y, to be defined inExample 4.5. We will also define a map (seeDefinition 4.4)

Spp_Γ:MHM(G)→Z[Q×Z]

with the property that

Spp_ψ(M, f, x) =Spp_Γ(LM),

provided that V = j^∗M is a polarized variation of Hodge structure, and M = j_∗j^∗M, wherej:X\Y →X is the inclusion.

This is the main result of the paper.

4. The Definition of Spp

In this sectionXis a two-dimensional analytic space,Y ⊂Xis a reduced divisor, x∈Y a normal singularity ofX. Assume that (X, Y) is contractible ontox. Let S(Y) be the set of irreducible components ofY atx.

Definition 4.1. Adecorated graphfor (Y, x) is a finite connected graph Γ, without edges connecting a vertex to itself, with set of verticesVand set of edgesE and the following data and conditions:

a) V =D t S withD,S non-empty and an injectionS(Y),→ S;

b) a map e:D →Zsuch that the matrixAonD × Dgiven by A(d, d⁰) =





e(d) ifd=d⁰ ;

0 ifd6=d⁰ and (d, d⁰)6∈ E;

1 ifd6=d⁰ and (d, d⁰)∈ E is negative definite;

c) a mapg:D →N;

d) a map m:S →Ntaking at least one non-zero value.

e) For any d ∈ D, let V_d ={v ∈ V |dist(v, d) = 1} be the set of neighbors of din Γ. Let Z^V be the free abelian group generated by {[v]}_v∈V. Define the groupG(Γ) as the quotient ofZ^V by the subgroup generated by the following relations:

e(d)[d] + X

v∈Vd

[v] = 0 (d∈ D)

or X

d⁰∈D

A(d, d⁰)[d⁰] + X

v∈Vd\D

[v] = 0 (d∈ D).

Let l be the composition Z^S ,→ Z^V → G(Γ), and let m : Z^S → Z be the linear extension of m (i.e., m[s] = m(s)). Then we assume that m can be extended toG(Γ), i.e., there existsm₀:G(Γ)→Zsuch thatm₀◦l=m.

Our maps are summarized in the following diagram:

0

Z^S Z

Z^V G(Γ)

?

coker(A)

→m

−→−→

↑

l

Notice that cokerAis a finite group of order detA, therefore ifAis unimodular l is an isomorphism and the assumption in e) is automatically satisfied.

Definition 4.2. For v ∈ V we define: mv =m0([v]), δv = #Vv, andMv ∈G(Γ) as the image of [v] by the natural projection. Ford∈ Dwe denotegd:=g(d).

It is a well-known fact that all the entries of the matrix−A⁻¹are strictly positive ifAis a matrix as inDefinition 4.1.b. In particular,m_d>0 for anyd∈ D.

Definition 4.3. Fix a character χ:G(Γ)→C^∗ of finite order. Letβv∈[0,1) be such that exp 2πiβv=χ([v]).

Ford∈ Dwe defineSppd(χ) as follows.

For eachv∈ Vd andk∈ {0, . . . , md−1}define:

R^kv_d ={−β_v+m_v

md(k+β_d)};

α^k_d={k+β_d md }.

We letR^k_d=P

v∈VdR^kv_d , and δ_d^k= #{v: R^kv_d 6= 0}. ThenSpp^k_d(χ) :=





−(α^k_d,0) + (δd−1)·(α^k_d+ 1,2) +gd(α^k_d,1) +gd(α^k_d+ 1,1) ifR^k_d = 0 (gd+R^k_d−1)(α^k_d,1) + (gd+δ_d^k−R^k_d−1)(α^k_d+ 1,1) + (δd−δ^k_d)(α^k_d+ 1,2)

else, and

Spp_d(χ) :=^mX^d−1

k=0

Spp^k_d(χ).

For e= (v, w)∈ E we define Spp_e(χ) as follows. Let m_e :=g.c.d.(m_v, m_w). The system of equations:

{βv}={mvγe/me} {β_w}={m_wγ_e/m_e}

either has a solutionγ_e∈R/Zor has not. We define Spp_e(χ) by:

Spp_e(χ) :=

P_me−1

k=0 ({^k+γ_me^e},0)−({^k+γ_me^e}+ 1,2) ifγ_eexists

0 otherwise

Finally, we let

Spp_Γ(χ) :=X

d∈D

Spp_d(χ) +X

e∈E˜

Spp_e(χ), where ˜E :=E ∩(D × D).

