One of the results proved in [2] (cf

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Doc. Math. J. DMV 65

The Local Monodromy

as a Generalized Algebraic Correspondence

Caterina Consani¹ with an Appendix by

Spencer Bloch Received: May 25, 1998 Revised: February 24, 1999 Communicated by Peter Schneider

Abstract. For an algebraic, normal-crossings degeneration over a local eld the local monodromy operator and its powers naturally dene Galois equivariant classes in the `-adic (middle dimensional) cohomology groups of some precise strata of the special ber of a normal-crossings model associated to the ber product degeneration.

The paper addresses the question whether these classes are algebraic.

It is shown that the answer is positive for any degeneration whose special ber has (locally) at worst triple points singularities. These algebraic cycles are responsible for and they explain geometrically the presence of poles of local Euler L-factors at integers on the left of the left-central point.

1991 Mathematics Subject Classication: 11G25, 11S40, 14C25, 14E10

Introduction

LetXbe a proper and smooth variety over a local eldKand let^X be a regular model ofX dened over the ring of integers^OK ofK. When^X is smooth over

OK, the Tate conjecture equates the`{adic Chow groups of algebraic cycles on the geometric special berXk of^X ^!Spec(^OK) with the Galois invariants in H²(XK;^Q_`()). One of the results proved in [2] (cf. Corollary 3.6) shows that the Tate conjecture for smooth and proper varieties over nite elds together with the monodromy{weight conjecture imply a generalization of the above result in the case of semistable reduction. Namely, let } ² Spec(^OK) be a

1Partially supported by the NSF grant DMS-9701302

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prime over which the special ber^XSpec(k(})) =Y is a reduced divisor with normal crossings in ^X (i.e. semistable ber). Then, assuming the above two conjectures, the `{adic groups of algebraic cycles modulo rational equivalence on the r{fold intersections of components ofY (r1) are related with Galois invariant classes on the Tate twistsH² ⁽^r ¹⁾(XK;^Q`( (r 1))).

An interesting case is when one replaces X by X K X, so that Galois invariant cycles may be identied with Galois equivariant maps H(XK;^Q`)^! H(XK;^Q`()). Examples of such maps are the powersNⁱ of the logarithm of the local monodromy around }. The operators Nⁱ : H(XK;^Q`) ^! H(XK;^Q`( i)) determine classes [Nⁱ] ² H²^d((X X)K;^Q`(d i)) (d = dim XK) invariant under the decomposition group. In this paper we study in detail the structure of [Nⁱ] when the special berY of ^X has at worst triple points as singularities. That is, we exhibit the corresponding algebraic cycles on the (normal crossings) special berT =^[iT_iof a resolution^Zof^X^OK^X. Denote by ~N = 1N+N1 the monodromy on the product, and letF be the geometric Frobenius. Then the classes [Nⁱ] naturally determine elements in Ker( ~N)^\H²^d((XX)_K;^Q_`(d i))^F⁼¹. Assuming the monodromy{weight conjecture on the product (i.e. the monodromy ltrationLonH((XX)_K;^Q_`) coincides{up to a shift{with the ltration by the weights of the Frobenius cf. [16]) and the semisimplicity of the action of the Frobenius on the inertia invariants, the following identications hold

(0.1) Ker( ~N)^\H²^d((XX)K;^Q`(d i))^F⁼¹

'

(gr^L₂₍_{d i}₎H²^d(T;^Q`))(d i)

F=1

'

Ker(⁽²⁽ⁱ⁺¹⁾⁾:H²⁽^{d i}⁾( ~T⁽²ⁱ⁺¹⁾;^Q`)(d i)^!H²⁽^{d i}⁾( ~T⁽²⁽ⁱ⁺¹⁾⁾;^Q`)(d i)) Image⁽²ⁱ⁺¹⁾

F=1

: Here ~T⁽^j⁾ denotes the normalization of the j{fold intersection on the closed ber T. These isomorphisms show that the classes [Nⁱ] have representatives in the cohomology groups of some precise strata of T. Moreover, the Tate conjecture and the semisimplicity of the action of the Frobenius on the smooth schemes ~T⁽^j⁾ would imply that these classes are algebraic. We refer to ^x 1, (1.6) for the description of the restriction mapsin (0.1).

