The Local Monodromy

as a Generalized Algebraic Correspondence

Caterina Consani^{1}

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 field the local monodromy operator and its powers naturally define Galois equivariant classes in the ℓ-adic (middle dimensional) cohomology groups of some precise strata of the special fiber of a normal-crossings model associated to the fiber 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 fiber 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 Classification: 11G25, 11S40, 14C25, 14E10

Introduction

Let X be a proper and smooth variety over a local field K and let X be a regular model of X defined over the ring of integers O_{K} of K. When X is smooth over O_{K}, the Tate conjecture equates the ℓ–adic Chow groups of algebraic cycles on the geometric special fiber Xk̄ of X → Spec(O_{K}) with the Galois invariants in H^{2∗}(XK̄, Q_{ℓ}(∗)). One of the results proved in [2] (cf. Corollary 3.6) shows that the Tate conjecture for smooth and proper varieties over finite fields together with the monodromy–weight conjecture imply a generalization of the above result in the case of semistable reduction. Namely, let ℘ ∈ Spec(O_{K}) be a prime over which the special fiber X ×Spec(k(℘)) = Y is a reduced divisor with normal crossings in X (i.e. semistable fiber). Then, assuming the above two conjectures, the ℓ–adic groups of algebraic cycles modulo rational equivalence on the r–fold intersections of components of Y (r ≥ 1) are related with Galois invariant classes on the Tate twists H^{2∗−(r−1)}(XK̄, Q_{ℓ}(∗ − (r − 1))). An interesting case is when one replaces X by X ×_{K} X, so that Galois invariant cycles may be identified with Galois equivariant maps H^{∗}(XK̄, Q_{ℓ}) → H^{∗}(XK̄, Q_{ℓ}(·)). Examples of such maps are the powers N^{i} of the logarithm of the local monodromy around ℘. The operators N^{i} : H^{∗}(XK̄, Q_{ℓ}) → H^{∗}(XK̄, Q_{ℓ}(−i)) determine classes [N^{i}] ∈ H^{2d}((X × X)K̄, Q_{ℓ}(d − i)) (d = dim XK̄) invariant under the decomposition group. In this paper we study in detail the structure of [N^{i}] when the special fiber Y of X has at worst triple points as singularities. That is, we exhibit the corresponding algebraic cycles on the (normal crossings) special fiber T = ∪_{i}T_{i} of a resolution Z of X ×_{O_{K}} X. Denote by Ñ = 1 ⊗ N + N ⊗ 1 the monodromy on the product, and let F be the geometric Frobenius. Then the classes [N^{i}] naturally determine elements in Ker(Ñ)∩H^{2d}((X×X)K̄, Q_{ℓ}(d−i))^{F =1}. Assuming the monodromy–weight conjecture on the product (i.e. the monodromy filtration L_{·} on H^{∗}((X ×X)K̄, Q_{ℓ}) coincides–up to a shift–with the filtration by the weights of the Frobenius cf. [16]) and the semisimplicity of the action of the Frobenius on the inertia invariants, the following identifications hold

1 Partially supported by the NSF grant DMS-9701302

(0.1) Ker(Ñ) ∩ H^{2d}((X × X)K̄, Q_{ℓ}(d − i))^{F =1}

≃ ((gr_{2(d−i)}^{L}H^{2d}(T, Q_{ℓ}))(d − i))^{F =1}
≃

[\frac{Ker(ρ^{(2(i+1))}:H^{2(d−i)}(T̃^{(2i+1)}, Q_{ℓ})(d−i)→H^{2(d−i)}(T̃^{(2(i+1))}, Q_{ℓ})(d−i))}{Image ρ^{(2i+1)}}]^{F =1}.

Here T̃^{(j)} denotes the normalization of the j–fold intersection on the closed fiber T. These isomorphisms show that the classes [N^{i}] 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 T̃^{(j)} would imply that these classes are algebraic. We refer to § 1, (1.6) for the description of the restriction maps ρ in (0.1). To better understand the geometry related to the desingulatization process Z → X ×_{O_{K}} X, and to avoid at first, 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 T̃^{(2i+1)} explain geometrically the presence of poles on specific local factors of the L–function related to the fiber product X × X. In fact, theorem 6.2 equates, under the assumption of the semisimplicity of the action of the Frobenius F on the inertia invariants H^{∗}((X × X)K̄, Q_{ℓ})^{I}, the rank of any of the groups in (0.1) with ord_{s=d−i} det(Id − F N(℘)^{−s}|H^{2d}((X × X)K̄, Q_{ℓ})^{I}). Here, N(℘) denotes the number of elements of the finite field k(℘). A study of the local geometry of the normal–crossings special fiber T shows that [N^{i}] are represented by certain natural “diagonal cycles” on T̃^{(2i+1)} together with a cycle supported on the exceptional part of the stratum that arises because the classes [N^{i}] must belong to the kernel of the restriction map ρ^{(2(i+1))} (cf. (0.1)). This result is obtained via the introduction of a generalized correspondence diagram for the map

(0.2) N^{i} : H^{∗}(Y, gr_{r+i}^{L}RΨ(Q_{X})) → H^{∗}(Y, (gr_{r−i}^{L}RΨ(Q_{X}))(−i)).

This morphism describes the monodromy action on the E_{1}–term of the spectral sequence of weights for the filtered complex of the nearby cycles (RΨ(Q_{X}), L_{·}) (cf. § 2, (2.1)). For i > 0, the classes [N^{i}] 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 fiber T (i.e. on T̃^{(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 [N^{i}] does not have weight zero in the ℓ–adic cohomology of the fiber product (X ×X)K̄, as one can easily check from (0.1). Nonetheless, we expect that each of these classes supplies a refined 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 fiber of the fiber product resolution Z. The correspondence diagram related to the map (0.2) is built up from the hypercohomology of the Steenbrink filtered resolution (A_{X}^{•} , L_{·}) of RΨ(Q_{X}). In § 3 we establish the necessary functoriality properties of the Steenbrink complex and its L_{·}–filtration. A difficult point in the description of the correspondence diagram is related to the definition of a product structure on the E_{1}–terms of the spectral sequence of weights. Example 3.1 points out a problem related to a canonical definition of a product structure for (A_{X}^{•} , L_{·}) in the filtered category. It comes out that the monodromy filtration L_{·} is not multiplicative on the level of the filtered complexes. A partial product, canonical only on the E_{2} = E_{∞}-terms is provided in the Appendix. This suffices 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 Études Scientifiques 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 in Winter 1998. Finally, it is my pleasure to thank Spencer Bloch for his constant 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 write f : X → S for a proper, surjective morphism and we let Y = f^{−1}(0) be its special fiber. We assume that f is smooth at every point of X^{∗} = X ∖ Y and that the special fiber Y is an algebraic divisor with normal–crossings. The local description of f near a closed point y ∈ Y is given by:

f(z_{1}, …, z_{m}) = z_{1}^{e_{1}} ⋯ z_{k}^{e_{k}}

for k ≤ m = dim X and {z_{1}, …, z_{m}} a local coordinate system on a neighborhood of y in X centered at y and e_{i} ∈ Z, e_{i} ≥ 1. The fibers of f have then dimension d = m − 1. A normal–crossings divisor as above is said to have semistable reduction (strict normal–crossings) if one has: e_{i} = 1 ∀i, in the local description of f. We fix a parameter t ∈ S. For t ≠ 0, let f^{−1}(t) = X_{t} be the fiber at t. Because the restriction of f at S^{∗} = S ∖ {0} is a C^{∞}, locally trivial fiber bundle, the positive generator of π_{1}(S^{∗}, t) ≃ Z induces an automorphism T_{t} of H^{∗}(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 (T_{t} − 1)^{i+1} = 0, on H^{i}(X_{t}, Z). The unipotency condition of the local monodromy is for example verified when g.c.d.(e_{i}, i ∈ [1, k]) = 1, ∀y ∈ Y (cf. op.cit. ). Under these conditions, the logarithm of the local monodromy is defined to be the finite sum:

N_{t} := log T_{t} = (T_{t} − 1) − \frac{1}{2}(T_{t} − 1)^{2} + \frac{1}{3}(T_{t} − 1)^{3} − ⋯

It is known (cf. [5]) that the automorphisms T_{t} of H^{i}(X_{t}, C) (t ∈ S^{∗}), are the fibers of an automorphism T of the fiber bundle R^{i}f_{∗}(Ω_{X/S}^{•}(log Y)) over S, whose fiber at 0 is described as T_{0} = exp(−2πiN_{0}). By definition, the endomorphism N_{0} is the residue at 0 of the Gauss-Manin connection ∇ on the “canonical prolongation” R^{i}f_{∗}(Ω_{X/S}^{•}(log Y)) of the locally free sheaf R^{i}f_{∗}(Ω_{X^{∗}/S^{∗}}^{•}). Because of the definition of T_{0}, it makes sense to think of a nilpotent map N := −\frac{1}{2πi} log T as the monodromy operator on the degeneration f : X → S. Via the canonical isomorphism (cf. [11], Thm. 2.18)(t ∈ S):

R^{i}f_{∗}(Ω_{X/S}^{•}(log Y)) ⊗_{O_{S}} k(t) →^{≃} H^{i}(X_{t}, Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{X_{t}})

where k(t) is the residue field of O_{S} at t, we can see the map N_{0} as an endomorphism of the hypercohomology of the relative de Rham complex Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y}. 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 RΨ(C). Namely, there exists a noncanonical quasi-isomorphism (i.e. depending on the choice of the parameter t on S) Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y} ≃ RΨ(CX̃^{∗}) := i^{−1}Rk_{∗}CX̃^{∗} (cf. [11], § 2). This isomorphism, composed with the canonical map Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y} → Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y^{red}} (Y^{red} = reduced, induced structure scheme on Y), induces a quasi-isomorphism (i^{−1}Rk_{∗}CX̃^{∗})_{un} ≃ Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y^{red}} (cf. op.cit. § 4). Here, we denote by (i^{−1}Rk_{∗}CX̃^{∗})_{un} the maximal subobject of i^{−1}Rk_{∗}CX̃^{∗} on which π_{1}(S^{∗}) acts with unipotent automorphisms. We refer to the following commutative diagram for the description of the maps:

X̃^{∗} ⟶^{k} X ⟵^{i} Y
↓  ↓^{f}  ↓
S̃^{∗} ⟶^{p} S ⟵ {0}.