R^k_d ∈N, thereforeSpp_Γ(χ)∈Z[Q×Z].

Definition 4.4. Letρ∈MHS(G(Γ)). The representationρ^pq splits into a direct sum of characters

ρ^pq=⊕^d(p,q)_i=1 χ^pq_i , d(p, q) = dimH^p,q. We define

Spp_Γ(ρ) :=X

p,q d(p,q)X

i=1

T(p, q)Spp_Γ(χ^pq_i ).

Example 4.5. Let X be a two-dimensional complex analytic space with normal singularity at x. Let Y ⊂ X be a (reduced) divisor such that the pair (X, Y) is contractible toxandX\Y is smooth. As inSection 4,S(Y) is the set of irreducible components ofY at x.

We now consider a holomorphic functionp:X →C and construct a decorated graph Γ, the decorated resolution graph of p with respect to (Y, x). The point x ∈ X is an isolated singular point of the reduced curve p⁻¹(0)∪Y. We let S denote the set of branches ofp⁻¹(0)∪Y atx; thenS(Y)⊂ S. Let π:U →X be an embedded good resolution ofp⁻¹(0)∪Y. ThenD:=π⁻¹(Y∪p⁻¹(0)) is a union of smooth curves on the two-dimensional complex manifold U. Let E = π⁻¹(x)

and letD be the set of irreducible components ofE, andV =D t S. We assume thatD 6=∅. Forv∈ V we letDv be the corresponding irreducible component ofE ifv∈ Dand the strict transform of the corresponding local irreducible component of Y ∪p⁻¹(0) for v ∈ S. The edges of Γ are pairs (v, w) for which v 6= w and E∩Dv∩Dw6=∅. We letg(d) = the genus ofDdande(d) =Dd·Dd ford∈ D. The matrixAas defined in (4.1.b) is then the intersection matrix of the components of E, which is negative definite. Finally we letm(v) be the order of zero ofpalongD_v forv∈ S, or even forv∈ V. Thenmvanishes on the relations (4.1.e) because the divisorπ^∗(p⁻¹(0)) on U is linearly equivalent to zero, hence has zero intersection product with eachD_d (d∈ D). The induced map with sourceG(Γ) ism₀.

EachM_v(v∈ V) can be represented inH₁(X\p⁻¹(0)∪Y) by an oriented circle in a transversal slice toD_v. They generate the subgroupG(Γ) ofH₁(X\p⁻¹(0)∪Y).

Actually, there exists an exact sequence

0→G(Γ)→ⁱ H₁(X\p⁻¹(0)∪Y)→H₁(E)→0.

SinceH1(E) is a torsion free group, the above sequence splits.

Notice that for anys∈ S we have exactly oned_s∈ Dsuch that (s, d_s)∈ E.

5. The Main Result

Assumption: In this section, X is a two-dimensional complex analytic space, x∈X a normal point on X, andY ⊂X a reduced divisor with x∈Y . Assume that X \Y is smooth and connected. Let V be a polarized variation of Hodge structure onX\Y such that its underlying representationρis abelian and quasi- unipotent. ConsiderK:=im ρ⊂Aut(H) and its torsion subgroupT. IfK/T 6= 0, we assume that there existw_j ∈ O_X (j= 1, . . . , s), such thatY =∪^s_j=1Z(w_j), and w= (w₁, . . . , w_s) :X\Y →(C^∗)^s induces an epimorphismw_∗:H₁(X\Y)→Z^s which fits in the following commutative diagram:

π1(X\Y) K

Z^s K/T

w_∗

ρ

ρf

-

-

? ?

For such aV, the limit mixed Hodge structureLV∈MHS(H1(X\Y)) exists by [9] (cf.Example 2.8). LetM=j∗Vwherej:X\Y →X is the natural inclusion.

Letp:X→Cbe a holomorphic function. Let Γ be a decorated resolution graph of pwith respect to (Y, x) (cf.Example 4.5) andSppΓ(LV) the invariant defined inDefinition 4.4via the composed mapG(Γ),→H1(X\p⁻¹(0)∪Y)→H1(X\Y).

Our key result is:

Theorem 5.1. Let X andMbe as above. Then:

Spp_ψ(M, p, x) =Spp_Γ(LV).

