New York Journal of Mathematics
New York J. Math. 1 (1995) 149{177.
Spectral Pairs, Mixed Hodge Modules, and Series
of Plane Curve Singularities
A. Nemethi and J.H.M. Steenbrink
M on a normal surface singularity ( ) and a holomorphic function germ : ( ) ! (C 0). For the
case that M has an abelian local monodromy group, we give a formula for
the spectral pairs of with values in M. This result is applied to generalize
the Sebastiani-Thom formula and to describe the behaviour of spectral pairs
in series of singularities.
Abstract. We consider a mixed Hodge module
X x
f
X x
f
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction
Mixed Hodge Modules and Spectral Pairs
The General Setup
The Denition of Spp;
The Main Result
5.1. The proof of Theorem 5.1
Examples
6.1. Abelian coverings
6.2. The case of the trivial mixed Hodge module
Topological Series of Curve Singularities
7.1. Geometric meaning
7.2. Topologically trivial series
7.3. Intrinsic invariants
Topological Series of Plane Singularities with Coe cients in a Mixed
Hodge Module
8.1. Limit mixed Hodge structures
8.2. Intrinsic meaning
8.3. Further computations
The Spectral Pairs of Series of Plane Singularities
150
151
154
154
157
158
162
162
164
165
166
166
167
167
168
168
168
169
Received December 15, 1994.
Mathematics Subject Classication. 14B, 14C.
Key words and phrases. singularity spectrum, series of singularities.
A. Nemethi was supported by the Netherlands Organisation for the Advancement of Scientic
Research N.W.O.
149
c 1995 State University of New York
ISSN 1076-9803/95
150
A. Nemethi and J.H.M. Steenbrink
9.1. General formula
9.2. The case of topologically trivial series
10. Spectral Pairs of Series of Composed Singularities
10.1. Some invariants of an ICIS with 2-dimensional base space
10.2. Topological series of composed singularities
11. A Generalized Sebastiani-Thom Type Result
References
1. Introduction
169
170
172
172
173
174
176
Spectral pairs were introduced rst in 17] as discrete invariants of the mixed
Hodge structure on the vanishing cohomology of an isolated hypersurface singularity. The spectral pairs which are considered in this article are dened following a
slightly dierent convention, as in 11]. This invariant encodes the dimensions of
the eigenspaces of the semisimple part Ts of the monodromy acting on each subquotient GrpW+q GrFp of the vanishing cohomology, and takes its values in the group
ring ZQ Z].
Instead of vanishing cycles with constant coe cients one may consider vanishing
cycles with coe cients in a mixed Hodge module 10]. We are led to consider
these in the study of composed functions f = p where : (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 such p at 0 with values in a mixed Hodge
module on (C2 0) in terms of a decorated graph associated with p;1 (0) , where
is the discriminant of the mixed Hodge module, under the assumption that the
latter has an abelian local monodromy group G. 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
G acts 9] and this action is used as input for the formula. The assumption about
abelian monodromy is always fullled 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 = 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 denition of spectral pairs and their basic properties. The avour of our result is described in Section 3 by reformulating the case of
a 1-dimensional base. The ingredients of the main formula are dened in Section 4,
whereas its statement and proof form the content of Section 5. Some illustrative
examples are treated in Section 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 rst author thanks the University of Nijmegen for
its hospitality in the academic year 1993-94, when this paper was nished.
Spectral Pairs
151
2. Mixed Hodge Modules and Spectral Pairs
Let X be a (separated and reduced) complex analytic space. In 10] the category
MHM (X ) of mixed Hodge modules on X is associated with X . This category is
stable under certain cohomological functors, for example under Hj f and Hj f ! associated with a morphism f of complex analytic spaces, and under Hj f associated
with a projective (or proper Kahler) morphism f . Moreover, if g is a holomorphic
function on X and X0 = g;1 (0), then the vanishing and the nearby cycle functors
'g g : MHM (X ) ! MHM (X0) are dened. All these functors are compatible
with the corresponding perverse cohomological functors on the underlying perverse
sheaves via the forgetful (exact) functor
rat : MHM (X ) ! Perv(QX )
which assigns to a mixed Hodge module the underlying perverse sheaf (with Q
coe cients). (For the denition of the functors 'g and g at the level of the
constructible sheaves, see 2].)
The vanishing and nearby cycle functors have a functor automorphism Ts of nite
order. It is provided by the Jordan decomposition T = Ts Tu of the monodromy
T.
One has the decompositions:
g = g1 g6=1 respectively 'g = 'g1 'g6=1
such that Ts is the identity on g1 and 'g1 and has no 1-eigenspace on g6=1 and
'g6=1 . One has the canonical morphisms:
can : g ! 'g and V ar : 'g ! g (;1)
compatible with the action of Ts , such that
'
(1)
can : g6=1 ;!
g6=1
is an isomorphism.
Let Db MHM (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
of Y . Then the cohomological functors are lifted to functors i i! i j j j! and
we have the functorial distinguished triangles for M 2 Db MHM (X ):
(2)
! j! j M !M ! i iM ;+1
;!
! i i!M !M ! j j M ;+1
;! :
The connection between the two sets of functors is the following. Set X0 = g;1(0)
and let i : X0 ! X be the corresponding immersion. Then for M 2 Ob MHM (X )
one has:
can ' M ! H0 i M ! 0
0 ! H;1 i M ! g1 M ;;!
g1
(3)
V
ar
0 ! H0 i!M ! 'g1 M ;;! g1 M(;1) ! H1 i!M ! 0
and Hk;1 i M = Hk i!M = 0 if k 62 f0 1g.
152
A. Nemethi and J.H.M. Steenbrink
On the other hand, if f : X ! Y is a proper morphism and g is a holomorphic
function on Y , then for any M 2 Ob MHM (X ) one has:
(4)
g Hj f M = Hj f gf M (and similarly for '):
Example 2.1. Assume that X is smooth. A mixed Hodge module M 2 MHM (X )
is called smooth if ratM is a local system (10]).
Example 2.2. The module M is called pure of weight n (or a polarizable Hodge
module of weight n) if GriW M = 0 for i 6= 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 polarizable Q-mixed Hodge structures (10] (3.9)).
If g1 and g2 are two holomorphic functions such that g1;1 (0) intersects g2;1(0)
transversally along X0 , then
g1 g2 = g2 g1 : MHM (X ) ! MHM (X0) (the same for ''s):
In this case g1 g2 M has two commuting monodromies T1 and T2 induced by the
-functors.
Moreover, consider the holomorphic functions g1 : : : gs such that s = dim X
and the intersection \si=1 gi;1 (0) is a regular point x 2 X , and the divisor si=1 gi;1(0)
in a neighbourhood of x has normal crossings. Then on g1 gs M 2 MHM (fxg)
the commuting set of monodromies T1 : : : Ts acts. We make the set of this type
of objects more explicit. For the denition of mixed Hodge structures, see 1].
Denition 2.4. For any abelian group G we let MHS (G) denote the category of
representations
: G ! AutMHS (H )
for H a mixed Hodge structure. For such we let pq denote the induced representation of G in AutC (H pq ), where H pq = GrpW+q GrFp HC .
Example 2.5. Let M be a mixed Hodge module on X , g : X ! C holomorphic
and x 2 g;1 (0). Then for all j 2 Z, we have the objects Hj ix g M and Hj ix 'g M
of MHS (Z), where the action of 1 2 Z is the semisimple part of the monodromy.
By the monodromy theorem, this is an automorphism of nite order.
Denition
2.6. For : Z ! Aut(H ) in MHS (Z) with nite order one denes:
hpq
:=
multiplicity
of t ; as a factor of the characteristic polynomial of pq (1)
(for 2 C)
and
X ]w;]
Spp() = he2i
( w) 2 ZQ Z]
w
where ] is the integral part of . Moreover, for g : X ! C holomorphic and
x 2 g;1(0) one denes for a mixed Hodge module M on X :
X
Spp (M g x) := (;1)j Spp Hj ixg M
Spp'(M g x) :=
j
X
j
(;1)j Spp Hj ix'g M:
Spectral Pairs
153
These take their values in ZQ Z].
Remark 2.7. In 17] the invariant SppSt(g 0) of spectral pairs was dened for
an isolated hypersurface singularity g : (Cn+1 0) ! (C 0). Its relation with the
invariants above is as follows:
X
if SppSt (g 0) = nw ( w) then
X
X
Spp'(QHCn+1 n + 1] g 0) = nw (n ; w) + nw (n ; w + 1):
62Z
2Z
Example 2.8. Let X be a smooth space-germ, Y X a reduced divisor, and
x 2 Y . Let V be a polarized variation of Hodge structure on X n Y such that its
underlying representation is abelian and quasi-unipotent. Then one obtains a limit
mixed Hodge structure LV at x equipped with a semi-simple action of H1 (X n Y ),
cf. 9], i.e., an object of MHS (H1 (X n Y )).
If Y has irreducible components Y1 : : : Ys , then H1 (X n Y ) is free abelian on
generators M1 : : : Ms , where Mj is represented by an oriented circle in a transverse
slice to Yj .