The space S̃^{∗} = {u ∈ C |Im u > 0} is the upper half plane, the map p : S̃^{∗} → S p(u) = exp(2πiu) = t, makes S̃^{∗} in a universal covering of S^{∗} and X̃^{∗} is the pullback X ×_{S} S̃^{∗} of X along p. The morphism k is the natural projection. It factorizes through X^{∗} by means of the injection j : X^{∗} → X. Finally, i is the closed embedding. Steenbrink, Guillen and Navarro Aznar and Masaiko Saito (cf. [11], [6], [12]) defined a mixed Hodge structure on the hypercohomology of the unipotent factor of the complex of the nearby cycles H^{∗}(X, Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y^{red}}). This is frequently referred as the limiting mixed Hodge structure. We will assume from now on that f is projective. Then, the weight filtration on the limiting mixed Hodge structure is the one induced by the nilpotent endomorphism N, namely by the logarithm of the unipotent Picard-Lefschetz transformation T that is already defined at the Q-level. This filtration, which one usually refers to as the monodromy–weight filtration L_{·}, is defined inductively. On the limiting cohomology H^{i}(X̃^{∗}, Q), it is increasing and has lenght at most 2i. By the local monodromy theorem N^{i+1} = 0, hence one sets L_{0} = Im N^{i} and L_{2i−1} = Ker N^{i}. The monodromy filtration L_{·} becomes a convolution product of the kernel and the image filtration relative to the endomorphism N. These filtrations are defined as

K_{l} H^{i}(X̃^{∗}, Q) := Ker N^{l+1}, I^{j} H^{i}(X̃^{∗}, Q) := Im N^{j}

and their convolution is

L = K ∗ I, L_{k} := ∑_{l−j=k} K_{l} ∩ I^{j}. (1.1)

It is a very interesting fact that there is no explicit construction of the monodromy-weight filtration L_{·} on Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y} itself. The filtration L_{·} is defined on a complex A_{C}^{•} which is a resolution of Ω_{X/S}^{•}(log Y)⊗_{O_{X}} O_{Y^{red}}. More precisely, the complex Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y^{red}} is isomorphic, in the derived category D^{+}(Y, C), to the complex A_{C}^{•} of O_{X}–modules supported on Y. The complex A_{C}^{•} is the simple complex associated to the double complex (p, q ≥ 0):

A_{C}^{p,q} := Ω_{X}^{p+q+1}(log Y)/W_{q}Ω_{X}^{p+q+1}(log Y)

where W_{∗}Ω_{X}^{•}(log Y) is the weight filtration by the order of log-poles (cf. [3], § 3). The differentials on it are defined as follows

d′ : A_{C}^{p,q} → A_{C}^{p+1,q}, d′(ω) = dω

is induced by the differentiation on the complex Ω_{X}^{•}(log Y) and

d′′ : A_{C}^{p,q} → A_{C}^{p,q+1}, d′′(ω) = θ ∧ ω

where θ := f^{∗}(\frac{dt}{t}) = ∑_{i=1}^{k} e_{i} \frac{dz_{i}}{z_{i}} is the form defining the quasi-isomorphism we mentioned before (cf. [11], § 4)

Ω_{X/S}^{•}(log Y) ⊗_{O_{X}} O_{Y^{red}} →^{∧ θ} A_{C}^{•}.

The total differential on A_{C}^{•} is d = d′ + d′′. The weight filtration W_{∗}Ω_{X}^{•}(log Y) induces a corresponding filtration on A_{C}^{•} (r ∈ Z):

(1.2) W_{r}A_{X,C}^{p,q} =: W_{r+q+1}Ω_{X}^{p+q+1}(log Y)/W_{q}Ω_{X}^{p+q+1}(log Y).

The filtration that W_{r}A_{C}^{•} induces on H^{∗}(Y, A_{C}^{•}) ≃ H^{∗}(X̃^{∗}, C) is the kernel filtration K (cf. (1.1))

K_{r}H^{∗}(X̃^{∗}, C) = W_{r}H^{∗}(Y,A_{C}^{•}) =: Im(H^{∗}(Y, W_{r}A_{C}^{•})→H^{∗}(Y, A_{C}^{•})) = Ker N^{r+1}.

The monodromy-weight filtration is then defined as

L_{r}A^{p,q} := W_{2q+r+1}Ω_{X}^{p+q+1}(log Y)/W_{q}Ω_{X}^{p+q+1}(log Y).

Via Poincaré residues, the related graded pieces have the following description

gr_{r}^{L}A_{C}^{•} ≃ ⨁_{k≥max(0,−r)}(a_{2k+r+1})_{∗}Ω• ; Ỹ^{(2k+r+1)}[−r − 2k]. (1.3)

Here, we have denoted by Ỹ^{(m)} the disjoint union of all intersections Y_{i_{1}} ∩ … ∩ Y_{i_{m}} for 1 ≤ i_{1} < … < i_{m} ≤ n (Y = Y_{1} ∪ … ∪ Y_{n}). We write (a_{m})_{∗} : Ỹ^{(m)} → X for the natural projection. The monodromy operator N is induced by an endomorphism ν̃ of A_{C}^{•} which is defined as (−1)^{p+q+1} times the natural projection

ν : A_{C}^{p,q} → A_{C}^{p−1,q+1}.

The endomorphism ν̃ is characterized by its behavior on the L-filtration, namely

ν̃(L_{r}A_{C}^{•}) ⊂ L_{r−2}A_{C}^{•}

and the induced map

ν̃^{r} : gr_{r}^{L}A_{C}^{•} → gr_{−r}^{L}A_{C}^{•} (1.4)

is an isomorphism for all r ≥ 0. The complex A_{C}^{•} contains the subcomplex W_{0}A_{C}^{•} = Ker(ν̃) that is known to be a resolution of C_{Y}. The filtration L and the Hodge filtration F on A_{C}^{•} induce resp. the kernel and F filtration on W_{0}A_{C}^{•}. The resulting mixed Hodge structure on H^{∗}(Y, C) is the canonical one. Similarly, the homology H_{∗}(Y, C) (i.e. H_{Y}^{∗} (X, C)) with its mixed Hodge structure is calculated by the hypercohomology of the complex Coker(ν̃). Because of the description given in (1.3), the spectral sequence of hypercohomology of the filtered complex (A_{C}^{•}, L) (frequently referred as the weight spectral sequence of RΨ(C)) has the E_{1} term given by

\begin{aligned}E_{1}^{−r,n+r} & = ⨁_{k≥max(0,−r)} H^{n−r−2k}(Ỹ^{(2k+r+1)}, C) \\ d_{1} & = ∑_{k}((−1)^{r+k}d′_{1} + (−1)^{k−r}d′′_{1}). \end{aligned} (1.5)

The explicit definition of the differentials, in the strict normal–crossings case (i.e. semistable degeneration), is the following:

\begin{aligned}d′_{1} & = ρ^{(r+2k+2)} = ∑_{u=1}^{r+2k+2}(−1)^{u−1}ρ_{u}^{(r+2k+2)} \\ d′′_{1} & = −γ^{(r+2k+1)} = ∑_{u=1}^{r+2k+1}(−1)^{u}γ_{u}^{(r+2k+1)} \end{aligned} (1.6)

where

ρ_{u}^{(r+2k+2)} = (δ_{u}^{(r+2k+2)})^{∗} : H^{n−r−2k}(Ỹ^{(2k+r+1)}, C) → H^{n−r−2k}(Ỹ^{(2k+r+2)}, C)
γ_{u}^{(r+2k+1)} = (δ_{u}^{(r+2k+1)})_{!} : H^{n−r−2k}(Ỹ^{(2k+r+1)}, C) → H^{n−r−2k+2}(Ỹ^{(2k+r)}, C)

are the restrictions, resp. the Gysin maps, induced by the inclusions (u, t ∈ Z)

δ_{u}^{(t)} : Y_{i_{1}} ∩ ⋯ ∩ Y_{i_{t}} → Y_{i_{1}} ∩ ⋯ ∩ (Y_{i_{u}})^{}ˆ ∩ ⋯ ∩ Y_{i_{t}}.

In the general normal–crossings case (i.e. fibrations locally described by f(z_{1}, …, z_{m}) = z_{1}^{e_{1}} ⋯ z_{k}^{e_{k}}, e_{i} ≥ 1), the definition of d′_{1} has to take into account multiplicity factors ±e_{i_{j}} before each map (δ_{j}^{(t)})^{∗}. The map d′_{1} is infact induced from a “wedging” operation with the form θ = ∑_{i=1}^{k} e_{i} \frac{dz_{i}}{z_{i}} (cf. last page). The definition of d′′_{1} 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 filtered double complex. This property distinguishes this weight spectral sequence from others as e.g. the spectral sequence of weights which defines the mixed Hodge structure on a quasi–projective smooth complex variety (cf. [3]). The complex A_{C}^{•} is the complex part of a cohomological mixed Hodge complex A_{Q}^{•} whose definition is less explicit than A_{C}^{•} and for which we refer to [7]. This rational complex induces on H^{·}(X̃^{∗}, Q) a rational mixed Hodge structure. The rational representative of the above spectral sequence (1.5) is

E_{1}^{−r,n+r} = ⨁_{k≥max(0,−r)} H^{n−r−2k}(Ỹ^{(2k+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 E_{2} = E_{∞} and they converge to H^{n}(X̃^{∗}, C) and H^{n}(X̃^{∗}, Q) respectively. For curves (i.e. d = 1), the degeneration of the weight spectral sequence provides the exact sequences

0 → E_{2}^{−1,2} → H^{0}(Ỹ^{(2)}, Q)(−1) →^{d_{1}^{−1,2}} H^{2}(Ỹ^{(1)}, Q) → H^{2}(X̃^{∗}, Q) → 0

and

0 → H^{0}(X̃^{∗}, Q) → H^{0}(Ỹ^{(1)}, Q) →^{d_{1}^{0,0}} H^{0}(Ỹ^{(2)}, Q) →^{α} H^{1}(X̃^{∗}, Q). (1.8)

The differentials d_{1}^{−1,2} and d_{1}^{0,0} are defined as in (1.6) and the map α in (1.8) is the edge map in the spectral sequence. We also have a non canonical decomposition

H^{1}(X̃^{∗}, Q) = H^{1}(Ỹ^{(1)}, Q) ⊕ E_{2}^{−1,2} ⊕ E_{2}^{1,0}.

with E_{2}^{1,0} = Im(α). Steenbrink proved that the L–filtration induced on the abutment of the spectral sequence of the nearby cycles is the Picard-Lefschetz filtration, hence it is uniquely described by the following properties

N(L_{n+r}H^{n}(X̃^{∗}, Q)) ⊂ (L_{n+r−2}H^{n}(X̃^{∗}, Q))(−1)

and

N^{r} : gr_{n+r}^{L}H^{n}(X̃^{∗}, Q) →^{≃} (gr_{n−r}^{L}H^{n}(X̃^{∗}, Q))(−r)

for r > 0. In the rest of the paper we will refer to it as the monodromy filtration.

2. The monodromy operator as algebraic cocycle

We keep the notations introduced in the last paragraph. As n varies in [0, 2d] (d = dimension of the fiber of f : X → S) and i ≥ 0, the power maps

N^{i} : H^{n}(X̃^{∗}, Q) → H^{n}(X̃^{∗}, Q)(−i)

induced by the endomorphism N : R^{n}f_{∗}(Ω_{X/S}^{•}(log Y)) → R^{n}f_{∗}(Ω_{X/S}^{•}(log Y)), define elements

N^{i} ∈ Hom(H^{·}(X̃^{∗}, Q), H^{·}(X̃^{∗}, Q)(−i)) which are invariant for the action of the local monodromy group π_{1}. They can be naturally identified with

N^{i}∈ ⨁_{n≥0}[H^{2d−n}(X̃^{∗}, Q)(d) ⊗ H^{n}(X̃^{∗}, Q)(−i)]^{π_{1}}=[H^{2d}(X̃^{∗} × X̃^{∗}, Q)(d−i)]^{π_{1}}.

The space X̃^{∗} ×_{S} X̃^{∗} is the generic fiber of the product degeneration X ×_{S} X → S. After a suitable sequence of blow-ups along Sing(Y × Y) ⊃ Sing(X ×_{S} X):

Z → ⋯ → X ×_{S} X → S

we obtain a normal–crossings degeneration h : Z → S with Z non singular and whose generic fiber is still X̃^{∗} ×X̃^{∗}. Its special fiber T = h^{−1}(0) = T_{1} ∪⋯∪T_{N} has normal crossings singularities. The local description of h along T looks like:

h(w_{1}, …, w_{2m}) = w_{1}^{e_{1}} ⋯ w_{r}^{e_{r}}

for {w_{1}, …, w_{2m}} a set of local parameters on Z and e_{1}, …, e_{r} non-negative integers. The semistable reduction theorem (cf. [9]) assures that modulo extensions of the basis S and up to a suitable sequence of blow-ups and down along subvarieties of the special fiber T, we may eventually obtain from h a semistable degeneration W → S with W_{0} = W_{0_{1}} ∪ … ∪ W_{0_{M}} as special fiber. Because of the assumption of the unipotency of the local monodromy on H^{∗}(X_{t}, C) (cf. § 1), the local monodromy σ of h will be also unipotent. We then call Ñ = log (σ). By the Künneth decomposition it results: Ñ = 1⊗N +N ⊗1 and we have:

N^{i} ∈ (H^{2d}(X̃^{∗} × X̃^{∗}, Q(d − i)))^{π_{1}}= Ker(Ñ) ∩ H^{2d}(X̃^{∗} × X̃^{∗}, Q(d − i)).