Recall that the spectrum Sppψ(M, p, x) of a mixed Hodge module M with irreducible one-dimensional support Y is zero if p|Y ≡ 0; otherwise it can be computed as follows. Since M is a polarizable admissible variation of Hodge structure on Y \ {x}, it has a limit mixed Hodge structure LM. The topologi- cal information from p is the degree deg(p|Y) of the map p|Y : Y → C. Then Sppψ(M, p, x) =cdeg(p|Y)(Spp(LM)) (cf.Example 3.1).

IfY_sis one of the irreducible components of the critical locus Y ofMand Γ is a decorated resolution graph of pwith respect to (Y, x), then deg(p|Y_s) = m(d_s) where (s, d_s)∈ E.

Theorem 5.2. Let X andMbe as in theAssumption. LetY =∪_s∈S(Y)Y_sbe the irreducible decomposition of (Y, x), i_Yv : Y_v →X andj :X\Y →X the natural inclusions. Then:

Spp_ψ(M, p,0) =Spp_Γ(Lj^∗M) + X

s∈S(Y) p|Ys6≡0

X

k

(−1)^kc_m(ds) Spp(LH^ki^!_YsM).

Proof. Use(2),Lemma 2.9, and Theorem 5.1.

5.1. The proof of Theorem 5.1. The proof is divided into three steps.

Step 1. Letπ:U →X be a resolution ofp⁻¹(0)∪Y as inExample 4.5.

Set N = H⁰π^∗M and NE = i^∗_Eψp◦πN, where iE : E = π⁻¹(0) → U is the natural inclusion. Now it is clear thatH⁰π∗N =Mmodulo terms with support in {0}, and supp H^jπ∗N ⊂ {0} ifj 6= 0. Therefore, ψpH^jπ∗N =ψpMifj = 0 and

= 0 ifj6= 0. Hence, by(4),

H^jπ_∗ψ_p◦πN = 0 if j6= 0.

(5)

Consider the following isomorphisms: i^∗₀ψ_pM=i^∗₀ψ_pH⁰π_∗N ⁽⁴⁾= i^∗₀H⁰π_∗ψ_p◦πN ⁽⁵⁾= i^∗₀π_∗ψ_p◦πN ^(∗)= π_∗i^∗_Eψ_p◦πN =π_∗N_E.

The relation (∗) follows fromi^∗₀π∗=π∗i^∗_E. For this, notice, thatπ∗=π! because πis proper ([10] (4.3.3)), and then use [loc. cit.] (4.4.3).

For eachd∈ D, let ˜Dd =Dd\ ∪d⁰∈D∩VdDd⁰ andkd : ˜Dd,→Dd its inclusion. For eache= (d, d⁰)∈E˜ denote by ie:Dd∩Dd⁰ →E the natural inclusion. Then by (2)one has the following distinguished triangle:

→ ⊕_e∈E˜(ie)∗i^!_eNE→ NE → ⊕d∈D(kd)∗k^∗_dNE→ By the additivity of the functorSpp, we obtain:

Spp(i^∗₀ψpM) =Spp(π∗NE) =X

e∈E˜

Spp(i^!_eNE) +X

d∈D

Spp(π∗(kd)∗k^∗_dNE).

(6)

(Everywhere, the action is the natural monodromy provided byψ.)

Fixd∈ D. Let D⁰_d= ˜Dd\St(p⁻¹(0)∪Y) =Dd\ ∪v∈VdDv, andjd :D⁰_d,→Dd

denotes the natural inclusion.

Lemma 5.3. The restriction toD⁰_dof the moduleN_E⁰ = (kd)∗k^∗_dNE is smooth(i.e., rat j_d^∗N_E⁰ is a local system). Moreover, it satisfies

N_E⁰ = (jd)∗(jd)^∗N_E⁰. (7)

Proof. By construction: j_d^∗N_E⁰ =j_d^∗NE=j_d^∗ψp◦πN.

SinceratN restricted toU\D is a local system, andD_d⁰ is a smooth divisor in U\ ∪v∈VdDv, the sheafrat j_d^∗ψp◦πN is a local system, too.

The obstruction to the isomorphism(7)lies in the pointsP ∈Dd∩(∪v∈SDv) = D˜d\D⁰_d.