Lemma 2.9. a) There is an unique way to extend the denition of Spp (M g x)
to M 2 Ob Db MHM (X ) in such a way that for any distinguished triangle
M0 ! M ! M00 ;+1
;!
one has
Spp (M g x) = Spp (M0 g x) + Spp (M00 g x):
one has
b) For u 2 OXx
Spp (M ug x) = Spp (M g x):
P
c) Spp (M g x) = Spp (GrW M g x) for M 2 Ob MHM (X ).
l
l
d) Let T (p q) : ZQ Z] ! ZQ Z] be the automorphism mapping ( w) to
( + p w + p + q). Then
Spp (M QHX (k) g x) = T (;k ;k)(Spp (M g x)):
e) We let Hj ixM 2 MHS (Z) with trivial representation. Then
Spp'(M g x) = Spp (M g x) +
X
j
(;1)j Spp(Hj ix M):
Z] (n 2 N) be the unique map which sends ( w)
f) LetPcn : ZQ Z] ! ZQ
;1 ( ] + fg+k w) (here ] (resp. f g = ; ]) is the integer part
to nk=0
n
(resp. the fractional part) of ). Then:
Spp (M f n x) = cn Spp (M f x):
The properties a){d) also hold with ' instead of .
The proof is left to the reader.
During the paper the notation QHX means aX QHpt where aX : X ! pt is the
constant function (see 10], p. 324).
154
A. Nemethi and J.H.M. Steenbrink
3. The General Setup
Assume a complex analytic space X and a point x 2 X are given. The invariant
Spp (M f x) depends on M and on f in a complicated way. We want to decom-
pose it into one step depending only on M, and a combinatorial step depending
mainly on f . To illustrate this, we rst treat the case where dim(X ) = 1.
Example 3.1. Let X be one-dimensional, x 2 X and f : X ! C non-constant
holomorphic with f (x) = 0. Assume that X is irreducible at x. Let M be a mixed
Hodge module on X . We will indicate how to compute Spp (M f x).
Let : X~ ! X be the normalization of X , and let t be a uniformizing parameter
~ and n 2 N . If
at x~ = ;1 (x). Then f = u tn for some germ u 2 OX
x~
N = H0 M, then by (4) and Lemma 2.9 b and f, one has:
Spp (M f x) = cn Spp (N t x~):
Moreover, Spp (N t x~) = Spp(LM Ts) with LM the limit mixed Hodge structure
of N at x~ (observe that the restriction of N to a punctured neighbourhood of x~ is an
admissible variation of mixed Hodge structure) and Ts is the semi-simple part of the
monodromy T . Hence Spp (M f x) = cn Spp(LM). Here LM depends only on
M, and n depends only on f . This means that the computation of Spp (M f x)
goes in two steps. The rst one, the computation of LM as an object of MHM (Z),
does not involve f . In the second step, only the multiplicity n of x~ as a zero of f matters.
We are going to generalize the previous example to the two-dimensional case.
The rst step, passage from M to LM, is possible if M has an abelian monodromy
group, which satises the condition of Example 2.8, and gives rise to an object
LM of MHM (G), where G = H1 (X n Y ) and Y is the critical locus of M. The
second step involves identication of the relevant discrete invariants of the function
f : X ! C at x. We will use the decorated resolution graph ; of f with respect to
Y , to be dened in Example 4.5. We will also dene a map (see Denition 4.4)
Spp; : MHM (G) ! ZQ Z]
with the property that
Spp (M f x) = Spp; (LM)
provided that V = j M is a polarized variation of Hodge structure, and M =
j j M, where j : X n Y ! X is the inclusion.
This is the main result of the paper.
4. The Denition of Spp;
In this section X is a two-dimensional analytic space, Y X is a reduced divisor,
x 2 Y a normal singularity of X . Assume that (X Y ) is contractible onto x. Let
S (Y ) be the set of irreducible components of Y at x.
Denition 4.1. A decorated graph for (Y x) is a nite connected graph ;, without
edges connecting a vertex to itself, with set of vertices V and set of edges E and the
following data and conditions:
a) V = D t S with D, S non-empty and an injection S (Y ) ,! S Spectral Pairs
155
b) a map e : D ! Z such that the matrix A on D D given by
8< e(d) if d = d0 if d 6= d0 and (d d0 ) 62 E A(d d0 ) = : 0
1
if d 6= d0 and (d d0 ) 2 E
is negative denite
c) a map g : D ! N
d) a map m : S ! N taking at least one non-zero value.
e) For any d 2 D, let Vd = fv 2 V j dist(v d) = 1g be the set of neighbors of
d in ;. Let ZV be the free abelian group generated by fv]gv2V . Dene the
group G(;) as the quotient of ZV by the subgroup generated by the following
relations:
X
e(d)d] + v] = 0 (d 2 D)
or
v2Vd
X
d 2D
A(d d0 )d0 ] +
0
X
v2Vd nD
v] = 0 (d 2 D):
Let l be the composition ZS ,! ZV ! G(;), and let m : ZS ! Z be the
linear extension of m (i.e., ms] = m(s)). Then we assume that m can be
extended to G(;), i.e., there exists m0 : G(;) ! Z such that m0 l = m.
Our maps are summarized in the following diagram:
coker(A)
;
;;;
;! G(;)
ZV ;!
?
" ;l ;
0
;;
m
ZS !
Z
Notice that cokerA is a nite group of order det A, therefore if A is unimodular
l is an isomorphism and the assumption in e) is automatically satised.
Denition 4.2. For v 2 V we dene: mv = m0(v]) v = #Vv , and Mv 2 G(;)
as the image of v] by the natural projection. For d 2 D we denote gd := g(d).
It is a well-known fact that all the entries of the matrix ;A;1 are strictly positive
if A is a matrix as in Denition 4.1.b. In particular, md > 0 for any d 2 D.
Denition 4.3. Fix a character : G(;) ! C of nite order. Let v 2 0 1) be
such that exp 2i v = (v]).
156
A. Nemethi and J.H.M. Steenbrink
For d 2 D we dene Sppd() as follows.
For each v 2 Vd and k 2 f0 : : : md ; 1g dene:
v
Rdkv = f; v + m
md (k + d )g
k = f k + d g:
8
<
:
d
md
kv
k
kv
k
d
v2Vd Rd , and d = #fv : Rd 6= 0g. Then Sppd () :=
;( kd 0) + ( d ; 1) ( kd + 1 2) + gd( kd 1) + gd( kd + 1 1)
if Rdk = 0
k
k
k
k
k
k
k
(gd + Rd ; 1)( d 1) + (gd + d ; Rd ; 1)( d + 1 1) + ( d ; d )( d + 1 2)
P
We let Rk =
else,
and
Sppd() :=
mX
d ;1
k=0
Sppkd ():
For e = (v w) 2 E we dene Sppe () as follows. Let me := g:c:d:(mv mw ). The
system of equations:
f g = fm =m g
v
v e e
f w g = fmw e =me g
either has a solution e 2 R=Z or has not. We dene Sppe () by:
Pme;1 k+e
k+e
e exists
Sppe () := 0 k=0 (f me g 0) ; (f me g + 1 2) ifotherwise
Finally, we let
X
X
Spp; () := Sppd() + Sppe ()
d2D
e2E~
where E~ := E \ (D D).
Rdk 2 N, therefore Spp; () 2 ZQ Z].
Denition 4.4. Let 2 MHS (G(;)). The representation pq splits into a direct
sum of characters
(pq) pq
pq = di=1
i d(p q) = dim H pq :
We dene
Spp;() :=
(pq)
X dX
pq i=1
T (p q)Spp;(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 to x and X n Y is smooth. As in Section 4, S (Y ) is the set of irreducible
components of Y at x.
We now consider a holomorphic function p : X ! C and construct a decorated
graph ;, the decorated resolution graph of p with respect to (Y x). The point
x 2 X is an isolated singular point of the reduced curve p;1 (0) Y . We let S
denote the set of branches of p;1 (0) Y at x then S (Y ) S . Let : U ! X be
an embedded good resolution of p;1 (0) Y . Then D := ;1 (Y p;1 (0)) is a union
of smooth curves on the two-dimensional complex manifold U . Let E = ;1 (x)
Spectral Pairs
157
and let D be the set of irreducible components of E , and V = D t S . We assume
that D 6= . For v 2 V we let Dv be the corresponding irreducible component of E
if v 2 D and the strict transform of the corresponding local irreducible component
of Y p;1 (0) for v 2 S . The edges of ; are pairs (v w) for which v 6= w and
E \ Dv \ Dw 6= . We let g(d) = the genus of Dd and e(d) = Dd Dd for d 2 D. The
matrix A as dened in (4.1.b) is then the intersection matrix of the components of
E , which is negative denite. Finally we let m(v) be the order of zero of p along Dv
for v 2 S , or even for v 2 V . Then m vanishes on the relations (4.1.e) because the
divisor (p;1 (0)) on U is linearly equivalent to zero, hence has zero intersection
product with each Dd (d 2 D). The induced map with source G(;) is m0 .