Let consider the monodromy filtration L_{·} relative to the degeneration h. We denote by Hom_{MH}(Q(0), V) (Hom(Q, V) shortly) the subgroup of Hodge cycles of pure weight (0, 0) of a bifiltered Q–vector space V: (V, L, F), endowed with the corresponding mixed Hodge structure. Then, we have the following

Proposition 2.1. For i ≥ 1

N^{i} ∈ Hom_{MH}(Q(0), Ker(Ñ) ∩ H^{2d}(X̃^{∗} × X̃^{∗}, Q(d − i))) ⊆

⊆ Hom_{MH}(Q(0), Ker(Ñ) ∩ (gr_{2(d−i)}^{L}H^{2d}(X̃^{∗} × X̃^{∗}, Q))(d − i)) ≃

≃ Hom_{MH}(Q(0), (gr_{2(d−i)}^{L}H^{2d}(T, Q))(d − i)) ≃ Hom(Q, A),

A := \frac{Ker(ρ^{2(i+1)} : H^{2(d−i)}(T̃^{(2i+1)}, Q)(d − i) → H^{2(d−i)}(T̃^{(2(i+1))}, Q)(d − i))}{Image ρ^{(2i+1)}}.

Here ρ is the restriction map on cohomology and by T̃^{(j)} we mean the disjoint union of all ordered j–fold intersections of the components of T (cf. §1).

Proof. The identification of N^{i} with a Hodge cycle is a consequence of N being a morphism in the category of Hodge structures. The first inclusion derives from the well known facts that Ker(Ñ) 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 (gr_{2(d−i)}^{L}H^{2d}(X̃^{∗} ×X̃^{∗}, Q))(d−i) of Ker(Ñ)∩⨁_{j}(gr_{j}^{L}H^{2d}(X̃^{∗} ×X̃^{∗}, 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 → gr_{2(d−i)}^{L}H^{2d}(T, Q) → gr_{2(d−i)}^{L}H^{2d}(X̃^{∗} × X̃^{∗}, Q)

↠^{N} gr_{2(d−i−1)}^{L}H^{2d}(X̃^{∗} × X̃^{∗}, Q)(−1)

Finally, the last isomorphism is a consequence of the description of the graded piece (gr_{2(d−i)}^{L}H^{2d}(T, Q))(d − i) as sub–Hodge structure of (gr_{2(d−i)}^{L}H^{2d}(X̃^{∗} × X̃^{∗}, Q))(d − i) (cf. op.cit. lemma 3.3).

Proposition 2.1 shows how the operators N^{i} can be detected by classes [N^{i}] in the cohomology of a fixed stratum of the special fiber T. Equivalently, we can say that N^{i} determine classes [N^{i}] ∈ H^{2d}(T, (gr_{−2i}^{L}RΨ_{h}(Q))(d − i)) in the (E_{1}^{2i,2(d−i)})(d−i)-term of the spectral sequence of weights for the degeneration h. Here we write gr_{−2i}^{L}RΨ_{h}(Q) for gr_{−2i}^{L}A_{W,Q}^{•}. The goal of this paper is to identify the class [N^{i}] with an algebraic cocycle related to the degeneration f : X → S. In all those cases that we will consider in the paper, this identification is obtained via a “correspondence-type” map (i ≥ 0)

N^{i} : H^{∗}(Y, gr_{r}^{L}A_{X,Q}^{•}) → H^{∗}(Y, (gr_{r−2i}^{L}A_{X,Q}^{•})(−i)) = H^{∗}(Y, gr_{r}^{L}(A_{X,Q}^{•}(−i)))

which makes the following diagram commute

H^{∗}(T, gr_{r}^{L}A_{Z,Q}^{•}) ⟶^{[N^{i}] ·} E_{1}^{−r+2i,∗+r+2(d−i)} ; ∥ ; H^{2d+∗}(T, (gr_{r−2i}^{L}A_{Z,Q}^{•})(d − i))
(p_{1})^{∗}↑  ↓^{(p_{2})_{∗}}
H^{∗}(Y, gr_{r}^{L}A_{X,Q}^{•}) ; ∥ ; E_{1}^{−r,∗+r} ⟶^{N^{i}}  H^{∗}(Y, gr_{r}^{L}(A_{X,Q}^{•}(−i))) ; ∥ ; E_{1}^{−r+2i,∗+r−2i}

(2.1)

The projections p_{1}, p_{2} : X̃^{∗} × X̃^{∗} → X̃^{∗} on the first and second factor, determine pullbacks and pushforwards on the hypercohomology as we shall describe in § 3. From the theory we will explain in the next paragraphs and in the Appendix it will follow that N^{i} has the expected shape. Namely, it is zero when N^{i} = 0 and it is the identity when N^{i} induces an isomorphism on E_{2}^{−r,∗+r}. Also, it will result that p_{1}^{∗}, (p_{2})_{∗} and [N^{i}] · all commute with the differential on E_{1}. That will imply an induced commutative diagram on E_{2}. For i = 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

Hom_{MH}(Q, \frac{Ker(ρ^{(2)} : H^{2d}(T̃^{(1)}, Q)(d) → H^{2d}(T̃^{(2)}, Q)(d))}{Im (−i^{∗} · i_{∗} : H_{2(d−1)}(T^{(1)}, Q)(d − 1) → H^{2d}(T̃^{(1)}, Q)(d)}) ≃

≃ Hom_{MH}(Q, \frac{Im(i^{∗} : H^{2d}(T, Q)(d) → H^{2d}(T̃^{(1)}, Q)(d))}{Im (−i^{∗} · i_{∗} : H_{2(d−1)}(T^{(1)}, Q)(d − 1) → H^{2d}(T̃^{(1)}, Q)(d)}).

Here the map i^{∗} (resp. i_{∗}) represents the pullback (resp. pushforward) relative to the embedding T^{(1)} → T. Proposition 2.1 shows this class as a Hodge cocycle in H^{2d}(X̃^{∗} × X̃^{∗}, 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 ΔX̃^{∗} ⊂ X̃^{∗} × X̃^{∗} seen as specialization of the cycle diagonal on X × X on the fiber product X̃^{∗} × X̃^{∗}. (cf. [8]). The cases described in the next paragraphs will also supply some evidence for our expectation that [N^{i}] can be always described by an algebraic (motivic) cocycle. Finally, notice that the calculation on the E_{1} involves the cohomology of individual components of the strata and it is therefore in some sense local, whereas E_{2} 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 and h : Z → S be the degeneration maps. Let assume that both Z and X are smooth over C and they have algebraic special fibers f^{−1}(0) = Y and h^{−1}(0) = T with normal crossings. We have the following commutative diagram

T  ⟶  Y
↓ i′  ↓ i
Z ⟶^{g} X
h ↘  ↙ f
S  .

Locally on the special fibers, f and h have the following description

f(z_{1}, …, z_{m}) = z_{1}^{e_{1}} ⋯ z_{k}^{e_{k}}; h(w_{1}, …, w_{M}) = w_{1}^{e′_{1}} ⋯ w_{K}^{e′_{K}}

for {z_{1}, …, z_{m}} and {w_{1}, …, w_{M}} local parameters resp. on X and Z, 1 ≤ k ≤ m, 1 ≤ K ≤ M and e_{1}, …, e_{k}; e′_{1}, …, e′_{K} integers. Because g^{−1}(Y) = T, at any point y ∈ g(T) ⊂ Y (y = g(t), for some t ∈ T) where the local description of Y is z_{1}^{e_{1}} ⋯ z_{k}^{e_{k}} = 0, the pullback sections g^{∗}(z_{i_{j}}) (∀ 1 ≤ i_{j} ≤ k) define divisors on Z supported on T (not necessarily reduced or irreducible). Let order the components of Y as Y = Y_{1} ∪ … ∪ Y_{k} and let denote by Ỹ^{(r)} the disjoint union of all intersections Y_{i_{1}} ∩…∩Y_{i_{r}} for 1 ≤ i_{1} < ⋯ < i_{r} ≤ k. There is a local system ϵ of rank one on Ỹ^{(r)} of standard orientations of r elements (cf. [3]). The canonical morphism

g^{∗}Ω_{X}^{•}(log Y) → Ω_{Z}^{•}(log T)

is a map of bifiltered complexes with respect to the weight and the Hodge filtrations on X and Z resp. (cf. op.cit. ). In particular it induces the following map of bicomplexes of sheaves supported on the special fibers (r ≥ 0)

g^{∗}(W_{r}A_{X,C}^{•}) → W_{r}A_{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. § 1). W_{r}A_{C}^{•} is the induced weight filtration on A_{C}^{•} (cf. (1.2)). Because the weight filtration on the complex A_{C}^{•} is induced by the weight filtration on the de Rham complex with log-poles, g induces a map in the derived category

g^{∗}(W_{r}RΨ_{f}(Q_{X})) → W_{r}RΨ_{h}(Q_{Z}).

Notice that g^{∗}(\frac{dz_{i_{j}}}{z_{i_{j}}}) ∈ W_{1}Ω_{Z}^{1}(log T), i.e. pullbacks preserve poles. Hence, we deduce the functoriality of the monodromy filtration

g^{∗}(L_{r}A_{X,C}^{•}) → L_{r}A_{Z,C}^{•}.

Because g^{−1} is an exact functor, g determines on the graded pieces a pullback map

g^{∗} : gr_{r}^{L}A_{X,C}^{•} → gr_{r}^{L}A_{Z,C}^{•}

where

gr_{r}^{L}A_{Z,C}^{•} ≃ ⨁_{k≥max(0,−r)}(a_{2k+r+1})_{∗}Ω• ; T̃^{(2k+r+1)}(ϵ^{2k+r+1})[−r − 2k].

The functor g^{−1} is also compatible with both differentials d′ and d′′ on A_{C}^{•}. Hence, g^{∗} induces a morphism of bifiltered mixed Hodge complexes (F^{·} = Hodge filtration cf. [3])

g^{∗} : (A_{X,C}^{•}, L, F) → (A_{Z,C}^{•}, L, F) which in turn induces a map between the spectral sequences of weights

g^{∗} : E_{1}^{−r,q+r}(X) = H^{q}(Y, gr_{r}^{L}A_{X}^{•}) → H^{q}(T, gr_{r}^{L}A_{Z}^{•}) = E_{1}^{−r,q+r}(Z).