TakeP=Dd∩Dvsuch thatp|Yv6≡0 (v∈ S). Then the assumptionM=j∗j^∗M and(2)giveψ_p◦πi^!_DvN = 0. Now(3)and the commutativity of the vanishing cycle functors complete the argument in this case.

IfP =D_d∩D_v such thatv∈ S andp|Y_v≡0, then see [11] (4.7).

Notice thatj_d^∗N_E⁰ =j_d^∗NE, and byLemma 5.3, (kd)∗k^∗_dNE= (jd)∗j_d^∗NE. Since the isomorphismH^•(Dd,(jd)∗j_d^∗NE) =H^•(D_d⁰, j_d^∗NE) is compatible with the mixed Hodge structures, one has:

Spp(π∗(kd)∗k^∗_dNE) =Spp(H^•(D⁰_d, j_d^∗NE)).

Step 2. The identitySpp(H^•(D⁰_d, j_d^∗NE)) =Sppd(LV).

By the additivity ofSpp(see 2.13.a) it is enough to prove X

l

Spp(H^•(D_d⁰, Gr_l^Wj_d^∗N_E)) =Spp_d(LV).

(8)

Since j_d^∗NE is smooth (Lemma 5.3), Vd,l :=Gr_l^Wj_d^∗NE is a polarizable variation of Hodge structure onD⁰_d.

Lemma 5.4. The representation associated with the local systemratVd,lis abelian and quasi-unipotent.

Proof. D_d⁰ has a neighbourhood homeomorphic toD⁰_d× {disc}.

The importance of this lemma appears in the following:

Lemma 5.5. Any polarizable variation of Hodge structure on a quasi-projective smooth curve C, whose underlying local system has a monodromy representation which is abelian and quasi-unipotent, is locally constant.

Proof. Since the monodromy representation on C is semi-simple ([1], (4.2.6)), it follows, that it is a direct sum of one-dimensional representations, which are finite.

Hence a finite cover ˜C ofC has trivial local and global monodromies. Therefore a global marking ˜C → D can be defined in the moduli space of Hodge structures.

This by Griffiths’ theorem [4] can be extended to the smooth closure of ˜C. But this extended map is trivial by the rigidity theorem [13].

Consider now the limit mixed Hodge structure LV ∈ MHS(H₁(X \Y)). Its representation defines a locally constant abelian variation M_∞ on X\Y. In [9], among other facts, the following is proved:

Lemma 5.6. Let (C, x) ⊂ (X, x) be a curve with C∩Y = {x}. Let L(M|C), respectivelyL(M_∞|C)be the limit mixed Hodge structures at xof the restrictions of M, respectively ofM_∞, toC. Then Gr^WL(M|C) =Gr^WL(M_∞|C).

Now, if we replace in the above construction M by M∞, then we obtain a variationV∞,d,l instead ofVd,l

Lemma 5.7. The variations of Hodge structure Vd,l and V∞,d,l on D_d⁰ are iso- morphic.

Proof. Both are abelian variations by Lemma 5.4, with flat Hodge bundles by Lemma 5.5. By construction, the underlying representations are the same. We have only to show that in a fixed pointP ∈ D⁰_d the stalks are isomorphic (by an isomorphism, which is compatible with the representations).

Let C be a transversal slice to D⁰_d at a point P ∈ D⁰_d and t a uniformizing parameter of (C, P). Then (V_d,l)_P 'Gr^W_l ψ_tmd(M|C)' ⊕^m_i=1^dGr^W_l L(M|C). Sim- ilar isomorphisms holds for the other variation, therefore the result follows from Lemma 5.6. The compatibility follows from the naturality of the constructions.

Therefore,(8)is equivalent to X

l

Spp(H^•(D_d⁰,V∞,d,l)[1]) =Sppd(LV).

(9)

Notice that both sides of(9)depend only on the limit mixed Hodge structureLV.

Now, (LV, ρ) ∈ MHS(H1(X \ Y)) splits in a direct sum ρ = ⊕p,q ⊕^d(p,q)_i=1 χ^p,q_i , d(p, q) = dimLV^p,q. The construction of V∞,d,l preserves this decompo- sition, thereforeV_∞,d,l=⊕_p+q=l⊕^d(p,q)_i=1 V^p,q,i_∞,d. We have to show that

Spp(H^•(D⁰_d,V_∞,d^p,q,i)[1]) =Spp_d(χ^p,q_i ).