Each Mv (v 2 V ) can be represented in H1 (X n p;1(0) Y ) by an oriented circle in
a transversal slice to Dv . They generate the subgroup G(;) of H1 (X n p;1 (0) Y ).
Actually, there exists an exact sequence
0 ! G(;) !i H1 (X n p;1 (0) Y ) ! H1 (E ) ! 0:
Since H1 (E ) is a torsion free group, the above sequence splits.
Notice that for any s 2 S we have exactly one ds 2 D such that (s ds ) 2 E .
5. The Main Result
Assumption : In this section, X is a two-dimensional complex analytic space,
x 2 X a normal point on X , and Y X a reduced divisor with x 2 Y . Assume
that X n Y is smooth and connected. Let V be a polarized variation of Hodge
structure on X n Y such that its underlying representation is abelian and quasiunipotent. Consider K := im Aut(H ) and its torsion subgroup T . If K=T 6= 0,
we assume that there exist wj 2 OX (j = 1 : : : s), such that Y = sj=1 Z (wj ), and
w = (w1 : : : ws ) : X n Y ! (C )s induces an epimorphism w : H1 (X n Y ) ! Zs
which ts in the following commutative diagram:
1 (X n Y )
?w
Zs
-
f
-
K
?
K=T
For such a V, the limit mixed Hodge structure LV 2 MHS (H1(X n Y )) exists by
9] (cf. Example 2.8). Let M = j V where j : X n Y ! X is the natural inclusion.
Let p : X ! C be a holomorphic function. Let ; be a decorated resolution graph
of p with respect to (Y x) (cf. Example 4.5) and Spp; (LV) the invariant dened
in Denition 4.4 via the composed map G(;) ,! H1 (X n p;1 (0) Y ) ! H1 (X n Y ).
Our key result is:
Theorem 5.1. Let X and M be as above. Then:
Spp (M p x) = Spp;(LV):
158
A. Nemethi and J.H.M. Steenbrink
Recall that the spectrum Spp (M p x) of a mixed Hodge module M with
irreducible one-dimensional support Y is zero if pjY 0 otherwise it can be
computed as follows. Since M is a polarizable admissible variation of Hodge
structure on Y n fxg, it has a limit mixed Hodge structure LM. The topological information from p is the degree deg(pjY ) of the map pjY : Y ! C. Then
Spp (M p x) = cdeg(pjY ) (Spp(LM)) (cf. Example 3.1).
If Ys is one of the irreducible components of the critical locus Y of M and ; is
a decorated resolution graph of p with respect to (Y x), then deg(pjYs ) = m(ds )
where (s ds ) 2 E .
Theorem 5.2. Let X and M be as in the Assumption. Let Y = s2S(Y )Ys be the
irreducible decomposition of (Y x), iYv : Yv ! X and j : X n Y ! X the natural
inclusions. Then:
X X k
Spp (M p 0) = Spp;(Lj M) +
(;1) cm(ds) Spp(LHk i!Ys M):
s2S (Y ) k
pjYs 60
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. Let : U ! X be a resolution of p;1 (0) Y as in Example 4.5.
Set N = H0 M and NE = iE p N , where iE : E = ;1 (0) ! U is the
natural inclusion. Now it is clear that H0 N = M modulo terms with support in
f0g, and supp Hj N f0g if j =
6 0. Therefore, p Hj N = p M if j = 0 and
= 0 if j =
6 0. Hence, by (4),
(5)
Hj p N = 0 if j =
6 0:
Consider the following isomorphisms: i0 p M = i0 p H0 N (4)
= i0 H0 p N (5)
=
(
)
i0 p N = iE p N = NE :
The relation () follows from i0 = iE . For this, notice, that = ! because
is proper (10] (4.3.3)), and then use loc. cit.] (4.4.3).
For each d 2 D, let D~ d = Dd n d 2D\Vd Dd and kd : D~ d ,! Dd its inclusion. For
each e = (d d0 ) 2 E~ denote by ie : Dd \ Dd ! E the natural inclusion. Then by
0
0
0
(2) one has the following distinguished triangle:
! e2E~(ie ) i!e NE ! NE ! d2D (kd ) kd NE !
By the additivity of the functor Spp, we obtain:
X
X
(6) Spp(i0 p M) = Spp( NE ) = Spp(i!e NE ) + Spp( (kd ) kd NE ):
e2E~
d2D
(Everywhere, the action is the natural monodromy provided by .)
Fix d 2 D. Let Dd0 = D~ d n St(p;1 (0) Y ) = Dd n v2Vd Dv , and jd : Dd0 ,! Dd
denotes the natural inclusion.
Lemma 5.3. The restriction to Dd0 of the module NE0 = (kd) kdNE is smooth (i.e.,
rat jd NE0 is a local system). Moreover, it satises
NE0 = (jd ) (jd ) NE0 :
(7)
Spectral Pairs
159
Proof. By construction: jdNE0 = jdNE = jdp N :
Since ratN restricted to U n D is a local system, and Dd0 is a smooth divisor in
U n v2Vd Dv , the sheaf rat jd p N is a local system, too.
The obstruction to the isomorphism (7) lies in the points P 2 Dd \ (v2S Dv ) =
~
Dd n Dd0 .
Take P = Dd \Dv such that pjYv 6 0 (v 2 S ). Then the assumption M = j j M
and (2) give p i!Dv N = 0. Now (3) and the commutativity of the vanishing cycle
functors complete the argument in this case.
If P = Dd \ Dv such that v 2 S and pjYv 0, then see 11] (4.7).
Notice that jd NE0 = jd NE , and by Lemma 5.3, (kd ) kd NE = (jd ) jd NE . Since
the isomorphism H (Dd (jd ) jd NE ) = H (Dd0 jd NE ) is compatible with the mixed
Hodge structures, one has:
Spp( (kd ) kd NE ) = Spp(H (Dd0 jd NE )):
Step 2. The identity Spp(H (Dd0 jd NE )) = Sppd(LV):
By the additivity of Spp (see 2.13.a) it is enough to prove
X
(8)
Spp(H (Dd0 GrlW jd NE )) = Sppd(LV):
l
Since jd NE is smooth (Lemma 5.3), Vdl := GrlW jd NE is a polarizable variation
of Hodge structure on Dd0 .
Lemma 5.4. The representation associated with the local system ratVdl is abelian
and quasi-unipotent.
Proof. Dd0 has a neighbourhood homeomorphic to Dd0 fdiscg.
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 nite.
Hence a nite cover C~ of C has trivial local and global monodromies. Therefore a
global marking C~ ! D can be dened in the moduli space of Hodge structures.
This by Gri ths' 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 2 MHS (H1 (X n Y )). Its
representation denes a locally constant abelian variation M1 on X n Y . In 9],
among other facts, the following is proved:
Lemma 5.6. Let (C x) (X x) be a curve with C \ Y = fxg. Let L(MjC ),
respectively L(M1 jC ) be the limit mixed Hodge structures at x of the restrictions
of M, respectively of M1 , to C . Then GrW L(MjC ) = GrW L(M1 jC ).
Now, if we replace in the above construction M by M1 , then we obtain a
variation V1dl instead of Vdl
160
A. Nemethi and J.H.M. Steenbrink
Lemma 5.7. The variations of Hodge structure Vdl and V1dl on Dd0 are iso-
morphic.
Proof. Both are abelian variations by Lemma 5.4, with at Hodge bundles by
Lemma 5.5. By construction, the underlying representations are the same. We
have only to show that in a xed point P 2 Dd0 the stalks are isomorphic (by an
isomorphism, which is compatible with the representations).
Let C be a transversal slice to Dd0 at a point P 2 Dd0 and t a uniformizing
d W
parameter of (C P ). Then (Vdl )P ' GrlW tmd (MjC ) ' m
i=1 Grl L(MjC ). Similar 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
(9)
Spp(H (Dd0 V1dl )1]) = Sppd(LV):
l
Notice that both sides of (9) depend only on the limit mixed Hodge structure LV.
(pq)
Now, (LV ) 2 MHS (H1 (X n Y )) splits in a direct sum = pq di=1
pq
pq
i d(p q) = dim LV . The construction of V1dl preserves this decompo(pq) pqi
sition, therefore V1dl = p+q=l id=1
V1d . We have to show that
pqi )1]) = Spp (pq ):
(10)
Spp(H (Dd0 V1
d i
d
In the sequel we omit the indices p q and i. Moreover, we can assume that is of
type (p q) = (0 0).
Lemma 5.8. The variation V1d is md-dimensional. It has a direct sum decompod ;1 k
sition m
k=0 Vd in one-dimensional locally constant variationsk of C-Hodge structure (of the same type (0 0)), such that the monodromy of Vd around the points
(Pd \ Pv )v2Vd is exp(;2iRdkv ). The monodromy action on Vdk given by the vanishing cycle functor is exp(2i kd ).
Proof. The verication is local in small neighbourhoods of the points Dd \ Dv (v 2
Vd). Here, in a suitable coordinate system = xmd ymv . The verication is left
to the reader.