On the rational level this morphism between spectral sequences is described by a direct sum of maps as

g^{∗} : H^{q−r−2k}(Ỹ^{(2k+r+1)}, Q)(−r − k) → H^{q−r−2k}(T̃^{(2k+r+1)}, Q)(−r − k). (3.1)

Both spectral sequences degenerate at E_{2} = E_{∞}. Keeping track of the multiplicities and the signs for these pullbacks can be rather hard. Let suppose that locally the defining equations for Y and T are t = ∏_{i} z_{i}^{e_{i}} and t = ∏_{j} w_{j}^{e′_{j}} respectively, and we are given strata Y_{I} = Y_{i_{1}} ∩ … ∩ Y_{i_{p}} (i_{1} < … < i_{p}) and Y_{J} = Y_{j_{1}} ∩ … ∩ Y_{j_{p}}. Then the computation of the multiplicities involved in g^{∗} : H^{∗}(Y_{I}, Q) → H^{∗}(T_{J}, Q) essentially amounts to determine the coefficients of \frac{dw_{j_{1}}}{w_{j_{1}}} ∧ … ∧ \frac{dw_{jp}}{w_{jp}} in g^{∗}(\frac{dz_{j_{1}}}{z_{j_{1}}} ∧ … ∧ \frac{dz_{jp}}{z_{jp}}). This technique will be frequently used in the paper. As an example, we describe the map (3.1) when f : X → S is a degeneration of curves with normal crossings singularities on its special fiber Y and Z is the blow-up of X at a closed point P ∈ Y. Let g : Z → X be the blowing up map. If P is a regular point in the special fiber, the number of components of the special fiber T of Z will simply increase by one (the exceptional divisor E) and the remaining components are the same as for Y. Hence g^{∗} : H^{0}(Ỹ^{(1)}, C) → H^{0}(T̃^{(1)}, C) is simply the map g^{∗}(1_{Y_{i}}) = 1_{T_{i}} + 1_{E} on the components. Let suppose instead that P is singular. Since the description of g^{∗} is local around each closed point, we may assume that the degeneration f is given, in a neighborhood of P, by the equation z_{1}^{e_{1}}z_{2}^{e_{2}} = t, being t a chosen parameter on the disk S and e_{1}, e_{2} positive integers. Let assume that e_{1} ≤ e_{2}. Then, locally around P: Ỹ^{(1)} = Y_{1} ∐ Y_{2}. Set-theoretically one has Y_{i} = {z_{i} = 0} (i = 1, 2) and Ỹ^{(2)} = Y_{1} ∩ Y_{2} = {P }. Then, T̃^{(1)} = T_{1} ∐ T_{2} ∐ T_{3} where T_{1} and T_{2} are the strict transforms of the two components Y_{i}, while T_{3} represents the exceptional divisor. We implicitly have fixed the standard orientation on Ỹ^{(r)} (e.g. Ỹ^{(2)} = Y_{1} ∩ Y_{2} = Y_{12}). On T̃^{(r)}, we choose the orientation for which the exceptional component T_{3} is always considered as the last one. There are only three graded complexes gr_{∗}^{L}A_{C}^{•} non zero both on X and Z. On X they have the following description

gr_{−1}^{L}A_{X,C}^{•} ≃ (a_{2})_{∗}Ω• ; Ỹ^{(2)}[−1]

gr_{0}^{L}A_{X,C}^{•} ≃ (a_{1})_{∗}Ω• ; Ỹ^{(1)}

and via the isomorphism (1.4) one has:

ν̃ : gr_{1}^{L}A_{X,C}^{•} →^{≃} gr_{−1}^{L}A_{X,C}^{•}.

Hence E_{1}^{1,q−1} = H^{q}(Y, gr_{−1}^{L}A_{X,C}^{•}) = 0 unless q = 1, in which case we get

g^{∗} : H^{0}(Ỹ^{(2)}, C) → H^{0}(T̃^{(2)}, C).

To understand the description of this map, one has to look at the local geometry of the blow-up at P. It is quite easy to check that Z is covered by two open sets, say Z = U ∪V. To make the notations easier, let call t_{1} = \frac{z_{1}}{z_{2}} and t_{2} = \frac{z_{2}}{z_{1}} . On U, described by t_{2}^{e_{2}} = _{\frac{t}{z_{1}^{e_{1}+e_{2}}}} , one has coordinates \{t_{2}, z_{1}\}, T_{2}^{red} = \{t_{2} = 0\}

and T_{3}^{red} = \{z_{1} = 0\}. On V, described by t_{1}^{e_{1}} = _{\frac{t}{z_{2}^{e_{1}+e_{2}}}} , one has coordinates

{t_{1}, z_{2}}, T_{1}^{red} = {t_{1} = 0} and T_{3}^{red} = {z_{2} = 0}. Then T̃^{(2)} = T_{13} ∐ T_{23}, here we denote T_{ij} = T_{i} ∩ T_{j}. On U we have g^{∗}(\frac{dz_{1}}{z_{1}} ∧ \frac{dz_{2}}{dz_{2}} ) = \frac{dz_{1}}{dz_{1}} ∧ \frac{dt_{2}}{t_{2}} , whereas on V one gets g^{∗}(\frac{dz_{1}}{z_{1}} ∧ \frac{dz_{2}}{dz_{2}} ) = \frac{dt_{1}}{dt_{1}} ∧ \frac{dz_{2}}{z_{2}} . Hence, keeping in account the fixed orientation among the components of T, the description of the pullback g^{∗}(1Ỹ^{(2)}) = g^{∗}(1_{Y_{12}}) is given by

g^{∗}(1_{Y_{12}}) = 1_{T_{13}} − 1_{T_{23}}.

The presence of a negative sign is due to the change of orientation. This description defines the above map g^{∗} on H^{0}. Similarly, we find that

g^{∗} : H^{0}(Ỹ^{(1)}, C) → H^{0}(T̃^{(1)}, C)

is given by g^{∗}(1_{Y_{1}}) = 1_{T_{1}} + 1_{T_{3}} and g^{∗}(1_{Y_{2}}) = 1_{T_{2}} + 1_{T_{3}}. The description of g^{∗} on the terms H^{1} goes in parallel. Let now consider the proper map that g induces on the closed fibers. For simplicity of notations we call it g : T → Y. Let d = (dim T − dim Y). The above arguments have shown that g induces a pullback map g^{∗} between the cohomologies of the strata: cf. (3.1). Since each stratum is a smooth projective complex variety (not connected), we can use the Poicaré duality to associate to each pullback in (3.1) that contributes to the definition of the map g^{∗} its dual so that we naturally obtain a dual pushforward on the E_{1}–terms of the spectral sequence of weights that is described by a direct sum of maps as

(3.2) g_{!} : H^{q−r−2(k−d)}(T̃^{(2k+r+1)}, Q)(−r − k + d)

→ H^{q−r−2k}(Ỹ^{(2k+r+1)}, Q)(−r − k).

On each stratum g_{!} is defined by the following formula

(\frac{1}{2π\sqrt{−1}})^{d−2k−r} ∫Ỹ^{(2k+r+1)} g_{!}(α) ∪ β = (\frac{1}{2π\sqrt{−1}})^{2d−2k−r} ∫T̃^{(2k+r+1)} α ∪ g^{∗}(β)

where ∫ denotes the morphism trace described by the cap–product with the fundamental class of each component of the stratum, for any chosen couple of elements α ∈ H^{q+2(2d−2k−r)}(T̃^{(2k+r+1)}, Q(2d − 2k − r)) and β ∈ H^{−q}(Ỹ^{(2k+r+1)}, Q), q ∈ Z, q ≥ 0. Notice that although we have a notion of bifiltered pullback

g^{∗} : (A_{X}^{•}, L, F) → (A_{Z}^{•}, L, F)

this does not imply a canonical definition of a product structure on A_{C}^{•} obtained via pullback along the diagonal map Δ : X → X ×_{S} X. In fact, the property of f : X → S to have normal crossings reduction is not preserved by the product map f × f : X ×_{S} X → S. The space X ×_{S} X is in general not even smooth over C! Finally, we remark that although the monodromy filtration is not multiplicative on the level of the filtered complexes (A_{C}^{•}, L) (the simple example shown below will motivate this claim), it becomes multiplicative on the limiting cohomology with its mixed Hodge structure.

Example 3.1.

Let f : P_{S}^{1} → S be a P^{1}-fibration over a disk S. We blow a closed point P ∈ P_{0}^{1} = Y in the fiber P_{0}^{1} over the origin {0}. The resulting map h : Z → S has a normal crossings special fiber h^{−1}(0) = T = T_{1} ∪ T_{2}, where T_{1} is the strict transform of Y and T_{2} is the exceptional component (i.e. P^{1}). The intersection Q = T_{1} ∩ T_{2} = T_{12} is transverse. Locally around Q, h has the following description

h(z_{1}, z_{2}) = z_{1}z_{2}.

Consider the subcomplex W_{0}(A_{Z,C}^{•}) of A_{Z,C}^{•} filtered by the monodromy filtration L induced on it by the one on A_{Z,C}^{•} (cf. § 1, (1.2)). Its hypercohomology computes H^{∗}(T, C) and it can be determined in terms of the homology of the complex

{C^{•} : H^{·}(T̃^{(1)}, C) →^{d} H^{·}(T̃^{(2)}, C)} =

{C^{•} : H^{·}(T_{1}, C) ⊕ H^{·}(T_{2}, C) →^{d} H^{·}(T_{12}, C)}

where C^{•} sits in degrees zero and one. The differential d on C^{•} is of “Čech type” i.e. it is an alternate sum of pullback maps as defined in (1.6). A product in the filtered derived category (A_{Z,C}^{•}, L) if any exists, should induce a product on C^{•}. The tensor product C^{•} ⊗ C^{•} is a complex sitting in degrees zero, one and two and it has the following description

{C^{•}⊗C^{•} : ⨁_{i,j∈[1,2]}(H^{·}(T_{i}, C)⊗H^{·}(T_{j}, C)) →^{d⊗d} ⨁_{i=1}^{2}{(H^{·}(T_{i}, C)⊗H^{·}(T_{12}, C))⊕

⊕ (H^{·}(T_{12}, C) ⊗ H^{·}(T_{i}, C))} →^{d⊗d} (H^{·}(T_{12}, C) ⊗ H^{·}(T_{12}, C))}.

However, there is no way to define canonically the product

μ : C^{•} ⊗ C^{•} → C^{•}.

In fact, let’s look for a possible description of it in each degree. In degree zero a product should satisfy

H^{·}(T_{1}, C) ⊗ H^{·}(T_{1}, C) ↦ H^{·}(T_{1}, C),
H^{·}(T_{2}, C) ⊗ H^{·}(T_{2}, C) ↦ H^{·}(T_{2}, C),
H^{·}(T_{i}, C) ⊗ H^{·}(T_{j}, C) ↦ 0, i, j = 1, 2.

In degree one, one could start by setting

H^{·}(T_{1}, C) ⊗ H^{·}(T_{12}, C) ↦ H^{·}(T_{12}, C),
H^{·}(T_{12}, C) ⊗ H^{·}(T_{2}, C) ↦ H^{·}(T_{12}, C),
H^{·}(T_{2}, C) ⊗ H^{·}(T_{12}, C) ↦ 0,
H^{·}(T_{12}, C) ⊗ H^{·}(T_{1}, C) ↦ 0.

Notice however, that this definition is not at all canonical, as one could alternatively set

H^{·}(T_{2}, C) ⊗ H^{·}(T_{12}, C) ↦ H^{·}(T_{12}, C),
H^{·}(T_{12}, C) ⊗ H^{·}(T_{1}, C) ↦ H^{·}(T_{12}, C),
H^{·}(T_{1}, C) ⊗ H^{·}(T_{12}, C) ↦ 0,
H^{·}(T_{12}, C) ⊗ H^{·}(T_{2}, C) ↦ 0.