To better understand the geometry related to the desingulatization process

Z!X

OK^X, and to avoid at rst, some technical complications connected to the theory of the nearby cycles in mixed characteristic, we start by investigating this problem in equal characteristic zero (i.e. for semistable degenerations over a disk). There, one can take full advantage of many geometric results based on the theory of the mixed Hodge structures. Under the assumption of the monodromy{weight conjecture and using some techniques of [16], our results generalize to mixed characteristic. The cycles we exhibit on ~T⁽²ⁱ⁺¹⁾ explain geometrically the presence of poles on specic local factors of the L{function

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related to the ber product X X. In fact, theorem 6.2 equates, under the assumption of the semisimplicity of the action of the FrobeniusF on the inertia invariants H((X X)K;^Q`)^I, the rank of any of the groups in (0.1) with ords=d idet(Id FN(}) ^s^jH²^d((X X)K;^Q`)^I). Here, N(}) denotes the number of elements of the nite eldk(}).

A study of the local geometry of the normal{crossings special ber T shows that [Nⁱ] are represented by certain natural \diagonal cycles" on ~T⁽²ⁱ⁺¹⁾ together with a cycle supported on the exceptional part of the stratum that arises because the classes [Nⁱ] must belong to the kernel of the restriction map ⁽²⁽ⁱ⁺¹⁾⁾ (cf. (0.1)). This result is obtained via the introduction of a generalized correspondence diagram for the map

Nⁱ:^H(Y;gr_Lr₊_i^R (^Q^X))^!^H(Y;(gr_{Lr i}^R (^Q^X))( i)): (0.2)

This morphism describes the monodromy action on theE1{term of the spectral sequence of weights for the ltered complex of the nearby cycles (^R (^Q^X);L) (cf. ^x 2, (2.1)). For i >0, the classes [Nⁱ] do not describe an algebraic correspondence in the classical sense. In fact, the algebraic cycles representing them are only supported on higher strata of the special berT (i.e. on ~T⁽²ⁱ⁺¹⁾) and they do not naturally determine classes in the cohomology of T. This is a consequence of the fact that for i >0, the cocycle [Nⁱ] does not have weight zero in the`{adic cohomology of the ber product (XX)K, as one can easily check from (0.1). Nonetheless, we expect that each of these classes supplies a rened information on the degeneration. Namely, we conjecture that the geometric description that we obtain up to triple points can be generalized to any kind of semistable singularity via a thorough combinatoric study of the toric singularities of the special ber of the ber product resolution^Z.

The correspondence diagram related to the map (0.2) is built up from the hypercohomology of the Steenbrink ltered resolution (A^X;L) of^R (^Q^X). In

x3 we establish the necessary functoriality properties of the Steenbrink complex and itsL{ltration. A dicult point in the description of the correspondence diagram is related to the denition of a product structure on theE1{terms of the spectral sequence of weights. Example 3.1 points out a problem related to a canonical denition of a product structure for (A^X;L) in the ltered category. It comes out that the monodromy ltration L is not multiplicative on the level of the ltered complexes. A partial product, canonical only on the E2=E¹-terms is provided in the Appendix. This suces for purposes of our paper.

Acknowledgments. I would like to thank Kazuya Kato for suggesting the study of the local monodromy as a Galois invariant class, eventually algebraic. I also would like to acknowledge an interesting conversation with Alexander Beilinson on some of the arguments presented here and few interesting remarks received from Ofer Gabber. I gratefully thank the Institut des Hautes Etudes Scien- tiques for the kind hospitality received during my stay there in January 1997 and the University of Cambridge where a revision of this work was carried out

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in Winter 1998. Finally, it is my pleasure to thank Spencer Bloch for his con- stant support and for many fruitful suggestions I received from him during the preparation of this paper.

1. Notations and techniques from mixed Hodge theory In this paragraph we introduce the main notations and recall some results on the mixed Hodge theory of a degeneration.