Proof of (10). By Lemma 5.8, we have to verify only:
(11)
Spp(H (Dd0 Vdk )1]) = Sppkd ():
In order to compute the left hand side of (11), we have to compute the dimensions
pq
k
hpq of the spaces GrpW+q GrFp H (Dd0 Vdk ). Then hpq
= h if = exp(2i d ), and
= 0 otherwise.
Let ! (log "d) be the complex of meromorphic dierentials on Dd with at worst
logarithmic poles along "d = v2Vd (Dd \ Dv ), and let V denote Deligne's canonical
extension of Vdk C ODd . Now, H (Dd0 Vdk ) = H(Dd K ), where K is the
complex fr : V ! !1 (log "d ) Vg. The Hodge ltration of this complex is Deligne's
\ltration b^ete" p , therefore the rst term of the Hodge spectral sequence is
pq
F E1 = H q (Dd K p ). This spectral sequence degenerates at E1 .
If Rdk = 0, then V = ODd . Hence E100 = C E101 = Cgd and E110 = Cgd + d ;1 .
This case corresponds to the trivial at bundle, therefore we recover exactly the
mixed Hodge structure of Dd n f d pointsg. In particular, H 0 = C has type (0 0),
0
0
Spectral Pairs
161
E101 has type (0 1), Cgd E110 has type (1 0), and C d ;1 = E110 =Cgd has type
(1 1).
Assume that Rdk 6= 0. Then F E1pq = 0 if p + q 6= 1 because degV = ;Rdk < 0
and deg(O(;") V ) = Rdk ; d < 0. Then by the Riemann-Roch Theorem,
dim E101 = gd + Rdk ; 1, and dim E110 = gd + d ; Rdk ; 1. This means that the
only nontrivial cohomological group is H 1 = H 1 (Dd0 Vdk ). In order to compute its
Hodge numbers hpq , we need the weight ltration too.
The weight ltration of the complex K is W;1 K = 0, W0 K = fr : V !
im rg, and W1 K = K . The extension W0 is a resolution of j Vdk and (by 20])
H 1 (Dd j ratVdk ) is pure of weight 1. On the other hand R1 j ratVdk = C d ; dk
and it induces in H 1 a quotient of weight 2. Therefore, the only nonzero Hodge
numbers are: h10 h01 and h11 . More precisely: h11 = d ; dk h01 = gd + Rdk ; 1,
and h10 = gd + dk ; Rdk ; 1. Now the expression for the spectral pairs readily
follows.
Example 5.9. Assume that d = 1. Then Rdk = 0 for any k. Therefore Sppkd =
;( kd 0) + gd( kd 1) + gd( kd + 1 1) for any k.
Example 5.10. Assume that gd = 0, d = 2 and Vd = fv wg. Then Rdk =
Rdkv + Rdkw 2 f0 1g. Using the obvious fact that for x y 2 R with x + y 2 Z
one has fxg + fyg = 1 if x 62 Z, and = 0 if x 2 Z, we obtain that Rdk = 1 if
mv (k + d ) 62 Z and = 0 otherwise.
v;m
d
Lemma 5.11. Let ld = g:c:d:(md mv ) = g:c:d:(mv mw ). The following assertions
are equivalent:
mv (k0 + d ) 2 Z.
a) There exists k0 2 Z such that v ; m
d
b) There exists d 2 R=Z such that
f g = fm =l g
v
v d d
f dg = fmdd=ldg:
c) There exists d 2 R=Z such that
f g = fm =l g
v
v d d
f w g = fmw d =ldg:
(Notice that b) or c) determines d uniquely.)
Moreover, if one of these conditions hold, then:
mv
(k ; k0 )mv 2 Z:
v ; m (k + d ) 2 Z ,
m
d
d
Proof. Use the relation v] + w] + edd] = 0 (cf. Denition 4.1.e).
Therefore, the lemma implies that if gd = 0 and d = 2, then
Pld;1 k+d
k+d
d exists
Sppd() = 0 k=0 ;(f ld g 0) + (f ld g + 1 2) ifotherwise.
We will see later that the expression Sppe() (e 2 E 0 ) is exactly of this type.
162
A. Nemethi and J.H.M. Steenbrink
Step 3. The computation of Spp(i!eNE ).
Fix a node e = (v w) 2 E~. Let us modify the resolution by a blowing up at
B . Dene D0 and
the point P = Dv \ Dw . The new resolution is 0 : U 0 ! U ;!
0
0
0
E similarly as in the rst case. Then D = D t fDeg, E = (E n fP g) t fPv Pw g,
where De is the new exceptional divisor, and Pv = De \ Dv , Pw = De \ Dw ,
where the strict transforms of Dv and Dw are denoted by the same symbols. Set
E 0 = (0 );1 (0), and consider the modules N 0 = H0 M and NE0 = H0 iE p N 0 .
Lemma 5.12. The mixed Hodge structures i!P NE , i!Pv NE0 and i!Pw NE0 are isomorphic in a way compatible with their monodromy action.
Proof. In a neighbourhood of P (resp. of Pv ) NE = p N (resp. NE0 = p N 0 ).
On the other hand, H (i!P p N ) = Hc (Fp N ), where Fp is the Milnor ber
of p in P . Similarly, H (i!Pv p N 0 ) = Hc (Fp N 0 ). But, for a suitable
representatives, one has an inclusion of Fp into Fp (induced by the blowing
up map) which is a homotopy equivalence, and it identies the sheaves N and N 0 .
Moreover, it preserves the monodromy action too. Since rat is an exact functor,
the lemma follows.
Now, if we write (6) for the resolutions and 0 , we obtain that
Spp(i!Pv NE0 ) + Spp(i!Pw NE0 ) + SppDe (0 ) = Spp(i!P NE ):
Using Lemma 5.12 one has Spp(i!e NE ) = ;SppDe (0 ). Now, the result follows from
the computation of Example 5.10.
0
0
0
0
0
0
0
6. Examples
6.1. Abelian coverings. Consider a connected normal surface singularity (X x)
()
and a covering : (X x) ! (C2 0) which is ramied over . Let = Sv=1
v
be the irreducible decomposition of . Consider a germ: p : (C2 0) ! (C 0). We
are interested in spectral pairs Spp (QHX 2] f x), where f is the composed map
f = p . The pair ( (X x)) is uniquely determined by the nonramied covering
X = ;1 (B n ) ! B n 16] (here (B ) is a good representative).
In particular, the mixed Hodge module M = H0 QHX 2] on (C2 0) is completely
determined by the exact sequence
1 ! 1 (X ) ! 1 (B n ) ;! G ! 1:
The critical locus of M is contained in and the representation of MjBn is the
induced representation by of the regular representation G of G.
Now, (4) assures, that H ix f QHX 2] = H i0 p M, therefore
Spp (QHX 2] f x) = Spp (M p 0):
Suppose, that G is abelian. Then Theorem 5.2 can be applied. The variation
MjBn is a at variation of Hodge structure. It can be identied with (QjGj F W ),
where GrFp QjGj = GrlW QjGj = 0 if p 6= 0 or l 6= 0. The underlying abelian
representation is
ab = G ab : H1 (B n Z) ! GL(jGj Q)
where ab : H1 (B n Z) ! G is induced by . Actually, this is the description of
LM, too. In particular Spp; (LM) is well-dened.
Spectral Pairs
163
Let v 2 S (). The variation i!v M has the following description: If j : B n !
B is the natural inclusion, then R0j j CX = CX and for any point Pv 2
v ; f0g one has dim(R1 j j CX )P = dim(R0 j j CX )P = av , where above
Pv there lie exactly av points of X . The above spaces can be recovered from the
representation ab via the expression CjvGj = ker(ab (Mv ) ; 1) CjGj. Let dv 2 D
be the unique vertex such that (v dv ) 2 E . Then CjvGj is a sub-Hodge structure of
CjGj with the automorphism ab(Mdv ]).
The above discussion shows that M ! j j M is one-to-one. In particular,
iHk i!v M = 0 if k 6= 1. The exact sequence (3) shows that H1 i!v M is of type
(1 1). Its restriction to v ; f0g is a locally constant variation it can be identied
with CjvGj(;1) with monodromy ab (Mdv ]). Similarly as above, its limit LH1 i!v M
has the same presentation too.
Finally, notice that degree(f jv ) = m(dv ).
Proposition 6.1.
X
Spp (QHX 2] f x) = Spp; (CjGj ab ) ;
cm(dv ) Spp(CjvGj(;1) ab (Mdv ])):
v2S ()
pjv 60
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 = fz1z2 = 0g.
Let Anq = C2 =Zn be the cyclic quotient singularity, where the action of G = Zn
is given by (u1 u2) 7! (u1 p u2 ), = exp(2i=n) and pq 1(mod n). Then
: Anq ! C2 given by zi = uni (i = 1 2) denes a covering which is ramied over
fz1z2 = 0g.