Finally, in degree two one would have

H^{·}(T_{12}, C)⊗H^{·}(T_{12}, C) ↦ H^{·}(T_{12}, C)

4. Semistable degenerations with double points

This section is mainly devoted to the determination of [N] for one–dimensional semistable fibrations with at worst double points as singularities. The description of [N] is obtained via the introduction of the algebraic correspondence-type square on the cohomology groups of the special fiber as described in (2.1). A one–dimensional double point degeneration is the simplest example of a normal crossings fibration. The generalization of these results to double points semistable degenerations of arbitrary dimension is done at the end of this paragraph where we also report as an example of application of these results the case of a Lefschetz pencil. We keep the same notations as in § 3, in particular we denote by f : X → S a semistable fibration of fiber dimension one. Its special fiber is denoted by Y. By definition, locally around a double point P ∈ Y the description of f looks like

f(z_{1}, z_{2}) = z_{1}z_{2}

for {z_{1}, z_{2}} local parameters on X at P. For one dimensional fiberings, the only group where the local monodromy may act non trivially is gr_{2}^{L}H^{1}(X̃^{∗}, Q), in which case the identity map on the E_{1}-terms of the weight spectral sequence (1.5)

E_{1}^{−1,2} = H^{0}(Ỹ^{(2)}, Q)(−1) →^{Id} H^{0}(Ỹ^{(2)}, Q)(−1) = E_{1}^{1,0}(−1)

determines an isomorphism of rational Hodge structures of weight two on the related graded groups E_{2} = E_{∞}. This isomorphism is induced by the action of the local monodromy N around the origin:

N : gr_{2}^{L}H^{1}(X̃^{∗}, Q) →^{≃} (gr_{0}^{L}H^{1}(X̃^{∗}, Q))(−1)

It is a well known consequence of the Clemens-Schmid exact sequence (considered as a sequence of mixed Hodge structures) that (cf. [9])

gr_{2}^{L}H^{1}(X̃^{∗}, Q) ≠ 0 ⇔

⇔ Ker(ρ^{(2)} : H^{1}(Ỹ^{(1)}, Q) → H^{1}(Ỹ^{(2)}, Q)) ≠ 0 ⇔ h^{1}(|Γ|) ≠ 0

where h^{1}(|Γ|) is the dimension of the first rational cohomology group of the geometric realization of the dual graph of Y. It follows from proposition 2.1 that [N] ∈ H^{2}(T, (gr_{−2}^{L}A_{Z,Q}^{•})) = H^{0}(T̃^{(3)}, Q) determines a Hodge class

(4.1) [N] ∈ Hom_{MH}(Q(0), gr_{0}^{L}H^{2}(T, Q)) ≃

≃ Hom_{MH}(Q(0), \frac{H^{0}(T̃^{(3)}, Q)}{Image(ρ^{(3)} : H^{0}(T̃^{(2)}, Q) → H^{0}(T̃^{(3)}, Q))}).

Here T is the special fiber of a normal–crossings degeneration h : Z → S. The variety Z is a smooth threefold over C obtained via resolution of the singularities of X ×_{S} X. Notice that no more than three components of T intersect at the same closed point since dim Z = 3. We shall determine the Hodge cycle [N] ∈ E_{1}^{2,0}(Z) = H^{0}(T̃^{(3)}, Q) by means of a “correspondence type” map

N : H^{∗}(Y, gr_{r}^{L}A_{X,Q}^{•}) → H^{∗}(Y, (gr_{r−2}^{L}A_{X,Q}^{•})(−1)) = H^{∗}(Y, gr_{r}^{L}(A_{X,Q}^{•}(−1)))

as we explained in (2.1). From the proof it will easily follow that the map N is zero for ∗ ≠ 1 and is the identity for ∗ = 1 = r. On the E_{2}-level it will induce (for ∗ = 1 = r) a commutative diagram

gr_{2}^{L}H^{1}(X̃^{∗} × X̃^{∗}, Q) ⟶^{[N] ·}  gr_{2}^{L}H^{3}(X̃^{∗} × X̃^{∗}, Q) = E_{2}^{1,2}
(p_{1})^{∗}↑  ↓^{(p_{2})_{∗}}
E_{2}^{−1,2} = gr_{2}^{L}H^{1}(X̃^{∗}, Q) ⟶^{N} (gr_{0}^{L}H^{1}(X̃^{∗}, Q))(−1) = (E_{2}^{1,0})(−1)

The pullback p_{1}^{∗} and pushforward (p_{2})_{∗} are defined as in § 3. The above diagram will determine uniquely both [N] ∈ Hom_{MH}(Q(0), gr_{0}^{L}H^{2}(T, Q)) and the product [N] ·. The following result defines the geometry of the model Z and the special fiber T after resolving the singularities of X ×_{S} X and Y × Y.

Lemma 4.1. Let z_{1}z_{2} = w_{1}w_{2} be a local description of X ×_{S} X around the point (P, P), with P ∈ Y = Y_{1} ∪ Y_{2} a double point of f and {w_{1}, w_{2}} a second set of regular parameters on X at P. After a blow-up of X ×_{S} X with center at the origin (z_{1}, z_{2}, w_{1}, w_{2}), the resulting degeneration h : Z → S is normal– crossings. Its special fiber T is the union of five irreducible components: T = ∪_{i=1}^{5}T_{i}. We number them so that the first four are the strict transforms of the irreducible components Y_{i}×Y_{j} of Y ×Y, namely T_{1} = (Y_{1}×Y_{1})^{}˜ , T_{2} = (Y_{1}×Y_{2})^{}˜ , T_{3} = (Y_{2} × Y_{1})^{}˜ , T_{4} = (Y_{2} × Y_{2})^{}˜ . The last one T_{5} represents the exceptional divisor of the blow-up. We have T̃^{(1)} = ∐_{i} T_{i}. The scheme Z is covered by four affine charts U_{j}. On each of them there are three non empty components T_{k}. The scheme T̃^{(3)} is the disjoint union of four zero dimensional schemes (closed points): T_{125} ∈ U_{2}, T_{135} ∈ U_{4}, T_{245} ∈ U_{3} and T_{345} ∈ U_{1}, each of whose supports projects isomorphically onto the diagonal Δ_{12} : Y_{12} → Y_{12} × Y_{12}.

Proof. The local description of X ×_{S} X around (P, P) is given by the equations z_{1}z_{2} = w_{1}w_{2} and z_{1}z_{2} = t, for t ∈ S a fixed parameter on the disk. We choose the standard orientation of the sets {z_{1}, z_{2}} and {w_{1}, w_{2}} and we write w′_{i_{1}} = \frac{w_{i}}{z_{1}} , w′_{i_{2}} = \frac{w_{i}}{z_{2}} , w_{ij} = \frac{w_{i}}{w_{j}}, z′_{i_{1}} = \frac{z_{i}}{w_{1}} , z′_{i_{2}} = \frac{z_{i}}{w_{2}} and z_{ij} = \frac{z_{i}}{z_{j}}, for i, j = 1, 2. After a single blow-up of X ×_{S} X at the origin (z_{1}, z_{2}, w_{1}, w_{2}), the resulting model Z is non singular as one can see by looking at the first of the following tables which describes Z on each of the four charts U_{j} who cover it. In the second table, we have collected for each U_{j}, the description of the non empty divisors T_{k} ∈ T^{(1)} there. We use the pullbacks p_{1}^{∗}(\frac{dz_{1}}{z_{1}} ∧ \frac{dz_{2}}{z_{2}} ) and p_{2}^{∗}(\frac{dw_{1}}{w_{1}} ∧ \frac{dw_{2}}{w_{2}} ) to define in the third table the pullbacks p_{i}^{∗}(1_{Y_{12}}) ∈ H^{0}(T̃^{(2)}, Q).

Open sets Loc. coordinates and relations
U_{1}  {w′_{1_{1}}, w′_{2_{1}}, z_{1}}, w′_{1_{1}}w′_{2_{1}} = z_{21}
U_{2}  {w′_{1_{2}}, w′_{2_{2}}, z_{2}}, w′_{1_{2}}w′_{2_{2}} = z_{12}
U_{3}  {z′_{1_{1}}, z′_{2_{1}}, w_{1}}, z′_{1_{1}}z′_{2_{1}} = w_{21}
U_{4}  {z′_{1_{2}}, z′_{2_{2}}, w_{2}}, z′_{1_{2}}z′_{2_{2}} = w_{12}

Open sets  Divisors
U_{1}  T_{3} = {w′_{1_{1}} = 0}, T_{4} = {w′_{2_{1}} = 0}, T_{5} = {z_{1} = 0}
U_{2}  T_{1} = {w′_{1_{2}} = 0}, T_{2} = {w′_{2_{2}} = 0}, T_{5} = {z_{2} = 0}
U_{3}  T_{2} = {z′_{1_{1}} = 0}, T_{4} = {z′_{2_{1}} = 0}, T_{5} = {w_{1} = 0}
U_{4}  T_{1} = {z′_{1_{2}} = 0}, T_{3} = {z′_{2_{2}} = 0}, T_{5} = {w_{2} = 0}

Open sets  p_{1}^{∗}(1_{Y_{12}})  p_{2}^{∗}(1_{Y_{12}})
U_{1}  −1_{T_{35}} − 1_{T_{45}}  −1_{T_{45}} + 1_{T_{35}} + 1_{T_{34}}
U_{2}  1_{T_{15}} + 1_{T_{25}}  −1_{T_{25}} + 1_{T_{15}} + 1_{T_{12}}
U_{3}  −1_{T_{45}} + 1_{T_{25}} + 1_{T_{24}}  −1_{T_{25}} − 1_{T_{45}}
U_{4}  −1_{T_{35}} + 1_{T_{15}} + 1_{T_{13}}  1_{T_{15}} + 1_{T_{35}}

The global description of the pullbacks p_{1}^{∗}(1_{Y_{12}}) and p_{2}^{∗}(1_{Y_{12}}) is

p_{1}^{∗}(1_{Y_{12}}) = (1_{T_{15}} + 1_{T_{25}} − 1_{T_{35}} − 1_{T_{45}}) + 1_{T_{13}} + 1_{T_{24}}
p_{2}^{∗}(1_{Y_{12}}) = (1_{T_{15}} − 1_{T_{25}} + 1_{T_{35}} − 1_{T_{45}}) + 1_{T_{12}} + 1_{T_{34}}.

Finally, notice that each U_{j} is isomorphic to A^{3} and in each of them one has three non empty components T_{k}.

The following result holds

Theorem 4.2. Let f : X → S be the semistable degeneration of curves as described above. Then, the following description of [N] ∈ H^{0}(T̃^{(3)}, Q) (cf. (4.1)) holds:

[N] = a_{125}1_{T_{125}} + a_{135}1_{T_{135}} + a_{245}1_{T_{245}} + a_{345}1_{T_{345}}

where the (rational) numbers a’s are subject to the following requirement:

−2a_{125} + 2a_{135} − 2a_{245} + 2a_{345} = 1.

The induced class [N] in gr_{0}^{L}H^{2}(T, Q) (i.e. modulo boundary relations via the restriction map ρ^{(3)} cf. (1.6)) determines a unique zero–cycle.

Proof. We determine [N] as a cocycle making the following square commute

\begin{matrix}gr_{2}^{L}H^{1}(X̃^{∗} × X̃^{∗}, Q) & ⟶^{[N] ·} & gr_{2}^{L}H^{3}(X̃^{∗} × X̃^{∗}, Q) = E_{2}^{1,2} \\ ^{(p_{1})^{∗}}↑ & & ↓^{(p_{2})_{∗}} \\ E_{2}^{−1,2} = gr_{2}^{L}H^{1}(X̃^{∗}, Q) & ⟶^{N} & (gr_{0}^{L}H^{1}(X̃^{∗}, Q))(−1) = (E_{2}^{1,0})(−1) \end{matrix} (4.2)

In terms of cohomologies of strata, we have to describe explicitly a representative of [N] in E_{1}^{2,0}(Z) that satisfies the commutativity of

\begin{matrix}H^{0}(T̃^{(2)}, Q)(−1) & ⟶^{[N] ·} & H^{2}(T̃^{(2)}, Q) \\ ^{p_{1}^{∗}}↑ & & ↓^{(p_{2})_{∗}} \\ E_{1}^{−1,2} = H^{0}(Ỹ^{(2)}, Q)(−1) & & H^{0}(Ỹ^{(2)}, Q)(−1) = E_{1}^{1,0}(−1). \end{matrix} (4.3)

With the notations used in lemma 4.1 the description of [N] is given by

[N] = a_{125}1_{T_{125}} + a_{135}1_{T_{135}} + a_{245}1_{T_{245}} + a_{345}1_{T_{345}}.