We denote by X a connected, smooth, complex analytic manifold and we let S be the unit disk. We writef :X^!S for a proper, surjective morphism and we let Y = f ¹(0) be its special ber. We assume that f is smooth at every point ofX=X^rY and that the special berY is an algebraic divisor with normal{crossings. The local description off near a closed pointy²Y is given by:

f(z1;::: ;zm) =z₁^e¹z_k^e^k

forkm= dimX and^fz1;::: ;zm^ga local coordinate system on a neighbor- hood of y in X centered at y and ei ² ^Z; ei 1. The bers of f have then dimensiond=m 1.

A normal{crossings divisor as above is said to have semistable reduction (strict normal{crossings) if one has: ei= 1 ⁸i, in the local description off.

We x a parametert²S. Fort⁶= 0, letf ¹(t) =Xtbe the ber att. Because the restriction off atS =S^r^f0^gis a C¹, locally trivial ber bundle, the positive generator of1(S;t)^'^Zinduces an automorphismT_tofH(X_t;^Z), called the local monodromy. We will always suppose throughout the paper that T_t is unipotent. This assumption, together with the local monodromy theorem (cf. [7], Theorem 2.1.2), implies that (Tt 1)ⁱ⁺¹ = 0, on Hⁱ(Xt;^Z).

The unipotency condition of the local monodromy is for example veried when g.c.d.(ei; i ² [1;k]) = 1, ⁸y ² Y (cf. op.cit. ). Under these conditions, the logarithm of the local monodromy is dened to be the nite sum:

Nt:= logTt= (Tt 1) 12(Tt 1)²+ 13(Tt 1)³

It is known (cf. [5]) that the automorphismsTtofHⁱ(Xt;^C) (t²S), are the bers of an automorphism T of the ber bundle ^Rⁱf(_X=S(logY)) over S, whose ber at 0 is described asT0= exp( 2iN0). By denition, the endomor- phismN0 is the residue at 0 of the Gauss-Manin connection^ron the \canonical prolongation"^Rⁱf(_X=S(logY)) of the locally free sheaf ^Rⁱf(_X_=S).

Because of the denition of T0, it makes sense to think of a nilpotent map N := ₂¹_i logT as the monodromy operator on the degenerationf :X ^!S. Via the canonical isomorphism (cf. [11], Thm. 2.18)(t²S):

Rif(_X=S(logY))Ô_Sk(t)^!^' ^Hⁱ(Xt;_X=S(logY)Ô_XÔXt)

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where k(t) is the residue eld of ÔS at t, we can see the map N0 as an endomorphism of the hypercohomology of the relative de Rham complex _X=S(logY)Ô_X ÔY. This complex represents in the derived category D⁺(Y;^C) of the abelian category of sheaves of ^C{vector spaces on Y, the complex of the nearby cycles ^R (^C). Namely, there exists a non- canonical quasi-isomorphism (i.e. depending on the choice of the parame- ter t on S) _X=S(logY)ÔX ÔY ^' ^R (^CX~) := i ¹^Rk^CX~ (cf. [11],

x2). This isomorphism, composed with the canonical map _X=S(logY)^OX

OY ^!_X=S(logY)Ô_XÔ_Y^red (Y^red= reduced, induced structure scheme on Y), induces a quasi-isomorphism (i ¹^Rk^CX~)un ^' _X=S(logY)Ô_X Ô_Y^red (cf. op.cit.^x4). Here, we denote by (i ¹^Rk^CX~)un the maximal subobject of i ¹^Rk^CX~ on which1(S) acts with unipotent automorphisms. We refer to the following commutative diagram for the description of the maps:

X~ ^k^! X ⁱ Y

?

y

?

yf ^?^?^y

S~ ^p^! S ^f0^g:

The space ~S=^fu²^C^jImu >0^gis the upper half plane, the mapp: ~S^!S p(u) = exp(2iu) =t, makes ~S in a universal covering of S and ~X is the pullbackXSS~ ofX alongp. The morphismkis the natural projection. It factorizes throughX by means of the injectionj :X ^!X. Finally,iis the closed embedding.

Steenbrink, Guillen and Navarro Aznar and Masaiko Saito (cf. [11], [6], [12]) dened a mixed Hodge structure on the hypercohomology of the unipotent factor of the complex of the nearby cycles ^H(X;_X=S(logY)^OX ^OY^red).