In this case, the covering transformation group is G = Zn and
ab : H1 (B n Z) = Z2 ! Zn
is ab (e1 ) = ^1 ab (e2 ) = ;cq:
On the other hand, for any v 2 S () one has CjvGj = C and the transformation
ab
(Mdv ]) is the identity. Therefore:
Spp (QHX 2] p x) = Spp;(Cn ab ) ;
dv ;1 X mX
v2S () k=0
pjv 60
1 + mk 2 :
dv
Example 6.3. Take p(z1 z2) = z1 + z2 in Example 6.2. For x 2 R take (x) = 0
if x 2 Z, and = 1 if x 62 Z. Then:
nX
;1 ;i qi ;
2 ; 1 ; qi :
Spp (QH 2] p x) = ;(0 0) +
2;
X
i=1
n
n
n
Remark 6.4. Above, we computed the spectral pairs of a composed function f .
By our general result, we can compute Spp (QHX 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).
A. Nemethi and J.H.M. Steenbrink
164
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 =
QHX 2]. The limit LM is the one-dimensional
mixed Hodge structure QH with
p
W
H
H
trivial action and Grl Q = GrF Q = 0 if l =
6 0 or p =
6 0. Denote the set of
spectral pairs merely by Spp (f ). By Theorem 5.2 one has: Spp (f ) = Spp;(QH ),
where ; is the decorated resolution graph of f .
Let h = rank H1 (;) be the number of independent cycles of ;. Alternatively,
h = rank H1 (E ) ; rank H1 (E~ ), where E~ is a smooth model for E = ;1 (x).
By a computation, we obtain an \almost symmetric" form of Spp (f ):
Proposition 6.5. Let g = Pd2V gd, Rdk = Pv2Vd fk mv =mdg, and R(n) =
f1 : : :n ; 1g. Then:
Spp (f ) = (h ; 1)(0 0) + (h + #S ; 1)(1 2) + g(0 1) + g(1 1)
X X
X X k
+
(1 + m(ke ) 2) +
(( m 0) + (2 ; mk 2))
s
s2S k2R (m(es ))
;
+
X X
(( mk 0) + (2 ; mk 2))
d
d2D k2R (md )
Rkd =0
X X
d2D k2R (md )
Rkd 6=0
X X
d2D k2R (md )
e
e
d
(Rdk ; 1) (( mk 1) + (2 ; mk 1))
+
e2E~ k2R (me )
d
d
gd (( mk 1) + (2 ; mk 1)):
d
d
Here m(es ) = g:c:d:(m(s) m(ds )), where ds is the unique vertex in D with (s ds ) 2
E.
Example 6.6. Let (X x) = fz2 = (x + y2)(x2 + y7)g (C3 0) and f (x y z) = z.
Then the decorated resolution graph is the following:
(;3)
4 s
1
9 s
(;1)
(;3)
s
3
-
1
s 4
(;3)
In parentheses are the numbers ed , the others are the multiplicities. All exceptional divisors are rational. The spectral pairs are:
Spp (f ) =
8
X
k=1
(2k=9 1) + (2=3 1) + (4=3 1) + 2 (1 2):
Remark 6.7. Using (3) one has: Spp'(f ) = Spp (f ) + (0 0).
Spectral Pairs
165
7. Topological Series of Curve Singularities
LetQp : (Cm2 0) ! (C 0) be a curve singularity with irreducible decomposition
p = rj=1 pj j , and (C2 0) a reduced one-dimensional analytic space-germ
with irreducible decomposition si=1 i . Let ; be the decorated resolution graph of
p;1 (0) . Let S () (resp. S (p)) be the subset of S corresponding to the strict
transforms St(i ) (resp. St(p;j 1(0))). Recall that the multiplicity of a vertex v 2 V
is the multiplicity of p on the divisor Dv . In particular, the multiplicities of the
vertices v 2 S are: fmj grj=1 corresponding to s 2 S (p), and ms = 0 corresponding
to s 2 S () n S (p) In the sequel we use the index notation j 2 f1 : : : rg for the
set S (p).
The schematic graph of ;, together with multiplicities, is:
0
..@slr
.;
..
.
(12)
0
- mr
..
.
..@sl1
.;
S () n S (p)
..
.
- m1
S (p)
D
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 to S () n S (p), those from
the right-hand side to S (p).
For each j 2 S (p), let dj 2 D be the unique vertex which is adjacent to j . The
corresponding exceptional divisor (Ddj ) is denoted by Ej and its multiplicity is
lj = mdj . The intersection point Dj \ Ej is denoted by Pj .
Denition 7.1. The topological series of curve singularities belonging to p, relative
to , consist of all curve singularities p0 such that there are decorated resolution
graphs ; of p;1 (0) , and ;0 of (p0 );1 (0) respectively, such that ; has form
(12), and ;0 has the following form:
0
..
.
0
..@s lr
.;
lr +mr
..@s l1
.;
l1 +m1
..
.
;@...
s
;@...
s
;r
...-
;1
...-
..
.
166
A. Nemethi and J.H.M. Steenbrink
The index sets D0 S 0 and E 0 are dened similarly as D S and E . For j 2
f1 : : : rg, the new index set of exceptional divisors (resp. of strict transforms or
arrows) of ;j is denoted by Dj (resp. Sj ). Therefore D0 = D t (tj Dj ) S 0 =
(S n S (p)) t (tj Sj ), and E 0 = E t (tj Ej ) t (tj fej g), where Ej are the edges in ;j ,
and ej is the new edge which joins ; and ;j .
It can happen that the box ;j is empty. In that case, instead of ;j we have just
an arrow as in the case of ;.
7.1. Geometric meaning. Let Uj be a small neighbourhood of Pj = Dj \ Ej .
Fix a coordinate system (x y) in Uj so that: Uj \ Dj = fy = 0g Uj \ Ej = fx = 0g
and p jUj = xlj ymj .
Denition 7.1 has the following meaning: p0 jUj = p~j (x y)xlj , where p~j is a
curve singularity at Pj such that:
a) the line fx = 0g is not in the tangent cone of fp~j = 0g,
b) the multiplicity of p~j at Pj is mj .
Q
P
jk
If p~j = rkj=1 pm
jk is the irreducible decomposition of p~j , then k mjk (x pjk )Pj =
mj . Here ( )Pj denotes the intersection multiplicity at the point Pj .
We dene the numbers fj grj=1 as follows. If j 2 S (p) n S (), then the index set
Sj is exactly f1 : : : rj g we set j = 0. If j 2 S (p) \ S () (i.e., fpj = 0g = sj ),
then one of the following conditions must hold:
1. St(sj ) = fpjk = 0g for some k (i.e., pjk (x y) = y), or
2. St(sj ) 6 fp~j = 0g.
In the rst case, Sj = f1 : : : rj g, and we set j = 0. In the second case, Sj =
f1 : : : rj gfsj g, and we take j = 1. This shows, that the index set S 0 () nS 0 (p0 )
(which is crucial in the main theorem 5.2) is (S () n S (p)) t tj: j =1 fsj g.
It is clear, that the diagram ;j is a part of the decorated resolution graph ;~ j
of p0 jUj (relative to fxy = 0g). Actually, the resolution graph ;~ j can be
obtained from ;j by adding an arrow corresponding to the strict transform of
fx = 0g = Ej \ Uj . The arrow takes the place of the new edge ej , which in ;0 joins
; and ;j . Notice, that the multiplicity of (the strict transform of) Ej is lj .
(13)
;~ j :
lj
ej
lj +mj
;@...
s
...-
Remark 7.2. A compatible denition of the topological series of curve singularities
can be given using Eisenbud-Neumann diagrams 3] (see 14] in the case = ,
and 6] in the general case).
7.2. Topologically trivial series.
Denition 7.3. 6] We say that p0 is an element of the \topologically trivial
series" belonging to p, relative to , if for any j 2 f1 : : : rg either p~j (x y) =
ymj a(x y) or p~j (x y) = (y + xbj b(x y))mj a(x y) where a b 2 OU j and bj 1.
In the denition, \topologically trivial" means that the topological types of the
germs p and p0 are the same, but the positions of their zero set with respect to are dierent.
Spectral Pairs
167
In the rst case (~pj = ymj a) one has j = 0 and the topological modication,
even relative to , is trivial. In the second case j = 1, and the diagram ;j is as
follows:
(14)
* mj
l
l
+
m
l
+
b
m
l
+(
b
;
1)
m
j
j
j
j
j
j
j
j
j
..@s
s
s
s
::: s
.;
E HH
0
| {z }
;
HHj
0
7.3. Intrinsic invariants. Notice that the choice of the graphs ; and ;0 is not
unique. Therefore, the numbers bj lj (or even the cycle Mdj ]) have no intrinsic,
geometric meaning. In our computations, we wish to express the spectral pairs in
terms of some intrinsic invariants. Actually, this paragraph makes the connection
between the set of numerical data used in the decoration of the resolution graph
and those used in the Eisenbud-Neumann diagrams.
Fix j 2 f1 : : : rg such that j = 1.
We dene
X
nj :=
;A;1 (dj di )mi i2S (p)nfj g
where A is the intersection matrix of ; as in Denition 4.1.
Let j be the multiplicity
of pj on Ej , i.e., pj jUj = x j y c(x y) c 2 OU j .