(10)

In the sequel we omit the indicesp, q andi. Moreover, we can assume thatχ is of type (p, q) = (0,0).

Lemma 5.8. The variationV∞,dismd-dimensional. It has a direct sum decompo- sition ⊕^m_k=0^d−1V^k_d in one-dimensional locally constant variations of C-Hodge struc- ture (of the same type (0,0)), such that the monodromy of V^k_d around the points (P_d∩P_v)_v∈Vd isexp(−2πiR^kv_d ). The monodromy action on V^k_d given by the van- ishing cycle functor is exp(2πiα^k_d).

Proof. The verification is local in small neighbourhoods of the pointsDd∩Dv(v∈ Vd). Here, in a suitable coordinate systemρ◦π=x^mdy^mv. The verification is left

to the reader.

Proof of (10). ByLemma 5.8, we have to verify only:

Spp(H^•(D_d⁰,V^k_d)[1]) =Spp^k_d(χ).

(11)

In order to compute the left hand side of(11), we have to compute the dimensions h^pq of the spacesGr^W_p+qGr^p_FH^•(D_d⁰,V^k_d). Thenh^pq_λ =h^pq if λ= exp(2πiα^k_d), and

= 0 otherwise.

Let Ω^•(log Σd) be the complex of meromorphic differentials onDdwith at worst logarithmic poles along Σd=∪v∈Vd(Dd∩Dv), and letV denote Deligne’s canonical extension of V^k_d ⊗C O_Dd⁰. Now, H^∗(D_d⁰,V^k_d) = H^∗(Dd, K^•), where K^• is the complex{∇:V →Ω¹(log Σd)⊗V}. The Hodge filtration of this complex is Deligne’s

“filtration bˆete” σ≥p, therefore the first term of the Hodge spectral sequence is

FE₁^pq=H^q(D_d, K^p). This spectral sequence degenerates atE₁.

IfR^k_d = 0, then V =O_D⁰_d. HenceE⁰⁰₁ =C, E⁰¹₁ =C^gd, and E₁¹⁰=C^gd+δd−1. This case corresponds to the trivial flat bundle, therefore we recover exactly the mixed Hodge structure ofDd\ {δd points}. In particular,H⁰=Chas type (0,0),

E₁⁰¹ has type (0,1), C^gd ⊂E₁¹⁰ has type (1,0), and C^δd−1 = E₁¹⁰/C^gd has type (1,1).

Assume thatR^k_d 6= 0. Then FE₁^pq = 0 if p+q 6= 1 because degV =−R^k_d < 0 and deg(O(−Σ)⊗ V^∗) = R^k_d −δ_d < 0. Then by the Riemann-Roch Theorem, dimE₁⁰¹ = gd+R^k_d −1, and dimE¹⁰₁ = gd+δd−R^k_d −1. This means that the only nontrivial cohomological group isH¹=H¹(D_d⁰,V^k_d). In order to compute its Hodge numbersh^pq, we need the weight filtration too.

The weight filtration of the complex K^• is W−1K^• = 0, W0K^• = {∇ : V → im∇}, andW1K^• =K^•. The extensionW0 is a resolution ofj∗V^k_d and (by [20]) H¹(Dd, j∗ratV^k_d) is pure of weight 1. On the other hand R¹j∗ratV^k_d = C^δd−δdk and it induces in H¹ a quotient of weight 2. Therefore, the only nonzero Hodge numbers are: h¹⁰, h⁰¹andh¹¹. More precisely: h¹¹=δ_d−δ_d^k, h⁰¹=g_d+R^k_d−1, and h¹⁰ = gd +δ_d^k −R^k_d −1. Now the expression for the spectral pairs readily

follows.

Example 5.9. Assume that δd = 1. ThenR^k_d = 0 for any k. ThereforeSpp^k_d =

−(α^k_d,0) +g_d(α^k_d,1) +g_d(α^k_d+ 1,1) for anyk.

Example 5.10. Assume that g_d = 0, δ_d = 2 and V_d = {v, w}. Then R^k_d = R^kv_d +R^kw_d ∈ {0,1}. Using the obvious fact that for x, y ∈ R with x+y ∈ Z one has {x}+{y} = 1 if x 6∈ Z, and = 0 if x ∈ Z, we obtain that R^k_d = 1 if β_v−^m_m^v_d(k+β_d)6∈Zand = 0 otherwise.