This is frequently referred as the limiting mixed Hodge structure.

We will assume from now on thatf is projective. Then, the weight ltration on the limiting mixed Hodge structure is the one induced by the nilpotent endo- morphismN, namely by the logarithm of the unipotent Picard-Lefschetz trans- formationT that is already dened at the ^Q-level. This ltration, which one usually refers to as the monodromy{weight ltrationL, is dened inductively.

On the limiting cohomologyHⁱ( ~X;^Q), it is increasing and has lenght at most 2i. By the local monodromy theorem Nⁱ⁺¹ = 0, hence one sets L0 = Im Nⁱ and L2i 1 = Ker Nⁱ. The monodromy ltration L becomes a convolution product of the kernel and the image ltration relative to the endomorphismN. These ltrations are dened as

K_l Hⁱ( ~X;^Q) := KerN^l⁺¹; I^j Hⁱ( ~X;^Q) := ImN^j and their convolution is

L=KI; Lk := ^X

l j=kKl^\I^j: (1.1)

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It is a very interesting fact that there is no explicit construction of the monodromy-weight ltrationLon _X=S(logY)Ô_XÔY itself. The ltration Lis dened on a complexA^Cwhich is a resolution of _X=S(logY)Ô_XÔ_Y^red. More precisely, the complex _X=S(logY)Ô_XÔ_Y^red is isomorphic, in the derived category D⁺(Y;^C), to the complex A^C of ÔX{modules supported on Y. The complexA^C is the simple complex associated to the double complex (p; q0):

A^p;q^C := ^p_X⁺^q⁺¹(logY)=Wq^p_X⁺^q⁺¹(logY)

where W_X(logY) is the weight ltration by the order of log-poles (cf. [3],^x3). The dierentials on it are dened as follows

d⁰ :A^p;q^C ^!A^p^C⁺¹^;q; d⁰(!) =d!

is induced by the dierentiation on the complex _X(logY) and d^{0 0}:A^p;q^C ^!A^p;q^C⁺¹; d⁰⁰(!) =^{^}!

where:=f(_dtt) =^P^k_i₌₁e_{i dz}_z_iⁱ is the form dening the quasi-isomorphism we mentioned before (cf. [11],^x4)

_X=S(logY)^O_X ^O_Y^red ^{^}^!A^C:

The total dierential onA^Cisd=d⁰+d⁰⁰. The weight ltrationW_X(logY) induces a corresponding ltration onA^C (r²^Z):

WrA^p;q_X;^C=:Wr+q+1^p_X⁺^q⁺¹(logY)=Wq^p_X⁺^q⁺¹(logY): (1.2)

The ltration that W_rA^C induces on ^H(Y;A^C) ^' ^H( ~X;^C) is the kernel ltrationK (cf. (1.1))

KrH( ~X;^C) =Wr^H(Y;A^C) =: Im

H

(Y;WrA^C)^!H(Y;A^C)

= KerN^r⁺¹: The monodromy-weight ltration is then dened as

L_rA^p;q:=W2q+r+1^p_X⁺^q⁺¹(logY)=W_q^p_X⁺^q⁺¹(logY):

Via Poincare residues, the related graded pieces have the following description gr_LrA^C^' ^M

kmax(0; r)(a2k+r+1)_Y_~⁽²k⁺r⁺¹⁾[ r 2k]: (1.3)

Here, we have denoted by ~Y⁽^m⁾the disjoint union of all intersectionsYi¹^\:::^\ Yim for 1i1< ::: < imn(Y =Y1^[:::^[Yn). We write (am): ~Y⁽^m⁾^!X for the natural projection.

The monodromy operatorN is induced by an endomorphism ~ ofA^Cwhich is dened as ( 1)^p⁺^q⁺¹ times the natural projection

:A^p;q^C ^!A^p^C ¹^;q⁺¹:

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The endomorphism ~ is characterized by its behavior on the L-ltration, namely

~(LrA^C)Lr 2A^C and the induced map

~^r:gr_LrA^C^!gr_LrA^C (1.4)

is an isomorphism for all r 0. The complex A^C contains the subcomplex W0A^C = Ker(~) that is known to be a resolution of ^CY. The ltration L and the Hodge ltrationF onA^C induce resp. the kernel andF ltration on W0A^C. The resulting mixed Hodge structure on H(Y;^C) is the canonical one. Similarly, the homologyH(Y;^C) (i.e. H_Y(X;^C)) with its mixed Hodge structure is calculated by the hypercohomology of the complex Coker(~).