For the standard choice of the orientations of {z_{1}, z_{2}} and {w_{1}, w_{2}} and the numbering of the T_{i}’s defined in lemma 4.1, the local description of the pullbacks p_{i}^{∗}(1_{Y_{12}}) for i = 1, 2 is given in the third table of the above lemma. Following the definition described in the Appendix (cf. (7.6)), the product [N] · p_{1}^{∗}(1_{Y_{12}}(−1)) is then the following

[N] · p_{1}^{∗}(1_{Y_{12}}(−1)) = (4.4)

= [N] · (1_{T_{15}}(−1) + 1_{T_{15}}(−1) − 1_{T_{35}}(−1) − 1_{T_{45}}(−1))
= a_{125}(g_{1}(1_{T_{125}} · 1_{T_{15}}(−1)) − g_{2}(1_{T_{125}} · 1_{T_{25}}(−1)))
+ a_{135}(g_{1}(1_{T_{135}} · 1_{T_{15}}(−1)) + g_{3}(1_{T_{135}} · 1_{T_{35}}(−1)))
+ a_{245}(g_{2}(1_{T_{245}} · 1_{T_{25}}(−1)) + g_{4}(1_{T_{245}} · 1_{T_{45}}(−1)))
+ a_{345}(−g_{3}(1_{T_{345}} · 1_{T_{35}}(−1)) + g_{4}(1_{T_{345}} · 1_{T_{45}}(−1)))
= a_{125}(1_{T_{25}} − 1_{T_{15}}) + a_{135}(1_{T_{35}} + 1_{T_{15}})
+ a_{245}(1_{T_{45}} + 1_{T_{25}}) + a_{345}(−1_{T_{45}} + 1_{T_{35}}).

The maps g_{1}, g_{2}, g_{3} and g_{4} are the pushforwards as introduced in the Appendix. The following formula illustrates the product 1_{T_{ijk}} · ∑_{l,m} 1_{T_{lm}}(−1) following the definition of it given in the Appendix:

1_{T_{ijk}} · ∑_{l,m} 1_{T_{lm}}(−1) = 1_{T_{ijk}} · (1_{T_{ik}}(−1) + 1_{T_{jk}}(−1))

= g_{i}(1_{T_{ijk}} · 1_{T_{ik}}(−1)) − g_{j}(1_{T_{ijk}} · 1_{T_{jk}}(−1)) = g_{i}(1_{T_{ijk}}) − g_{j}(1_{T_{ijk}})

∈ Image(⨁_{t} g_{t} : H^{0}(T̃^{(3)}, Q)(−1) → H^{2}(T̃^{(2)}, Q)).

In (4.4), we have denoted, for simplicity of notations, the difference g_{i}(1_{T_{ijk}}) − g_{j}(1_{T_{ijk}}) with 1_{T_{jk}} − 1_{T_{ik}}. The map g_{i} represents the pushforward on cycles deduced from the embedding g_{i} : T_{ijk} → T_{jk}. The definition of g_{j} is similar. Therefore, via the local definition of the pushforward (p_{2})_{∗} along the affine charts (cf. §3 and third table in lemma 4.1), we obtain:

(p_{2})_{∗}([N] · p_{1}^{∗}(1_{Y_{12}}(−1))) = (−2a_{125} + 2a_{135} − 2a_{245} + 2a_{345})1_{Y_{12}}(−1).

The commutativity of (4.3) and hence of (4.2) is then equivalent to the requirement

−2a_{125} + 2a_{135} − 2a_{245} + 2a_{345} = 1.

Hence, the operator [N] is determined as a cocycle in H^{0}(T̃^{(3)}, Q) by the setting

\begin{aligned}[N] = & a_{125}1_{T_{125}} + a_{135}1_{T_{135}} + a_{245}1_{T_{245}} + a_{345}1_{T_{345}}; \\ & − 2a_{125} + 2a_{135} − 2a_{245} + 2a_{345} = 1. \end{aligned} (4.5)

Up to boundary relations by means of the restriction map ρ^{(3)} which connects the elements 1_{T_{125}} with 1_{T_{245}} and 1_{T_{135}} with 1_{T_{345}}, (4.5) determines a unique zero–cycle in the quotient E_{2}^{2,0}(Z) (cf. (4.1)). Of course, if N = 0, this class may be trivial.

Remark 4.3.

The description of [N] ∈ E_{1}^{2,0}(Z) as well as the relation among the coefficients a_{ijk} in (4.5) is not unique in E_{1}. In fact, it depends on the choice of the desingularization process, as well as on the ordering of the components T_{k} ∈ T̃^{(1)}. For example, for the ordering of T_{k} for which T_{1} represents in each chart the exceptional divisor of the blow-up (T_{2} = (Y_{1} × Y_{1})^{}˜ , T_{3} = (Y_{1} × Y_{2})^{}˜ , T_{4} = (Y_{2} × Y_{1})^{}˜ , T_{5} = (Y_{2} × Y_{2})^{}˜ ), the setting (4.5) becomes

\begin{aligned}[N] = & a_{123}1_{T_{123}} + a_{124}1_{T_{124}} + a_{135}1_{T_{135}} + a_{145}1_{T_{145}}; \\ & − a_{123} + a_{124} − a_{135} + a_{145} = 1. \end{aligned}

If instead we choose to desingularize X ×_{S} X via a blowing-up along z_{1} = w_{1} = 0 and we set the order among the T_{k}’s so that the exceptional divisor is represented in each chart by the last component (i.e. T_{1} = (Y_{1} × Y_{2})^{}˜ , T_{2} = (Y_{2} × Y_{1})^{}˜ , T_{3} = (Y_{2} × Y_{2})^{}˜ , T_{4} = (Y_{1} × Y_{1})^{}˜ ,), then we would get

\begin{aligned}[N] = & a_{134}1_{T_{134}} + a_{234}1_{T_{234}}; \\ & − a_{134} + a_{234} = 1. \end{aligned}

It is a consequence of the uniqueness of the product structure on the corresponding E_{2}–terms that all these different settings determine a unique description of [N] ∈ E_{2}^{2,0}(Z). In what it follows we support some evidence for our belief that the description of [N] for a double points degeneration of higher fiber dimension (i.e. locally described by f(z_{1}, …, z_{n}) = z_{i}z_{j}, cf. below) is deducible from the case worked out for curves. As already remarked, the description of [N] in the cohomology of the strata of the special fiber of the fiber product resolution is of local nature, i.e. it can be described locally around each double point. For a higher dimensional double points degeneration [N] should be again described in terms of a “diagonal” cocycle whose support projects isomorphically onto the diagonal Δ_{12} ∈ Y_{12} × Y_{12} as it was shown in theorem 4.2. In general, that “diagonal” cocycle would be formally locally a bundle over the corresponding diagonal cocycle which comes up for a degeneration of curves. This is a consequence of the local description of the degeneration map around a double point. We give now some details for these ideas. Let f : X → S be a semistable degeneration with double points of fiber dimension d over the disk S. Then, locally in a neighborhood of a double point P on Y, f has the following description

f(z_{1}, …, z_{n}) = z_{i}z_{j}

for {z_{1}, …, z_{n}} a set of regular parameters on X at P and suitable indices i < j in I = {1, …, n}. Let Y = Y_{1} ∪ Y_{2} be the local description of Y in a neighborhood of P ∈ Y_{1} ∩ Y_{2} = Y_{12}. Locally around P, {z_{1}, …, ẑ_{i}, …, ẑ_{j}, …, z_{n}} are free parameters for this description. Hence, the special fiber is locally around the point, formally isomorphic to A^{d−1} × Ŷ with Ŷ = Ŷ_{1} ∪ Ŷ_{2} of dimension 1. In a formal neighborhood of P, Y is defined by Spec(C{{z_{1}, …, ẑ_{i}, …, ẑ_{j}, … z_{n}}}[z_{i}, z_{j}]/z_{i}z_{j}). The model X is formally locally isomorphic to A^{d−1} × X̂, with X̂ of fiber dimension 1 and special fiber Ŷ. The formal description of X ×_{S} X is similar, namely X ×_{S} X ≃ A^{d−1} × A^{d−1} × (X̂ ×_{S} X̂). Keeping the same notations introduced before, we get a formal local description of the stratum T̃^{(3)} (containing the

cocycle [N]) as A^{d−1}×A^{d−1}× ˆT̃^{(3)}, with ˆ T̃^{(3)} collection of points. [N] is a cycle

(of dimension d − 1) in T̃^{(3)} formally, locally described by Δ_{A^{d−1}} × ˆ T̃^{(3)}. This scheme is isomorphic to the formal completion of the diagonal Δ_{Y_{12}} ⊂ Y_{12}×Y_{12}, i.e. Δ̂_{Y_{12}} ≃ A^{d−1} × Ŷ_{12}, (dim Ŷ_{12} = 0). In this way, the description of [N] would be deduced from a formal local description of the Lefschetz pencil of fiber dimension one f : X̂ → C{{t}}. Hence, one would get a formal local class representative of N as a bundle over the diagonal cocycle which describes [N] in theorem 4.2. What said so far supports evidence for the following

Conjecture 4.4. Let f : X → S be a semistable double points degeneration of fiber dimension d. Then, the local monodromy operator is described by a unique algebraic cocycle of codimension d-1 in the stratum T̃^{(3)} (dim T̃^{(3)} = 2(d − 1)) i.e.

[N] ∈ \frac{CH^{d−1}(T̃^{(3)})}{Image(ρ^{(3)} : CH^{d−1}(T̃^{(2)}) → CH^{d−1}(T̃^{(3)}))}.

The formal local description of [N] is given by the algebraic cycle Δ_{A^{d−1}} × ˆT̃^{(3)}.

Notice that for a double point degeneration of fiber dimension d > 1, [N] may represent the monodromy map acting non trivially on different graded pieces of the limiting cohomology. However, they are all of type gr_{q+1}^{L}H^{q}(X̃^{∗}, Q) = E_{2}^{−1,q+1}(X) for q ∈ [0, d]. In fact, for double point degenerations we have always N = 0 on gr_{q}^{L}H^{q}(X̃^{∗}, Q), and H^{∗}(Y, gr_{i}^{L}A_{X,C}^{•}) = 0 for i ≠ −1, 0, 1 because no more than two components of Y intersect simultaneusly at the same closed point. As an example of application of these results we consider the case of a Lefschetz pencil of fiber dimension at least three. The description of [N] is the same to the one just described for a degeneration with double points. We will only show how to reduce in this case the study of [N] to the previous one. A Lefschetz pencil of fiber dimension greater than one is not even normal-crossings because the special fiber is irreducible and singular. We will only consider the case of odd fiber dimension since Lefschetz pencils of even fiber dimension have trivial monodromy always. Let f′ : X → S be such a pencil and let n = 2m + 1 be the dimension of its fiber. Locally, in a neighborhood of the singular point of the special fiber Y, the pencil f′ is described by

f′(z_{0}, …, z_{n}) = ∑_{ν=0}^{m} z_{ν}z_{ν+1+m}

where as usual {z_{0}, …, z_{n}} represents a set of regular parameters on X. It is clear from the definition that the special fiber Y is irreducible and singular at the origin (z_{0}, …, z_{n}). However, after a single blow-up at that point we get a normal-crossings degeneration f : X → S with special fiber locally described by Y = Y_{1} ∪ Y_{2}. The component Y_{1} is the exceptional divisor of the blow-up, a projective space of dimension n which intersects the strict transform Y_{2} of Y along a quadric hypersurface Y_{12} of dimension 2m. The component Y_{1} appears with multiplicity e_{1} = 2 whereas Y_{2} is reduced (i.e. e_{2} = 1). Let h : X → X be the blow-up map. It is a (proper) map of S-schemes, therefore it induces a morphism

g^{∗}RΨ_{f′}(Q_{X}) → RΨ_{f}(Q_{X})

of complexes of nearby cycles. This morphism induces in turn a homomorphism between the corresponding hypercohomologies

g^{∗} : H^{i}(Y, RΨ_{f′}(Q)) → H^{i}(Y, RΨ_{f}(Q))