Lemma 5.11. Letl_d=g.c.d.(m_d, m_v) =g.c.d.(m_v, m_w). The following assertions are equivalent:

a) There existsk₀∈Zsuch thatβ_v−^m_m^v_d(k₀+β_d)∈Z.

b) There existsγd∈R/Zsuch that

{β_v}={m_vγ_d/l_d} {β_d}={m_dγ_d/l_d}.

c) There existsγ_d∈R/Zsuch that

{βv}={mvγd/ld} {βw}={mwγd/ld}.

(Notice that b) or c) determinesγ_d uniquely.) Moreover, if one of these conditions hold, then:

βv−mv

md(k+βd)∈Z⇔ (k−k0)mv

md ∈Z.

Proof. Use the relation [v] + [w] +ed[d] = 0 (cf.Definition 4.1.e).

Therefore, the lemma implies that ifgd= 0 andδd= 2, then Spp_d(χ) = P_ld−1

k=0 −({^k+γ_ld^d},0) + ({^k+γ_ld^d}+ 1,2) ifγd exists

0 otherwise.

We will see later that the expressionSppe(χ) (e∈ E⁰) is exactly of this type.

Step 3. The computation ofSpp(i^!_eNE).

Fix a node e = (v, w) ∈ E. Let us modify the resolution by a blowing up at˜ the pointP =Dv∩Dw. The new resolution is π⁰ :U⁰ →U −→^π B. DefineD⁰ and E⁰ similarly as in the first case. ThenD⁰ =D t {D_e},E⁰ = (E \ {P})t {P_v, P_w}, where D_e is the new exceptional divisor, and P_v = D_e∩D_v, P_w = D_e∩D_w, where the strict transforms ofD_v and D_w are denoted by the same symbols. Set E⁰= (π⁰)⁻¹(0), and consider the modulesN⁰ =H⁰π^0∗MandN_E⁰ =H⁰i^∗_E0ψp◦π⁰N⁰. Lemma 5.12. The mixed Hodge structuresi^!_PNE,i^!_PvN_E⁰ andi^!_PwN_E⁰ are isomor- phic in a way compatible with their monodromy action.

Proof. In a neighbourhood ofP (resp. ofPv)NE=ψp◦πN (resp. N_E⁰ =ψp◦π⁰N⁰).

On the other hand,H^∗(i^!_Pψ_p◦πN) = H^∗_c(F_p◦π,N), whereF_p◦π is the Milnor fiber of p◦π in P. Similarly, H^∗(i^!_Pvψp◦π⁰N⁰) = H^∗_c(Fp◦π⁰,N⁰). But, for a suitable representatives, one has an inclusion of F_p◦π⁰ into F_p◦π (induced by the blowing up map) which is a homotopy equivalence, and it identifies the sheavesN andN⁰. Moreover, it preserves the monodromy action too. Since rat is an exact functor,

the lemma follows.

Now, if we write(6)for the resolutionsπandπ⁰, we obtain that Spp(i^!_PvN_E⁰) +Spp(i^!_PwN_E⁰) +Spp_De(χ⁰) =Spp(i^!_PN_E).

UsingLemma 5.12one hasSpp(i^!_eNE) =−SppDe(χ⁰). Now, the result follows from the computation ofExample 5.10.

6. Examples

6.1. Abelian coverings. Consider a connected normal surface singularity (X, x) and a coveringφ : (X, x)→(C²,0) which is ramified over ∆. Let ∆ =∪^S_v=1^(∆)∆v

be the irreducible decomposition of ∆. Consider a germ: p: (C²,0)→(C,0). We are interested in spectral pairs Sppψ(Q^H_X[2], f, x), where f is the composed map f =p◦φ. The pair (φ,(X, x)) is uniquely determined by the nonramified covering X^∗=φ⁻¹(B\∆)→B\∆ [16] (here (B,∆) is a good representative).

In particular, the mixed Hodge moduleM=H⁰φ∗Q^H_X[2] on (C²,0) is completely determined by the exact sequence

1→π₁(X^∗)→π₁(B\∆,∗)−→^τ G→1.

The critical locus ofM is contained in ∆ and the representation ofM|_B\∆ is the induced representation byτ of the regular representation ρ_G ofG.