Because of the description given in (1.3), the spectral sequence of hypercohomology of the ltered complex (A^C;L) (frequently referred as the weight spectral sequenceof^R (^C)) has theE1term given by

E₁^r;n⁺^r= ^M

kmax(0; r)H^{n r} ²^k(~Y⁽²^k⁺^r⁺¹⁾;^C) d1=^X

k (( 1)^r⁺^kd⁰1+ ( 1)^{k r}d⁰⁰1): (1.5)

The explicit denition of the dierentials, in the strict normal{crossings case (i.e. semistable degeneration), is the following:

d⁰₁=⁽^r⁺²^k⁺²⁾=^r⁺²^X^k⁺²

u=1 ( 1)^u ¹⁽_u^r⁺²^k⁺²⁾ d^{0 0}₁= ⁽^r⁺²^k⁺¹⁾=^r⁺²^X^k⁺¹

u=1 ( 1)^u_u⁽^r⁺²^k⁺¹⁾ (1.6)

where

⁽_u^r⁺²^k⁺²⁾= (⁽_u^r⁺²^k⁺²⁾):H^{n r} ²^k(~Y⁽²^k⁺^r⁺¹⁾;^C)^!H^{n r} ²^k(~Y⁽²^k⁺^r⁺²⁾;^C) _u⁽^r⁺²^k⁺¹⁾= (⁽_u^r⁺²^k⁺¹⁾)! :H^{n r} ²^k(~Y⁽²^k⁺^r⁺¹⁾;^C)^!H^{n r} ²^k⁺²(~Y⁽²^k⁺^r⁾;^C) are the restrictions, resp. the Gysin maps, induced by the inclusions (u;t²^Z)

⁽_u^t⁾:Yi¹^\^\Yit ^!Yi¹^\^\(Yiu)^{^}^\^\Yit:

In the general normal{crossings case (i.e. brations locally described by f(z1;:::;zm) = z₁^e¹z^e_k^k, ei 1), the denition of d⁰1 has to take into ac- count multiplicity factors eij before each map (_j⁽^t⁾). The map d⁰₁ is infact induced from a \wedging" operation with the form = ^P^k_i₌₁ei dzziⁱ (cf. last page). The denition ofd⁰⁰₁ is analogous to the one given in the strict normal{

crossings case.

Notice that the weight spectral sequence (1.5) is built up from a ltered double complex. This property distinguishes this weight spectral sequence from others

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as e.g. the spectral sequence of weights which denes the mixed Hodge structure on a quasi{projective smooth complex variety (cf. [3]).

The complexA^Cis the complex part of a cohomological mixed Hodge complex A^Q whose denition is less explicit thanA^Cand for which we refer to [7]. This rational complex induces onH( ~X;^Q) a rational mixed Hodge structure. The rational representative of the above spectral sequence (1.5) is

E₁^r;n⁺^r= ^M

kmax(0; r)H^{n r} ²^k(~Y⁽²^k⁺^r⁺¹⁾;^Q)( r k): (1.7)

The index in the round brackets outside the cohomology refers to the Tate twist.

Both these spectral sequences degenerate at E2 =E¹ and they converge to Hⁿ( ~X;^C) andHⁿ( ~X;^Q) respectively.

For curves (i.e. d= 1), the degeneration of the weight spectral sequence pro- vides the exact sequences

0^!E₂¹^;²^!H⁰(~Y⁽²⁾;^Q)( 1)^d¹^!¹^;²H²(~Y⁽¹⁾;^Q)^!H²( ~X;^Q)^!0 and

0^!H⁰( ~X;^Q)^!H⁰(~Y⁽¹⁾;^Q)^d^!⁰¹^;⁰H⁰(~Y⁽²⁾;^Q)^! H¹( ~X;^Q): (1.8)

The dierentialsd₁¹^;² andd⁰₁^;⁰ are dened as in (1.6) and the mapin (1.8) is the edge map in the spectral sequence. We also have a non canonical decomposition

H¹( ~X;^Q) =H¹(~Y⁽¹⁾;^Q)E₂¹^;²E₂¹^;⁰: withE₂¹^;⁰= Im().