In order to work with the resolution A_{Q}^{•} of RΨ_{f}(Q) which carries the monodromy filtration, we have to consider Y with its reduced structure (the exceptional divisor has multiplicity e_{1} = 2 as algebraic cycle on X). Because g.c.d.(e_{1}, e_{2}) = 1 ∀y ∈ Y the action of the local monodromy on the complex of sheaves RΨ_{f}(Y, Q) is unipotent (cf. § 1). That implies that the monodromy operator acts unipotently on cohomology. Because f′ is a Lefschetz pencil of fiber dimension n, the only group where N acts non trivially is H^{n}(X̃^{∗}, Q). Also, [N] determines an element in (H^{2n}(X̃^{∗}× X̃^{∗}, Q(n − 1)))^{π_{1}} and because the generic fibers of f′ and f are the same, we may as well consider [N] ∈ H^{2n}(Y × Y, RΨ_{f}(Q))^{π_{1}}. The map f is locally described by z_{i}^{2}q(z_{0}, …, ẑ_{i}, …, z_{n}) = t for some i ∈ [0, n], t being a local parameter on S and q(z_{0}, …, ẑ_{i}, …, z_{n}) an irreducible quadratic polynomial. Via the extension of the basis S′ → S τ ↦ \sqrt{t}, the degeneration f is deformed to w_{i}z_{i} = τ, with w_{i} = \frac{τ}{z_{i}} and w_{i}^{2} = h. It is clear that this procedure does not affect the special fibers (i.e. the reduced closed fibers are the same). Hence, after a possible normalization of the resulting model, we obtain a double point semistable degeneration h : Z → S. Let T = T_{1} ∪ T_{2} be its special fiber. Then [N] can be seen as a Hodge cycle in H^{2n}(T × T, RΨ_{h}(Q))^{π_{1}} = Ker(Ñ) ∩ H^{2n}(X̃^{∗} × X̃^{∗}, Q), for Ñ = 1 ⊗ N + N ⊗ 1. The geometric description of [N] is then the same as the one we have shown before. The class [N] represents the monodromy operator acting non trivially only on gr_{n+1}^{L}H^{n}(X̃^{∗}, Q).

5. Semistable degenerations with triple points

A semistable degeneration with triple points is the first case where both the operators N and N^{2} may be non trivial. In this paragraph we will mainly consider a triple point degeneration of surfaces. The description of [N] and [N^{2}] for higher dimensional triple points degenerations can be deduced from the one for surfaces using the same kind of arguments described in the last paragraph for double points degenerations of higher fiber dimension. Let f : X → S be a surfaces degeneration with reduced normal crossings and with triple points on its special fiber Y. We keep the basic notations as in the previous sections. Then, locally around a triple point P ∈ Y we may assume that f has the following description:

f(z_{1}, z_{2}, z_{3}) = z_{1}z_{2}z_{3}.

As usual, {z_{1}, z_{2}, z_{3}} is a regular set of parameters on X at P. Globally on X, the special fiber can be the union of more than three components i.e. Y = Y_{1} ∪ … ∪ Y_{N}, but at most three of them intersect at the same closed point. The Clemens–Schmid exact sequence of mixed Hodge structures describes the behavior of the operators N and N^{2} in terms of some invariants on the special fiber. Namely

Lemma 5.1. (Monodromy criteria) Let f : X → S be a semistable degeneration of surfaces, then

N = 0 on H^{1}(X̃^{∗}, Q) ⇔ h^{1}(|Γ|) = 0
N = 0 on H^{2}(X̃^{∗}, Q) ⇔ h^{2}(|Γ|) = 0 and ρ^{(2)} : H^{1}(Ỹ^{(1)}, Q) ↠ H^{1}(Ỹ^{(2)}, Q)
N^{2} = 0 on H^{2}(X̃^{∗}, Q) ⇔ h^{2}(|Γ|) = 0.

Here h^{i}(|Γ|) denotes the dimension of the ith-cohomology group of the geometric realization of the dual graph of Y.

Proof. cf. [9].

A degeneration of K-3 surfaces with special fiber made by rational surfaces intersecting along a cycle of rational curves, is an example for which both N and N^{2} are non zero (cf. [9]). Let us suppose that at least one of the groups gr_{2}^{L}H^{1}(X̃^{∗}, Q) and gr_{3}^{L}H^{2}(X̃^{∗}, Q) is non zero (for the above example it is well known that gr_{2}^{L}H^{1}(X̃^{∗}, Q) = 0, as H^{1}(X̃^{∗}, Q) = 0). The map N acts on them as an isomorphism of pure Hodge structures

N : gr_{2}^{L}H^{1}(X̃^{∗}, Q) →^{≃} (gr_{0}^{L}H^{1}(X̃^{∗}, Q))(−1)

N : gr_{3}^{L}H^{2}(X̃^{∗}, Q) →^{≃} (gr_{1}^{L}H^{2}(X̃^{∗}, Q))(−1).

The only group where N^{2} behaves as an isomorphism is gr_{4}^{L}H^{2}(X̃^{∗}, Q). The map N^{2} is defined by the composition

gr_{4}^{L}H^{2}(X̃^{∗}, Q) →^{N} (gr_{2}^{L}H^{2}(X̃^{∗}, Q))(−1) →^{N} (gr_{0}^{L}H^{2}(X̃^{∗}, Q))(−2).

The sequence is not exact in the middle. The map N on the left is injective and the one on the right surjects (gr_{2}^{L}H^{2}(X̃^{∗}, Q))(−1) onto (gr_{0}^{L}H^{2}(X̃^{∗}, Q))(−2). Its kernel, in term of the spectral sequence of weights is

(Im(H^{2}(Y, gr_{1}^{W} Ω_{X}^{•+1}(log Y)) ⊗ Q → H^{2}(Y, A_{X,Q}^{•})))(−1) ≃

≃ \frac{Ker(ρ^{(2)} : H^{2}(Ỹ^{(1)}, Q)(−1) → H^{2}(Ỹ^{(2)}, Q)(−1))}{Im(γ^{(2)} : H^{0}(Ỹ^{(2)}, Q)(−2) → H^{2}(Ỹ^{(1)}, Q)(−1))}.

We first consider N and its related class [N]. Both gr_{2}^{L}H^{1}(X̃^{∗}, Q) and gr_{3}^{L}H^{2}(X̃^{∗}, Q) are described in terms of cohomology classes on Ỹ^{(2)} (cf. (1.7)). The study of the correspondence-diagram (2.1) is similar for them. Namely, once one has found an algebraic cycle representing [N], it certainly makes both the correspondence diagrams commute. For degenerations of surfaces it follows from proposition 2.1 that

[N] ∈ (gr_{2}^{L}H^{4}(T, Q))(1) ≃ \frac{Ker(ρ^{(4)} : H^{2}(T̃^{(3)}, Q)(1) → H^{2}(T̃^{(4)}, Q)(1))}{Im(ρ^{(3)} : H^{2}(T̃^{(2)}, Q)(1) → H^{2}(T̃^{(3)}, Q)(1))} (5.1)

where h : Z → S is a normal crossings degeneration with special fiber T and generic fiber X̃^{∗} × X̃^{∗} obtained via resolution of the singularities of X ×_{S} X. Similarly, one has

[N^{2}] ∈ gr_{0}^{L}H^{4}(T, Q) ≃ \frac{H^{0}(T̃^{(5)}, Q)}{Im(ρ^{(5)} : H^{0}(T̃^{(4)}, Q) → H^{0}(T̃^{(5)}, Q))}. (5.2)

Both [N] and [N^{2}] have the further property to be Hodge cycles in the cohomologies of the corresponding strata. The following lemma determines the geometry of the model Z and the special fiber T after resolving the singularities of X ×_{S} X and Y × Y.

Lemma 5.2. Let z_{1}z_{2}z_{3} = w_{1}w_{2}w_{3} be a local description of X ×_{S} X around the point (P, P), being P ∈ Y = ∪_{i=1}^{3}Y_{i} a triple point of f and {w_{1}, w_{2}, w_{3}} a second set of regular parameters on X at P. After three blows-up of X ×_{S} X with centers at z_{i} = 0 = w_{i} (i = 1, 2, 3) the resulting degeneration h : Z → S is normal–crossings. Its special fiber T is the union of nine irreducible components: T = ∪_{i=1}^{9}T_{i}. We number them so that the first six are the strict transforms of the irreducible components Y_{i} × Y_{j} of Y × Y: T_{1} = (Y_{1} × Y_{2})^{}˜ , T_{2} = (Y_{1} ×Y_{3})^{}˜ , T_{3} = (Y_{2} ×Y_{1})^{}˜ , T_{4} = (Y_{2} ×Y_{3})^{}˜ , T_{5} = (Y_{3} ×Y_{1})^{}˜ , T_{6} = (Y_{3} ×Y_{2})^{}˜ . The last three components are the exceptional divisors of the three blows-up: T_{7} = (Y_{1} × Y_{1})^{}˜ , T_{8} = (Y_{2} × Y_{2})^{}˜ , T_{9} = (Y_{3} × Y_{3})^{}˜ . We have T̃^{(1)} = ∐_{i} T_{i}. The scheme Z is covered by eight affine charts, on each of them there are at most five non empty components T_{i}. Among the components T_{ijk} whose disjoint union defines the scheme T̃^{(3)}, T_{178} and T_{378} contain resp. the curves “diagonal” δ̃_{12} and δ_{12} whose supports project isomorphically onto the diagonal Δ_{12} : Y_{12} → Y_{12} ×Y_{12}. Similarly, T_{279} and T_{579} contain resp. δ̃_{13} and δ_{13} whose support projects isomorphically onto Δ_{13} : Y_{13} → Y_{13} × Y_{13}. Finally, T_{489} and T_{689} contain δ̃_{23} and δ_{23} whose support is isomorphic to Δ_{23}. The exceptional surface T_{789}–intersection of the three exceptional divisors of h–is isomorphic to the blow-up Bl of P^{1} × P^{1} at the points {(0, 1) × (1, 0)} and {(1, 0) × (0, 1)}. Finally, the scheme T̃^{(5)} is the disjoint union of six irreducible components (points). They are: T_{12789}, T_{16789}, T_{24789}, T_{34789}, T_{35789}, T_{56789}. Their support maps isomorphically onto the (point) diagonal Δ_{123} : Y_{123} → Y_{123} × Y_{123}.

Proof. The local description of X ×_{S} X at (P, P) is given by the equations z_{1}z_{2}z_{3} = w_{1}w_{2}w_{3} and z_{1}z_{2}z_{3} = t, for t ∈ S a fixed parameter on the disk. We choose the standard orientation of the sets {z_{1}, z_{2}, z_{3}} and {w_{1}, w_{2}, w_{3}} and we write w′_{i} = \frac{w_{i}}{z_{i}}, z′_{i} = \frac{z_{i}}{w_{i}} for i = 1, 2, 3. After three blows-up of X ×_{S} X along the subvarieties z_{i} = 0 = w_{i}, the resulting model Z is non singular as one can see by looking at the first of the following tables which describes Z on each of the eight charts U_{j} who cover it. In the second table, we have collected for each U_{j}, the description of the non empty divisors T_{k} ∈ T^{(1)} there and the third table shows the “diagonal” curves δ and δ̃ defined in each chart. The remaining charts describe the pullbacks p_{1}^{∗}(\frac{dz_{i}}{z_{i}} ∧ \frac{dz_{j}}{z_{j}}), p_{2}^{∗}(\frac{dw_{i}}{w_{i}} ∧ \frac{dw_{j}}{w_{j}}), p_{1}^{∗}(\frac{dz_{1}}{z_{1}} ∧ \frac{dz_{2}}{z_{2}} ∧ \frac{dz_{3}}{z_{3}} ) and p_{2}^{∗}(\frac{dw_{1}}{w_{1}} ∧ \frac{dw_{2}}{w_{2}} ∧ \frac{dw_{3}}{w_{3}} ) in terms of the related descriptions by cocycles classes in the corresponding cohomologies.