Now,(4)assures, that H^•i^∗_xψ_fQ^H_X[2] =H^•i^∗₀ψ_pM, therefore Spp_ψ(Q^H_X[2], f, x) =Spp_ψ(M, p,0).

Suppose, that G is abelian. Then Theorem 5.2 can be applied. The variation M|_B\∆is a flat variation of Hodge structure. It can be identified with (Q^|G|, F, W), where Gr^p_FQ^|G| = Gr_l^WQ^|G| = 0 if p 6= 0 or l 6= 0. The underlying abelian representation is

ρ^ab=ρG◦τ^ab:H1(B\∆,Z)→GL(|G|,Q)

whereτ^ab:H1(B\∆,Z)→Gis induced byτ. Actually, this is the description of LM, too. In particularSppΓ(LM) is well-defined.

Letv∈ S(∆). The variationi^!_∆vMhas the following description: Ifj:B\∆→ B is the natural inclusion, then R⁰j_∗j^∗φ_∗C_X = φ_∗C_X and for any point P_v ∈

∆_v− {0} one has dim(R¹j_∗j^∗φ_∗C_X)_P = dim(R⁰j_∗j^∗φ_∗C_X)_P =a_v, where above Pv there lie exactly av points ofX. The above spaces can be recovered from the representationρ^ab via the expressionC^|G|v = ker(ρ^ab(Mv)−1)⊂C^|G|. Letdv∈ D be the unique vertex such that (v, dv)∈ E. ThenC^|G|v is a sub-Hodge structure of C^|G| with the automorphismρ^ab([M_dv]).

The above discussion shows that M → j_∗j^∗M is one-to-one. In particular, i_∗H^ki^!_∆vM = 0 if k6= 1. The exact sequence (3)shows that H¹i^!_∆vM is of type (1,1). Its restriction to ∆_v− {0}is a locally constant variation; it can be identified withC^|G|v (−1) with monodromyρ^ab([M_dv]). Similarly as above, its limitLH¹i^!_∆vM has the same presentation too.

Finally, notice that degree(f|∆_v) =m(d_v).

Proposition 6.1.

Spp_ψ(Q^H_X[2], f, x) =Spp_Γ(C^|G|, ρ^ab)− X

v∈S(∆) p|∆v6≡0

c_m(dv)Spp(C^|G|_v (−1), ρ^ab([M_dv])).

Example 6.2. The case of Hirzebruch-Jung singularities. Let (z1, z2) be a local coordinate system in (C²,0). We make the above formula more explicit in the case when ∆ ={z1z2= 0}.

LetAn,q =C²/Znbe the cyclic quotient singularity, where the action ofG=Zn

is given by (u1, u2) 7→ (ζu1, ζ^pu2), ζ = exp(2πi/n) and pq ≡ 1(mod n). Then φ:An,q→C² given byzi=uⁿ_i (i= 1,2) defines a covering which is ramified over {z1z2= 0}.

In this case, the covering transformation group isG=Z_n, and τ^ab:H1(B\∆,Z) =Z²→Zn

isτ^ab(e₁) = ˆ1, τ^ab(e₂) =−q.c

On the other hand, for anyv∈ S(∆) one hasC^|G|v =Cand the transformation ρ^ab([Mdv]) is the identity. Therefore:

Spp_ψ(Q^H_X[2], p◦φ, x) =Spp_Γ(Cⁿ, ρ^ab)− X

v∈S(∆) p|∆v6≡0

mXdv−1 k=0

1 + k

m_dv,2

.

Example 6.3. Take p(z₁, z₂) =z₁+z₂in Example 6.2. Forx∈Rtakeδ(x) = 0 ifx∈Z, and = 1 ifx6∈Z. Then:

Sppψ(Q^H_X[2], p◦φ, x) =−(0,0) +

n−1X

i=1

2−

−i n

− qi

n

,2−δ 1−q

n i

. Remark 6.4. Above, we computed the spectral pairs of a composed function f. By our general result, we can compute Spp_ψ(Q^H_X[2], f, x) of an arbitrary function f : (X, x) → (C,0), provided that we know the decorated resolution graph of f (cf. the second part of this section).