Steenbrink proved that theL{ltration induced on the abutment of the spectral sequence of the nearby cycles is the Picard-Lefschetz ltration, hence it is uniquely described by the following properties

N(Ln+rHⁿ( ~X;^Q))(Ln+r 2Hⁿ( ~X;^Q))( 1) and N^r:gr_Ln₊_rHⁿ( ~X;^Q)^!^' (gr_{Ln r}Hⁿ( ~X;^Q))( r)

forr >0. In the rest of the paper we will refer to it as the monodromy ltration.

2. The monodromy operator as algebraic cocycle

We keep the notations introduced in the last paragraph. As nvaries in [0;2d] (d= dimension of the ber off :X^!S) andi0, the power maps

Nⁱ:Hⁿ( ~X;^Q)^!Hⁿ( ~X;^Q)( i)

induced by the endomorphismN :^Rⁿf(_X=S(logY))^!^Rⁿf(_X=S(logY)), dene elements

Nⁱ²Hom(H( ~X;^Q);H( ~X;^Q)( i))

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which are invariant for the action of the local monodromy group1. They can be naturally identied with

Nⁱ²^M

n0

H²^{d n}( ~X;^Q)(d)Hⁿ( ~X;^Q)( i)

¹

=

H²^d( ~XX^~;^Q)(d i)

¹

: The space ~X_SX^~is the generic ber of the product degenerationX_SX ^! S. After a suitable sequence of blow-ups along Sing(Y Y)Sing(X_SX):

Z ^!^!X_SX^!S

we obtain a normal{crossings degenerationh:Z ^!SwithZ non singular and whose generic ber is still ~XX^~. Its special berT =h ¹(0) =T1^[^[TN

has normal crossings singularities. The local description ofhalongT looks like:

h(w1;::: ;w2m) =w^e₁¹w^e_r^r

for^fw1;::: ;w2m^ga set of local parameters onZ ande1;::: ;ernon-negative integers.

The semistable reduction theorem (cf. [9]) assures that modulo extensions of the basisSand up to a suitable sequence of blow-ups and down along subvarieties of the special berT, we may eventually obtain fromha semistable degeneration W ^!S withW0=W0¹^[:::^[W0_M as special ber.

Because of the assumption of the unipotency of the local monodromy on H(Xt;^C) (cf.^x1), the local monodromyofhwill be also unipotent. We then call ~N = log (). By the Kunneth decomposition it results: ~N= 1N+N1 and we have:

Nⁱ²H²^d( ~XX^~;^Q(d i))

¹

= Ker( ~N)^\H²^d( ~XX^~;^Q(d i)): Let consider the monodromy ltration L relative to the degenerationh. We denote byHomMH(^Q(0);V) (Hom(^Q;V) shortly) the subgroup of Hodge cycles of pure weight (0;0) of a biltered^Q{vector spaceV: (V;L;F), endowed with the corresponding mixed Hodge structure. Then, we have the following Proposition 2.1. Fori1

Nⁱ²HomMH

Q(0);Ker( ~N)^\H²^d( ~XX^~;^Q(d i))

HomMH

Q(0);Ker( ~N)^\(gr₂₍^L_{d i}₎H²^d( ~XX^~;^Q))(d i)

'

'HomMH

Q(0);(gr^L₂₍_{d i}₎H²^d(T;^Q))(d i)

'Hom(^Q;A); A:= Ker(²⁽ⁱ⁺¹⁾:H²⁽^{d i}⁾( ~T⁽²ⁱ⁺¹⁾;^Q)(d i)^!H²⁽^{d i}⁾( ~T⁽²⁽ⁱ⁺¹⁾⁾;^Q)(d i))

Image ⁽²ⁱ⁺¹⁾ :

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Here is the restriction map on cohomology and by ~T⁽^j⁾we mean the disjoint union of all ordered j{fold intersections of the components ofT (cf.^x1).