P
Then the relation dj ] = v2S ;A;1 (dj dv )v] gives a relation between the set of
multiplicities of pj on the components of D, namely: j = ;A;1 (dj dj ). On
the other hand, the same relation applied for the set of multiplicities of p gives:
lj = nj + j mj . Therefore:
(15)
bj mj + lj = aj mj + nj where aj := bj + j :
Notice
that S (p) = S (p0 ). Order the irreducible factors of p0 in such a way that
Q
0
p = j (p0j )mj , and Z (p0j ) in Uj is fy + xbj b(x y) = 0g
Lemma 7.4. The numbers aj nj and aj mj + nj have the following expression in
terms of intersection multiplicities at the origin:
Y
aj = m(pj p0j )
nj = m(pj (p0i )mi )
i6=j
aj mj + nj = m(pj p0 ):
Proof. m(pj p0j ) = deg(pj jZ (p0j )) = deg(x j yjy + xbj = 0) = bj + j = aj : The
others follow by similar argument.
8. Topological Series of Plane Singularities with Coe cients
in a Mixed Hodge Module
Let M 2 MHM (C2 0) be mixed Hodge module with singular locus such that
V = MjBn is a polarisable variation of Hodge structure. Here (B ) are some
representatives as usual.
Consider a curve singularity p and let p0 be an element of the topological series belonging to p, relative to . Our purpose is to compare Spp (M p 0) and
Spp (M p0 0). Since p0 is a modication of p in the neighbourhoods fUj grj=1 ,
168
A. Nemethi and J.H.M. Steenbrink
we expect that the dierence of their spectral pairs depends only on the germs
p~j : (Uj Pj ) ! (C 0) and the restriction of H0 M on rj=1 Uj . Since the restriction on Uj has abelian representation over Uj ; fxy = 0g, by our general principle,
we expect that we need only the corresponding limit mixed Hodge structures at the
points fPj gj (and the graphs of the germs fp~j gj ).
8.1. Limit mixed Hodge structures. Let : U ! B be the resolution of
p;1 (0) as in Section 7. Let V be the lifting of V on U n D and Vj its
restriction to Uj n D (j 2 S (p)). If j 2 S (p) n S (), then Gj := H1 (Uj n D Z) = Z,
and it is generated by e1 = Mdj ]. If j 2 S (p) \ S (), then Gj = Z2 , and it is
generated by e1 = Mdj ] and e2 = Mj ].
In both cases the variation Vj has abelian representation, therefore its limit
mixed Hodge structure LVj exists (13]). Set Lj V := GrW LVj 2 MHS (Gj ). Its
representation is denoted by j .
8.2. Intrinsic meaning. We show, that Lj V can be recovered in a more direct
way, without using the resolution .
Let Zj = fpj = 0g (C2 0). Let j : (C 0) ! (Zj 0) be the normalization of
Zj , and let x be a local uniformization of (C 0). Then Hj := GrW x H0 j pj M 2
MHM (f0g) is a mixed Hodge structure with two semi-simple monodromy actions:
Tjh (called the horizontal monodromy) is induced by pj and Tjv (called the vertical
monodromy) is induced by x .
Now, if is the resolution as above, then jSt(Zj ) : St(Zj ) ! Zj is the normalization of Zj . Moreover, if (x y) are the coordinates in Uj as in Section 7.1, then x is a
local coordinate in St(Zj ). By our notation (cf. Section 7.3) pj jUj = x j y c(x y),
where c 2 OU j . Then by the properties of Section 2, Hj = GrW x pj N ,
where N = H0 M. Obviously, this depends only on NjUj nfxy=0g = Vj hence
Hj = GrW x xj ycVj .
Notice that LVj = x y Vj , and this limit depends on the choice of the coordinates (x y). The ambiguity is modulo the action of exp( C ), where C is the
complex monodromy cone. Therefore Lj V = Hj .
By a local computation:
(16)
Tjh = j (Mj ]) and Tjv = j (Mdj ] ; j Mj ]):
Notice, that if j 2 S (p) n S (), then Tjh = j (Mj ]) is the identity and the only
essential action is Tjv = j (Mdj ]) (cf. Section 8.1).
8.3. Further computations. Similarly as in Section 8.2, we can consider the
mixed Hodge structure H~ j = GrW x H0 j 'pj M 2 MHM (f0g) with semi-simple
monodromies T~jh (induced by 'pj ) and T~jv (induced by x ).
Let Hj = Hj6=1 Hj1 be the eigenspace decomposition given by Tjh similarly
H~ j = H~ j6=1 H~ j1 provided by T~jh.
On the other hand, if j = 1 (e.i. Zj = sj , but p0 jsj 60 ), then j 2 S () n
S (p0 ) and to apply Theorem 5.2 we need the limit mixed Hodge structures Vjk =
GrW LHk i!sj M, too. Their monodromy is denoted by Tjk .
We have the following relations between our limit mixed Hodge structures and
their semi-simple actions:
Spectral Pairs
169
Proposition 8.1. Let j 2 S (p) \ S (). Then:
can : (Hj6=1 Tjh Tjv )!
~ (H~ j6=1 T~jh T~jv ) is an isomorphism in MHS (Z2 ), and
V ar (H T v )(;1) ! (V 1 T 1 ) ! 0
0 ! (Vj0 Tj0 ) ! (H~ j1 T~jv ) ;;!
j1 j
j j
is an exact sequence in MHS (Z).
Proof. Use (1), (2) and (3) and the fact that the vanishing cycle functor is exact.
9. The Spectral Pairs of Series of Plane Singularities
9.1. General formula. Let M 2 MHM (C2 0) be a polarizable mixed Hodge
module with critical locus . For simplicity, we assume that M restricted to the
complement of is pure (but this is not essential because of Lemma 2.9.c). Let
p : (C2 0) ! (C 0) be a curve singularity, and p0 a germ in the topological series
belonging to p relative to .
Fix j 2 S (p). In Section 8.1 we constructed a representation (Hj j ) 2 MHS (Gj ).
If ;~ j denotes the decorated resolution graph of p0 jUj : (Uj Pj ) ! (C 0) (relative to fxy = 0g), and ej its edge which joins ; and ;j (cf. Section 7.1), then
Spp;~ j (j ) and Sppej (j ) are dened (cf. Denitions 4.3{4.4). (The latter one is
P
T (p q)Sppe (pq ), with the notations of Denition 4.4.) We let
pqi
j
i
Spp;~ j ej (j ) := Spp;~ j (j ) + Sppej (j ):
Theorem 9.1. Let j and sj be as in Section 7.1. Then one has:
X
Spp (M p0 0) ; Spp (M p 0) =
Cj where
Cj = Spp;~ j ej (j ) + j
j 2S (p)
k
(;1) cdeg(p jsj ) Spp(Vjk Tjk ):
X
k
0
Notice also that deg(p0 jsj ) is equal to the intersection multiplicity mPj (y p0 jUj ).
Proof. Using (2) and Lemma 2.9 we can assume that M = jV, where j : B n !
B is the inclusion. Consider the resolution 0 corresponding to the graph ;0 . Notice
that in the rst step of Section 5.1 we did not use the commutativity assumption.
Therefore, the relation (6) holds for both p and p0 . If d 2 D (or e 2 E~) then in a
neighbourhood of Dd (or of Pe , respectively) the functions p0 and p dier only
by an invertible function. Therefore, their contributions are equal. Using again the
rst step of Section 5.1 applied to p0 jUj , one has:
Spp(i0 p M) ; Spp(i0 p M) = Sppej (j ) + Spp(iPj p M):
Now, since Gj is abelian, the result follows from Theorem 5.1.
0
0
We end this subsection with the following remark. By Lemma 2.9.e, one has:
Spp'(M p0 0) ; Spp'(M p 0) = Spp (M p0 0) ; Spp (M p 0)
for any M, p and p0 .
A. Nemethi and J.H.M. Steenbrink
170
9.2. The case of topologically trivial series. In the sequel denotes the characteristic map of R n Z, i.e., : R ! f0 1g is dened by (x) = 0 if x 2 Z and = 1
otherwise.
Fix a character : Z2 ! C of nite order. Choose ! and such that (e1 ) =
exp(2i!) and (e2 ) = exp(2i). Given a triple (n m a) of integers (where n 0,
m > 0 and a > 0), we dene in ZQ Z] the elements C () and C' () as follows.
For any k 2 f0 : : : n + am ; 1g we let
a! :
k = k n++ +
am
Then dene C () :=
n+X
am;1
(1 ; f;!g + fk g + f;! + mk g ; fmk g
k=0
2 ; (!) ; (mk ) + (;! + mk ))
and dene C' () :=
n+X
am;1
k=0
(f!g + fk g + f;! + mk g ; fmk g (!) ; (mk ) + (;! + mk )):
For 2 MHS (Z2 ) dene
C () :=
(pq)
X dX
pq i=1
T (p q)C (pq
i )
(and similarly C' ()), where the characters pq
i 's are as in Denition 4.4.
Example 9.2. 1) Assume that m = 1. Since ! and are dened modulo an
integer, we can take some representatives ! 2 0 1) and 2 n! n! + 1).