6.2. The case of the trivial mixed Hodge module. Let (X, x) be a normal surface singularity and f : (X, x) →(C,0) an analytic germ. Assume that M= Q^H_X[2]. The limit LM is the one-dimensional mixed Hodge structure Q^H with trivial action and Gr_l^WQ^H =Gr^p_FQ^H = 0 if l 6= 0 or p 6= 0. Denote the set of spectral pairs merely bySpp_ψ(f). ByTheorem 5.2one has: Spp_ψ(f) =Spp_Γ(Q^H), where Γ is the decorated resolution graph off.

Let h=rank H₁(Γ) be the number of independent cycles of Γ. Alternatively, h=rank H₁(E)−rank H₁( ˜E), where ˜E is a smooth model forE=π⁻¹(x).

By a computation, we obtain an “almost symmetric” form ofSpp_ψ(f):

Proposition 6.5. Let g = P

d∈Vgd, R_d^k = P

v∈Vd{k·mv/md}, and R^∗(n) = {1, . . . n−1}. Then:

Sppψ(f) = (h−1)(0,0) + (h+ #S −1)(1,2) +g(0,1) +g(1,1)

+X

s∈S

X

k∈R^∗(m(es))

(1 + k

m(es),2) +X

e∈E˜

X

k∈R^∗(me)

(( k

me,0) + (2− k me,2))

−X

d∈D

X

k∈R^∗(md) R^k_d=0

(( k

md,0) + (2− k md,2))

+X

d∈D

X

k∈R^∗(md) R^k_d6=0

(R_d^k−1) (( k

m_d,1) + (2− k m_d,1))

+X

d∈D

X

k∈R^∗(md)

gd(( k

md,1) + (2− k md,1)).

Herem(e_s) =g.c.d.(m(s), m(d_s)), whered_sis the unique vertex inDwith(s, d_s)∈ E.

Example 6.6. Let (X, x) ={z²= (x+y²)(x²+y⁷)} ⊂(C³,0) andf(x, y, z) =z.

Then the decorated resolution graph is the following:

s s

s s

-

1

(−3) 1

(−1)

(−3)

(−3) 4

9

3

4

In parentheses are the numbers ed, the others are the multiplicities. All excep- tional divisors are rational. The spectral pairs are:

Spp_ψ(f) =X⁸

k=1

(2k/9,1) + (2/3,1) + (4/3,1) + 2·(1,2).

Remark 6.7. Using(3)one has: Spp_ϕ(f) =Spp_ψ(f) + (0,0).

7. Topological Series of Curve Singularities

Let p: (C²,0) → (C,0) be a curve singularity with irreducible decomposition p = Q_r

j=1p^m_j ^j, and ∆ ⊂ (C²,0) a reduced one-dimensional analytic space-germ with irreducible decomposition∪^s_i=1∆_i. Let Γ be the decorated resolution graph of p⁻¹(0)∪∆. Let S(∆) (resp. S(p)) be the subset of S corresponding to the strict transformsSt(∆_i) (resp. St(p⁻¹_j (0))). Recall that the multiplicity of a vertexv∈ V is the multiplicity ofp◦πon the divisorD_v. In particular, the multiplicities of the verticesv∈ S are: {m_j}^r_j=1 corresponding to s∈ S(p), andm_s= 0 corresponding to s∈ S(∆)\ S(p) In the sequel we use the index notation j ∈ {1, . . . , r} for the setS(p).

The schematic graph of Γ, together with multiplicities, is:

-

-

@

@ 0

0 ...

...

...

... ...

m_r

m1

S(∆)\ S(p) D S(p)

s

s l_r

l1

(12)

The vertices indexed by D are in the box. Those indexed by S are drawn as arrows. The arrows from the left-hand side corresponds toS(∆)\ S(p), those from the right-hand side toS(p).

For eachj∈ S(p), letdj ∈ D be the unique vertex which is adjacent toj. The corresponding exceptional divisor (Ddj) is denoted by Ej and its multiplicity is lj =mdj. The intersection pointDj∩Ej is denoted byPj.

Definition 7.1. The topological series of curve singularities belonging top, relative to ∆, consist of all curve singularitiesp⁰ such that there are decorated resolution graphs Γ ofp⁻¹(0)∪∆, and Γ⁰ of (p⁰)⁻¹(0)∪∆ respectively, such that Γ has form (12), and Γ⁰ has the following form:

Γr

Γ₁

- -

- -

@ @

@ @ 0

0 ...

... ...

...

...

...

...

... ...

s s

s s

lr+mr

l1+m1

lr

l1