Proof. The identication of Nⁱ with a Hodge cycle is a consequence of N being a morphism in the category of Hodge structures. The rst inclusion derives from the well known facts that Ker( ~N) has monodromic weight at most zero and that its Hodge cycles are included (Hom being a functor left exact on the second place) in the corresponding ones for the graded piece (gr₂₍^L_{d i}₎H²^d( ~XX^~;^Q))(d i) of Ker( ~N)^\^L_j(gr_LjH²^d( ~XX^~;^Q))(d i).

The second isomorphism comes from the local invariant cycle theorem, namely from the following exact sequence of pure Hodge structures (cf. [2], lemma 3.3 and corollary 3.4)

0^!gr₂₍^L_{d i}₎H²^d(T;^Q)^!gr₂₍^L_{d i}₎H²^d( ~XX^~;^Q)

N

gr^L₂₍_{d i} ₁₎H²^d( ~XX^~;^Q)( 1) Finally, the last isomorphism is a consequence of the description of the graded piece (gr₂₍^L_{d i}₎H²^d(T;^Q))(d i) as sub{Hodge structure of (gr^L₂₍_{d i}₎H²^d( ~X X~;^Q))(d i) (cf. op.cit. lemma 3.3).

Proposition 2.1 shows how the operators Nⁱ can be detected by classes [Nⁱ] in the cohomology of a xed stratum of the special ber T. Equivalently, we can say thatNⁱ determine classes [Nⁱ]²^H²^d(T;(gr^L₂_i^R h(^Q))(d i)) in the (E₁²^i;²⁽^{d i}⁾)(d i)-term of the spectral sequence of weights for the degeneration h. Here we writegr^L2i^R h(^Q) forgr^L2iA_W;^Q.

The goal of this paper is to identify the class [Nⁱ] with an algebraic cocycle related to the degenerationf :X ^!S. In all those cases that we will consider in the paper, this identication is obtained via a \correspondence-type" map (i0)

Nⁱ:^H(Y;gr_LrA_X;^Q)^!^H(Y;(gr_Lr ₂_iA_X;^Q)( i)) =^H(Y;gr_Lr(A_X;^Q( i))) which makes the following diagram commute

H

(T;gr_LrA_Z;^Q) ^[^Nⁱ^]^!

E₁^r⁺²^i;⁺^r⁺²⁽^{d i}⁾

k

H2d+(T;(gr_Lr ₂_iA_Z;^Q)(d i))

(p¹)^x^?^? ^?^?^y(p²)

H

(Y;gr_LrA_X;^Q)

k

E₁^r;⁺^r

Nⁱ

! H

(Y;gr_Lr(A_X;^Q( i)))

k

E₁^r⁺²^i;⁺^r ²ⁱ (2.1)

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The projectionsp1; p2: ~XX^~^!X^~ on the rst and second factor, determine pullbacks and pushforwards on the hypercohomology as we shall describe in ^x3.

From the theory we will explain in the next paragraphs and in the Appendix it will follow thatNⁱ has the expected shape. Namely, it is zero whenNⁱ= 0 and it is the identity when Nⁱ induces an isomorphism on E₂^r;⁺^r. Also, it will result that p1, (p2) and [Nⁱ] all commute with the dierential on E1. That will imply an induced commutative diagram onE2.

Fori= 0, i.e. when the correspondence map is the identity, proposition 2.1 can be slightly generalized, using the theory developed in [2] (cf. lemma 3.3 and corollary 3.4) and in [1] so that the identity operator is seen as an element in

HomMH

Q; ^Ker(⁽²⁾ :H²^d( ~T⁽¹⁾;^Q)(d)^!H²^d( ~T⁽²⁾;^Q)(d)) Im ( ii:H2(d 1)(T⁽¹⁾;^Q)(d 1)^!H²^d( ~T⁽¹⁾;^Q)(d)

'