Then
n+X
a;1
C' () =
(k (!) ; (k ) + (;! + k )):
k=0
2) Assume that m = 1 and n = 0. Then
k + aX
;1 k
+
(!) ; ! +
+
C () =
!+
'
a
k=0
a
k + a
:
Now we return to our representations (Hj j ) and (H~ j ~j ), (j 2 S (p)). Recall
that in Gj = Z2 one has: e1 = Mdj ] and e2 = Mj ]. We dene the representation
j : Z2 ! AutMHS (Hj ) by the following change of bases: j (e1 ) = j (e2 ) and
j (e2 ) = j (e1 ; j e2 ). Then j (e1 ) = Tjh and j (e2 ) = Tjv by (16). Similarly we
dene ~j .
The relation between the above invariants is given in the following lemma.
Lemma 9.3. Let (Hj j ), (H~ j ~j ) 2 MHS (Z2) and (Vjk Tjk ) be as in Section 8.
Dene C and C' using the triplet (nj mj aj ). Then
X
C (j ) + (;1)k cnj +aj mj Spp(Vjk Tjk ) = C' (~j ):
k
Spectral Pairs
171
Proof. Notice that if (e1) 6= 1 (i.e., ! 62 Z) then C () = C'(). Otherwise,
C' () = T (;1 ;1)C () = cn+am Spp(C (e2 )). Now use Proposition 8.1.
Now we consider p0 , a germ in the topological series belonging to p relative to ,
as in Section 7.2. From the rst step of Section 5.1, it is clear that the correction Cj
is zero provided that j 2 S (p) nS (). In the sequel we assume that j 2 S (p) \S ().
We also assume that p~j = (y + xbj b(x y))a(x y) otherwise Cj = 0 again.
We will compute Cj in a more direct way. First notice that our relations do not
depend on the choice of the resolution graph. In the case of p0 , we will take the
graph ;0 , but for p we will use the graph of modied by bj blowing-ups. This
graph will be denoted by ;00 , and its schematic form is the following:
(17)
..@slj
.;
lj +mj
s
| {z }
s
: : l:j +(bj ;s 1)mj lj +bj ms j
E
00
- mj
;
The arrow corresponds to the strict transform of Zj = sj .
By a similar argument to that in the proof of Theorem 9.1, one has:
(18)
Cj = SppE (j ) ; SppE (j ) +
0
X
k
(;1)k cnj +aj mj Spp(Vjk Tjk )
where the vertices E 00 and E 0 are drawn on the corresponding graphs, (cf. (17) and
(14)).
Lemma 9.4. SppE (j ) ; SppE (j ) = C (j ):
Proof. We can assume that j is one dimensional. Let j(e1) = exp(2i!) and
0
j (e2 ) = exp(2i).
In the case of p (graph ;00 ) the invariants are:
{ the degree of E 00 is 2
{ the multiplicities of the adjacent vertices are: nj + (aj ; 1)mj and mj { the monodromies around the adjacent vertices are: exp(2i( + (aj ; 1)!))
and exp(2i!).
In the case of p0 (graph ;0 ) the invariants are:
{ the degree of E 0 is 3
{ the multiplicities of the adjacent vertices are: nj + (aj ; 1)mj , mj and 0
{ the monodromies around the adjacent vertices are: exp(2i( + (aj ; 1)!)),
1 and exp(2i!).
The multiplicity of E 0 and E 00 is equal to nj + aj mj , and the monodromy around
them is exp(2i( + aj !)).
The verication is left to the reader.
The main result of this section is the following:
Theorem 9.5. Let M be a pure mixed Hodge module Qwith mcritical
locus , p0 a
j
germ of the topologically trivial series belonging to p = j pj , relative to . Let
A. Nemethi and J.H.M. Steenbrink
172
H~ j be the limit mixed Hodge structure of 'pj M with semisimple monodromies Tjh
(induced by 'pj ) and Tjv (induced by the monodromy of 'pj M). Then
X
Spp (M p0 0) ; Spp (M p 0) =
C'j (H~ j Tjh Tjv )
j : j =1
where C'j is C' with parameters (nj mj aj ).
Proof. Use Theorem 9.1, (18), Lemma 9.4 and Lemma 9.3.
Corollary 9.6. Let M and p p0 as in Theorem 9.5. Assume that p is irreducible.
Set
(pq)
(H~ j Tjh Tjv ) = pq di=1
(Hipq exp(2i!ipq ) exp(2ipq
i ))
pq
where Hi is one-dimensional C-Hodge structure of type (p q), and !ipq pq
i 2 0 1).
The integer a denotes the intersection multiplicity m(p p0 ). Then:
Spp (M p0 0) ; Spp (M p 0) =
pq aX
;1 pq
pq X
T (p q)
!pq + k + i (!pq ) ; !pq + k + i + k + i
:
pqi
k=0
i
a
i
i
a
a
Proof. If p is irreducible, then S (p) has only one element whose multiplicity is one,
and n = 0. Now apply Example 9.2.
10. Spectral Pairs of Series of Composed Singularities
10.1. Some invariants of an ICIS with 2-dimensional base space. Let :
(X x) ! (C2 0) be an analytic germ such that both (X x) and (;1 (0) 0) are
isolated complete intersection singularities, and dim X = n 2. By 18] there
exists a projective extension & : X& ! (C2 0) of such that the central ber
& x) ! (C2 0)
X& 0 = &;1 (0) has only x as singular point and the germ & : (X
&
&
is analytically equivalent to . Fix a good representative : X ! B of & such
that a suitable restriction of it is a good representative : X ! B of . We let
C X be the critical locus of and = (C ) the (reduced) discriminant locus
with irreducible decomposition si=1 i . Over i lie the irreducible components
Ci1 : : : Citi of C . The degree of the rectriction : Cil ! i is dil .
Let i : U ! i ( resp. il : U ! Cil ) be the normalization and zi (resp. zil ) a
uniformizing parameter in U .
Let pi be a generator of the ideal of i . Then the support of the mixed Hodge
module 'pi QHX n] is a subset of X&0 C . For any i 2 f1 : : : sg and l 2 f1 : : : ti g
we dene (H~ il T~ilh T~ilv ) 2 MHS (Z2 ) by
H~ il := GrW zil H 0 il 'pi QHX n]:
The (horizontal) monodromy Tilh is induced by 'pi , the other (vertical) one Tilv
by zil . By the properties of the vanishing cycle functors, they commute.
The structure (H~ il T~ilh) has another interpretation too. For this, take a point
P 2 i ; f0g and a transversal slice S to i at P . Choose P 0 2 ;1 (P ) \ Cil .
Then : (;1 (S ) P 0 ) ! (S P ) denes an isolated hypersurface singularity. Its
limit mixed Hodge structure on its reduced cohomology is exactly H~ il and T~ilh is
the semi-simple part of its monodromy.
Spectral Pairs
173
If H is a mixed Hodge structure with commuting automorphisms T h and T v ,
dene cm (H ) as m
i=1 H with automorphisms:
cm (T h )(x1 : : : xm ) = (T h(x1 ) : : : T h(xm )) and
cm (T v )(x1 : : : xm ) = (T v (xm ) x1 : : : xm;1 ):
The new structure in HMS (Z2 ) is denoted by cm (H T h T v ).
Dene the mixed Hodge module M 2 MHM (C2 0) by M = H0 & QHX n].
Then the critical locus of M is and MjBn is pure. In Section 8.3 we associated
the set of invariants (H~ i T~ih T~iv )si=1 2 MHS (Z2 ) with such a module, namely
H~ i = GrW zi H0 i 'pi M. The horizontal monodromy T~ih is induced by 'pi , the
vertical one T~iv by zi .
Lemma 10.1. The following isomorphism holds:
h v
i cd (H
(H~ i T~ih T~iv ) = tl=1
il ~ il T~il T~il ):
Proof. Use (4).
10.2. Topological series of composed singularities. Let be as above and
p : (C2 0) ! (C 0) an arbitrary analytic germ.
Denition 10.2. The topological series of composed singularities belonging to f =
p consists of all composed singularities f 0 = p0 such that p0 is in the topological
series of curve singularities belonging to p relative to the discriminant locus of
.
For simplicity, we will consider only germs f 0 = p0 , where p0 belongs to a topologically trivial series of p, relative to . For these germs we extend Theorem 9.5
and Corollary 9.6. The interested reader can formulate easily the corresponding
result which extends Theorem 9.1 and
holdsmfor any p0 .
Q
r
We recall our notations. Let p = Qj=1 pj j be the irreducible decomposition of
p. Let p0 be as above. We write p0 = rj=1 (p0j )mj , where we ordered the irreducible
factors of p0 as in Lemma 7.4. For any j 2 S (p) \ S () (with j = 1) consider the
numerical invariants
Y
aj = m(pj p0j ) and nj = m(pj (p0i )mi ):
i6=j
Theorem 10.3. With the notation Spp (f ) = Spp (QHX n] f x) one has:
X
Spp (f 0 ) ; Spp (f ) =
j C'j (H~ j T~jh T~jv )
j 2S (p)\S ()
where C'j is C' with parameters (nj mj aj ).
Remark 10.4. Lemma 10.1 implies that
ti
X
h
v
~
~
~
C'j (Hj : Tj Tj ) = C' (nj djl mj aj djl )(H~ jl T~jlh T~jlv ):
l=1
Proof of Theorem 10.3. First notice, that Spp (f 0) ; Spp (f ) = Spp'(f 0) ;
Spp'(f ) (by Lemma 2.9.e).
Dene N = iX 0 'p QHX n]. Let j : X& 0 nfxg ,! X&0 be the natural inclusion. Then
j N = j 'p QHX n], because 'p QHX n] is supported in X&0 C and GrW j N is
A. Nemethi and J.H.M. Steenbrink
174
a constant variation of mixed Hodge structure with stalk isomorphic to 'p QHB 2].
Similarly we consider N 0 dened by p0 . Then GrW j N ' GrW j N 0 . Therefore,
by the rst distinguished triangle of (2), one has:
Spp'(f 0 ) ; Spp'(f ) = Spp(& N 0 ) ; Spp(& N ):
But & N = i0 'p & QHX . Since for j 6= 0 the module Hj & QHX is smooth
Spp'p Hj & QHX = Spp'p Hj & QHX
by Theorem 5.1. Therefore
Spp'(f 0 ) ; Spp'(f ) = Spp(M p0 0) ; Spp(M p 0):
Now apply Theorem 9.5.
Corollary 10.5. Assume that p is irreducible, and fp = 0g = j . Let
(pql)
(H~ jl T~jlh T~jlv ) = pq di=1
(Hipql exp(2i!ipql ) exp(2ipql
i ))
for any l = 1 : : : tj , where Hipql is one-dimensional C-Hodge structure of type
0
(p q), and !ipql pql
i 2 0 1). Let dl be the degree of : Cjl ! j and a = m(p p ).
0
Then Spp (f ) ; Spp (f ) =
adX
pql
pql
l ;1
pql
X
T (p q)
(!pql + k + i (!pql ) ; (!pql + k + i ) + ( k + i )):
0
lipq
k=0
i
adl
i
i
adl
adl
Proof. Apply Corollary 9.6 and Remark 10.4.
Example 10.6. The Yomdin's series (see 21, 15, 19, 11]). Let f : (Cn 0) !
(C 0) (n 2) be an analytic germ with one-dimensional critical locus ", which
has an irreducible decomposition tl=1 "l . The mixed Hodge module 'f QHX n] is
supported on " and its restriction on "l n f0g is an admissible variation of mixed
Hodge structure. Denote its limit by K~ l . Two natural semi-simple automorphisms
act on K~ l : T~lh induced by 'f , and T~lv induced by the monodromy of the restriction
'f QXH n]j"l n 0.
Let l : (Cn 0) ! (C 0) be a generic linear form. Then the pair := (f l) :
(Cn 0) ! (C2 0) denes an ICIS, and (") = 1 is one of the irreducible components of . Then
(K~ l T~lh T~lv ) = (H~ 1l T~1hl T~1vl ):
Notice that f = p where p(z1 z2 ) = z1 . Then for su ciently large a p0 (z1 z2) =
z1 + z2a is in the topologically trivial series belonging to p, relative to .
Therefore Spp (f + la ) ; Spp (f ) follows from Remark 10.4.
11. A Generalized Sebastiani-Thom Type Result
If (Hi W i Fi ) (i = 1 2) are mixed Hodge structures,
H1 H2 carries a
P then
1 W 2 and Hodge
mixed Hodge structure
with
weight
ltration
W
=
W
k
p
q
p
+
q
=
k
P
ltration F k = p+q=k F1p F2q . This mixed Hodge structure is still denoted by
H1 H2 .
A graded mixed Hodge structure is a nite direct sum H = k H k of mixed
Hodge structures fH k gk . Their category is denoted by GMHS . There is a natural
extension of the tensor product to GMHS by H1 H2 := k (p+q=k H1p H2q ).
Spectral Pairs
175
There is a natural tensor product MHS (G1 ) MHS (G2 ) ! MHS (G1 G2 ),
which associates with two elements, say i : Gi ! AutMHS (Hi ) (i = 1 2), an
element = 1 2 : G1 G2 ! AutMHS (H1 H2 ) dened by (g1 g2 ) =
1 (g1 ) 2 (g2 ).
If GMHS (G) denotes the graded version of MHS (G) (i.e., the category of representations = k k , where k 2 MHS (G)), then the above tensor product has
an extension
GMHS (G1 ) GMHS (G2 ) ! GMHS (G1 G2 )
given by 1 2 = k (p+q=k p1 q2 ).
There is an extension SppP: GMHS (Z) ! ZQ Z] of the map dened in
Denition 2.6 by Spp( ) = k (;1)k Spp(k ). Moreover if ; is as in Section 3,
then Spp; also extends by the same formula as above.
The map ( w) ( !) = ( + w + !) extends to a bilinear map ZQ Z] Z
ZQ Z] ! ZQ Z]. In general it is not true that Spp(1 2) = Spp(1) Spp(2)
but still one has the following result.
Lemma 11.1. Set (Hi i) 2 MHS (Z) such that 1 = 1H1 . Let cn (n 2 N) be
the map dened in Lemma 2.9.f. Then one has:
a) Spp(1 2 ) = Spp(1) Spp(2P
).
b) If 2 ZQ Z] such that = ( !) satises 2 Z, then
cn ( ) = cn ( ):
The easy proof is left to the reader.
In the sequel it is convenient to dene the map c1 as the zero map.
Let g : (X x) ! (C 0) be an analytic germ. Denote j Hj i0 g QHX n] by
(Hg Tg ) 2 GMHS (Z). Here Tg = g (1) is the semisimple part of the monodromy.
The decomposition g = g1 g6=1 gives a decomposition Hg1 Hg6=1 of Hg .
Therefore the spectrum Spp (g) := Spp(Hg Tg ) can be written as the sum of the
corresponding spectral pairs Spp1(g) and Spp6=1(g). There are similar notations
for ' instead of . By Lemma 2.9.e one has:
Spp (g) = Spp'(g) + (;1)n;1 (0 0):
We introduce the notation:
Spp!(g) = Spp'1(g) ; T (1 1)Spp1(g) = (1 ; T (1 1))Spp1 + (;1)n (0 0):
Consider an analytic germ p : (C2 0) ! (C 0) and the space-germ ( 0) given
by fcd = 0g, where (c d) are the local coordinates in (C2 0). Let ; be the decorated
resolution graph of p with respect to ( 0) (cf. Example 4.5). Notice that G(;) =
ZS = H1(C2 n ((p;1(0) )) Z). The group H1(C2 n Z) = Z2 is generated by
Mc] and Md ], where Mc (resp. Md) is an oriented circle in a transversal slice to
fc = 0g (resp. fd = 0g).
Let g : (X x) ! (C 0) and h : (Y y) ! (C 0) be two analytic germs which
dene ICIS's, and p : (C2 0) ! (C 0) an arbitrary analytic germ. Dene f :
(X Y x y) ! (C 0) by f (x y) = p(g(x) h(y)).
Consider the tensor product Hg Hh 2 GMHS (Z2 ). By the identication of
2
Z with H1(C2 n Z) one has:
(Mc ]) = Tg id and (Md]) = id Th :
176
A. Nemethi and J.H.M. Steenbrink
Via the map G(;) ! H1 (C2 n Z), induced by the inclusion, Hg Hh becomes
an element of GMHS (G(;)).
Let r(c) be the intersection multiplicity m0 (p c) if fc = 0g is not a factor of p,
and = 1 otherwise. Symmetrically dene r(d). The main result of this section is
the following.
Theorem 11.2. Let h g p and f be as above. Then:
Spp (f ) = Spp;(Hh Hg ) + cr(d)Spp (g) Spp!(h) + cr(c)Spp (h) Spp!(g):
& X& 0 ) ! (C 0) of the germ g such
Proof. Consider a projective extension g& : (X
& x) ! (C 0)
that x 2 X&0 = g&;1 (0) is the only singular point of g& and g& : (X
analytically equivalent to g. (For the existence of g&, see 18].) Similarly, let h& :
(Y& Y&0 ) ! (C 0) be a projective extension of h with the above properties. Now use
several times the rst distinguished triangle (2) applied to the natural stratication
of X&0 Y&0 , and apply Theorem 5.2. The details are left to the reader.
For the corresponding formula at the zeta function level, see 7, 8].
Example 11.3. LetPg and h be as above. P
Then one has:
a) If Spp (g) = ( w) and Spp(h) = ( !), then
X
Spp (g(x) h(y)) = T (1 1) ; 1] ( ] + ] + w + !):
]= ]
b) For the sum f (x y) = g(x) + h(y), it is simpler to rewrite Theorem 11.2 in
terms of SppSt (see Remark 2.7):
SppSt(f (x) + g(y)) = T (1 0)SppSt(g) SppSt(h)
(cf. 12]).
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L
Department of Mathematics, The Ohio State University, Columbus, OH 43210,
U.S.A
nemethi@math.ohio-state.edu
Department of Mathematics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands
steenbri@sci.kun.nl