'HomMH

Q; ^Im(i:H²^d(T;^Q)(d)^!H²^d( ~T⁽¹⁾;^Q)(d)) Im ( ii:H2(d 1)(T⁽¹⁾;^Q)(d 1)^!H²^d( ~T⁽¹⁾;^Q)(d)

: Here the mapi (resp.i) represents the pullback (resp. pushforward) relative to the embeddingT⁽¹⁾^!T. Proposition 2.1 shows this class as a Hodge cocycle in H²^d( ~XX^~;^Q(d)). That agrees with the classical theory of algebraic correspondences describing the identity map via an algebraic correspondence with the cycle diagonal. Namely, the identity is determined by the diagonal X~ X~X^~ seen as specialization of the cycle diagonal on^X^X on the ber product ~XX^~. (cf. [8]).

The cases described in the next paragraphs will also supply some evidence for our expectation that [Nⁱ] can be always described by an algebraic (motivic) cocycle. Finally, notice that the calculation on theE1involves the cohomology of individual components of the strata and it is therefore in some sense local, whereasE2 introduces relations among components of strata, so that any calculation on it becomes of global nature. That is the reason why the description of the monodromy cycle is carried out mainly at a local level in this paper.

3. Functoriality of the Steenbrink complex and remarks on products

Let g : Z ^! X be a morphism between two connected, complex analytic manifolds over a disk S. Let f :X ^!S andh: Z ^!S be the degeneration maps. Let assume that both Z and X are smooth over ^C and they have algebraic special bersf ¹(0) =Y andh ¹(0) =T with normal crossings. We have the following commutative diagram

T ^! Y

#i⁰ ^#i

Z ^g^! X

h^& ^.f

S :

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Locally on the special bers,f andhhave the following description f(z1;::: ;zm) =zê₁¹z_kê^k; h(w1;::: ;wM) =wê₁⁰¹wê_K⁰^K

for ^fz1;::: ;z_m^g and ^fw1;::: ;w_M^g local parameters resp. on X and Z, 1 km; 1KM ande1;::: ;e_k; e⁰₁;:::;e⁰_K integers.

Because g ¹(Y) =T, at any pointy ²g(T) Y (y =g(t), for some t² T) where the local description ofY isz^e₁¹z_k^e^k= 0, the pullback sectionsg(zij) (⁸1ijk) dene divisors onZ supported onT (not necessarily reduced or irreducible).

Let order the components ofY asY =Y1^[:::^[Yk and let denote by ~Y⁽^r⁾the disjoint union of all intersectionsYi¹^\:::^\Yir for 1i1<< irk. There is a local system of rank one on ~Y⁽^r⁾of standard orientations of relements (cf. [3]). The canonical morphism

g_X(logY)^!_Z(logT)

is a map of biltered complexes with respect to the weight and the Hodge ltrations onX andZresp.(cf. op.cit. ). In particular it induces the following map of bicomplexes of sheaves supported on the special bers (r0)

g(WrA_X;^C)^!WrA_Z;^C

where A^C is the Steenbrink complex which represents in the derived category the maximal subobject of the complex of nearby cycles where the action of the monodromy is unipotent (cf. ^x 1). WrA^C is the induced weight ltration on A^C (cf. (1.2)). Because the weight ltration on the complexA^Cis induced by the weight ltration on the de Rham complex with log-poles,g induces a map in the derived category

g(Wr^R f(^QX))^!Wr^R h(^QZ):

Notice thatg(^dz_z_ij^ij)²W1¹_Z(logT), i.e. pullbacks preserve poles. Hence, we deduce the functoriality of the monodromy ltration

g(LrA_X;^C)^!LrA_Z;^C:

Becauseg ¹ is an exact functor,gdetermines on the graded pieces a pullback map g:gr_LrA_X;^C^!gr_LrA_Z;^C

where

gr_LrA_Z;^C^' ^M

kmax(0; r)(a2k+r+1)_T_~⁽²k⁺r⁺¹⁾(²^k⁺^r⁺¹)[ r 2k]: The functor g ¹ is also compatible with both dierentials d⁰ and d^{0 0} on A^C. Hence, g induces a morphism of biltered mixed Hodge complexes (F = Hodge ltration cf. [3])

g: (A_X;^C;L;F)^!(A_Z;^C;L;F)