Open sets  Loc. coordinates and relations
U_{1}  {w′_{1}, w′_{2}, w′_{3}, z_{1}, z_{2}, z_{3}}, w′_{1}w′_{2}w′_{3} = 1
U_{2}  {w′_{1}, w′_{2}, z_{1}, z_{2}, w_{3}}, w′_{1}w′_{2} = z′_{3}
U_{3}  {w′_{1}, w′_{3}, z_{1}, z_{3}, w_{2}}, w′_{1}w′_{3} = z′_{2}
U_{4}  {z′_{2}, z′_{3}, z_{1}, w_{2}, w_{3}}, z′_{2}z′_{3} = w′_{1}
U_{5}  {w′_{2}, w′_{3}, z_{2}, z_{3}, w_{1}}, w′_{2}w′_{3} = z′_{1}
U_{6}  {z′_{1}, z′_{3}, z_{2}, w_{1}, w_{3}}, z′_{1}z′_{3} = w′_{2}
U_{7}  {z′_{1}, z′_{2}, z_{3}, w_{1}, w_{2}}, z′_{1}z′_{2} = w′_{3}
U_{8}  {z′_{1}, z′_{2}, z′_{3}, w_{1}, w_{2}, w_{3}}, z′_{1}z′_{2}z′_{3} = 1

Documenta Mathematica 4 (1999) 65–108

Open sets  Divisors
U_{1}  T_{7} = {z_{1} = 0}, T_{8} = {z_{2} = 0}, T_{9} = {z_{3} = 0}
U_{2}  T_{5} = {w′_{1} = 0}, T_{6} = {w′_{2} = 0}, T_{7} = {z_{1} = 0},

T_{8} = {z_{2} = 0}, T_{9} = {w_{3} = 0}

U_{3}  T_{3} = {w′_{1} = 0}, T_{4} = {w′_{3} = 0}, T_{7} = {z_{1} = 0},

T_{8} = {w_{2} = 0}, T_{9} = {z_{3} = 0}

U_{4}  T_{3} = {z′_{2} = 0}, T_{5} = {z′_{3} = 0}, T_{7} = {z_{1} = 0},

T_{8} = {w_{2} = 0}, T_{9} = {w_{3} = 0}

U_{5}  T_{1} = {w′_{2} = 0}, T_{2} = {w′_{3} = 0}, T_{7} = {w_{1} = 0},

T_{8} = {z_{2} = 0}, T_{9} = {z_{3} = 0}

U_{6}  T_{1} = {z′_{1} = 0}, T_{6} = {z′_{3} = 0}, T_{7} = {w_{1} = 0},

T_{8} = {z_{2} = 0}, T_{9} = {w_{3} = 0}

U_{7}  T_{2} = {z′_{1} = 0}, T_{4} = {z′_{2} = 0}, T_{7} = {w_{1} = 0},

T_{8} = {w_{2} = 0}, T_{9} = {z_{3} = 0}

U_{8}  T_{7} = {w_{1} = 0}, T_{8} = {w_{2} = 0}, T_{9} = {w_{3} = 0}

Open sets  “Diagonal” curves
U_{1}  none
U_{2}  δ_{13} = {w′_{1} = z_{1} = w_{3} = 0, w′_{2} = 1} ⊂ T_{579}, δ_{13} ∩ T_{8} ≠ ∅

δ_{23} = {w′_{2} = z_{2} = w_{3} = 0, w′_{1} = 1} ⊂ T_{689}, δ_{23} ∩ T_{7} ≠ ∅

U_{3}  δ_{12} = {w′_{1} = z_{1} = w_{2} = 0, w′_{3} = 1} ⊂ T_{378}, δ_{12} ∩ T_{9} ≠ ∅

δ̃_{23} = {w′_{3} = z_{3} = w_{2} = 0, w′_{1} = 1} ⊂ T_{489}, δ̃_{23} ∩ T_{7} ≠ ∅

U_{4}  δ_{12} = {z′_{2} = z_{1} = w_{2} = 0, z′_{3} = 1} ⊂ T_{378}, δ_{12} ∩ T_{9} ≠ ∅

δ_{13} = {z′_{3} = z_{1} = w_{3} = 0, z′_{2} = 1} ⊂ T_{579}, δ_{13} ∩ T_{8} ≠ ∅

U_{5}  δ̃_{12} = {w′_{2} = z_{2} = w_{1} = 0, w′_{3} = 1} ⊂ T_{178}, δ̃_{12} ∩ T_{9} ≠ ∅

δ̃_{13} = {w′_{3} = z_{3} = w_{1} = 0, w′_{2} = 1} ⊂ T_{279}, δ̃_{13} ∩ T_{8} ≠ ∅

U_{6}  δ̃_{12} = {z′_{1} = z_{2} = w_{1} = 0, z′_{3} = 1} ⊂ T_{178}, δ̃_{12} ∩ T_{9} ≠ ∅

δ_{23} = {z′_{3} = z_{2} = w_{3} = 0, z′_{1} = 1} ⊂ T_{689}, δ̃_{23} ∩ T_{7} ≠ ∅

U_{7}  δ̃_{13} = {z′_{1} = z_{3} = w_{1} = 0, z′_{2} = 1} ⊂ T_{279}, δ̃_{13} ∩ T_{8} ≠ ∅

δ̃_{23} = {z′_{2} = z_{3} = w_{2} = 0, z′_{1} = 1} ⊂ T_{489}, δ̃_{23} ∩ T_{7} ≠ ∅

U_{8}  none

Denote by v_{Y_{ij}} a class in H^{∗}(Y_{ij}, C) and by v_{T_{lk}} a class in H^{∗}(T̃^{(2)}, C). Then we have

Open sets  p_{1}^{∗}(v_{Y_{12}})  p_{2}^{∗}(v_{Y_{12}})
U_{1}  v_{T_{78}}  v_{T_{78}}
U_{2}  v_{T_{78}}  v_{T_{56}} + v_{T_{58}} − v_{T_{67}} + v_{T_{78}}
U_{3}  −v_{T_{37}} − v_{T_{47}} + v_{T_{78}}  v_{T_{38}} + v_{T_{78}}
U_{4}  −v_{T_{37}} + v_{T_{78}}  v_{T_{78}} + v_{T_{38}} + v_{T_{58}}
U_{5}  v_{T_{18}} + v_{T_{28}} + v_{T_{78}}  v_{T_{78}} − v_{T_{17}}
U_{6}  v_{T_{18}} + v_{T_{78}}  v_{T_{78}} − v_{T_{17}} − v_{T_{67}}
U_{7}  v_{T_{24}} + v_{T_{28}} − v_{T_{47}} + v_{T_{78}}  v_{T_{78}}
U_{8}  v_{T_{78}}  v_{T_{78}}

Local Monodromy 93

Hence, the global description of the pullbacks p_{1}^{∗}(v_{Y_{12}}) and p_{2}^{∗}(v_{Y_{12}}) are

p_{1}^{∗}(v_{Y_{12}}) = (v_{T_{18}} + v_{T_{28}} − v_{T_{37}} − v_{T_{47}} + v_{T_{78}}) + v_{T_{24}}
p_{2}^{∗}(v_{Y_{12}}) = (−v_{T_{17}} + v_{T_{38}} + v_{T_{58}} − v_{T_{67}} + v_{T_{78}}) + v_{T_{56}}.
Open sets  p_{1}^{∗}(v_{Y_{13}})  p_{2}^{∗}(v_{Y_{13}})
U_{1}  v_{T_{79}}  v_{T_{79}}
U_{2}  −v_{T_{57}} − v_{T_{67}} + v_{T_{79}}  v_{T_{79}} + v_{T_{59}}
U_{3}  v_{T_{79}}  v_{T_{79}} − v_{T_{47}} + v_{T_{39}} + v_{T_{34}}
U_{4}  −v_{T_{57}} + v_{T_{79}}  v_{T_{79}} + v_{T_{39}} + v_{T_{59}}
U_{5}  v_{T_{19}} + v_{T_{29}} + v_{T_{79}}  v_{T_{79}} − v_{T_{27}}
U_{6}  v_{T_{16}} + v_{T_{19}} − v_{T_{67}} + v_{T_{79}}  v_{T_{79}}
U_{7}  v_{T_{29}} + v_{T_{79}}  v_{T_{79}} − v_{T_{27}} − v_{T_{47}}
U_{8}  v_{T_{79}}  v_{T_{79}}

Hence we have the global descriptions

p_{1}^{∗}(v_{Y_{13}}) = (v_{T_{19}} + v_{T_{29}} − v_{T_{57}} − v_{T_{67}} + v_{T_{79}}) + v_{T_{16}}
p_{2}^{∗}(v_{Y_{13}}) = (−v_{T_{27}} + v_{T_{39}} − v_{T_{47}} + v_{T_{59}} + v_{T_{79}}) + v_{T_{34}}.
Open sets  p_{1}^{∗}(v_{Y_{23}})  p_{2}^{∗}(v_{Y_{23}})
U_{1}  v_{T_{89}}  v_{T_{89}}
U_{2}  −v_{T_{58}} − v_{T_{68}} + v_{T_{89}}  v_{T_{89}} + v_{T_{69}}
U_{3}  v_{T_{39}} + v_{T_{49}} + v_{T_{89}}  v_{T_{89}} − v_{T_{48}}
U_{4}  v_{T_{35}} + v_{T_{39}} − v_{T_{58}} + v_{T_{89}}  v_{T_{89}}
U_{5}  v_{T_{89}}  v_{T_{89}} − v_{T_{28}} + v_{T_{19}} + v_{T_{12}}
U_{6}  −v_{T_{68}} + v_{T_{89}}  v_{T_{89}} + v_{T_{19}} + v_{T_{69}}
U_{7}  v_{T_{49}} + v_{T_{89}}  v_{T_{89}} − v_{T_{28}} − v_{T_{48}}
U_{8}  v_{T_{89}}  v_{T_{89}}

Documenta Mathematica 4 (1999) 65–108

Finally we have

p_{1}^{∗}(v_{Y_{23}}) = (v_{T_{39}} + v_{T_{49}} − v_{T_{58}} − v_{T_{68}} + v_{T_{89}}) + v_{T_{35}}
p_{2}^{∗}(v_{Y_{23}}) = (v_{T_{19}} − v_{T_{28}} − v_{T_{48}} + v_{T_{69}} + v_{T_{89}}) + v_{T_{12}}.

Using the above tables we deduce the following

Open sets  p_{1}^{∗}(1_{Y_{123}})  p_{2}^{∗}(1_{Y_{123}})
U_{1}  1_{T_{789}}  1_{T_{789}}
U_{1}  1_{T_{578}} + 1_{T_{678}} + 1_{T_{789}}  1_{T_{569}} + 1_{T_{589}} − 1_{T_{679}} + 1_{T_{789}}
U_{3}  −1_{T_{379}} − 1_{T_{479}} + 1_{T_{789}}  −1_{T_{348}} + 1_{T_{389}} + 1_{T_{478}} + 1_{T_{789}}
U_{4}  1_{T_{357}} − 1_{T_{379}} + 1_{T_{578}} + v_{T_{789}}  1_{T_{389}} + 1_{T_{589}} + 1_{T_{789}}
U_{5}  1_{T_{189}} + 1_{T_{289}} + 1_{T_{789}}  1_{T_{127}} − 1_{T_{179}} + 1_{T_{278}} + v_{T_{789}}
U_{6}  −1_{T_{168}} + 1_{T_{189}} + 1_{T_{678}} + 1_{T_{789}}  −1_{T_{179}} − 1_{T_{679}} + v_{T_{789}}
U_{7}  1_{T_{249}} + 1_{T_{289}} − 1_{T_{479}} + 1_{T_{789}}  1_{T_{278}} + 1_{T_{478}} − v_{T_{789}}
U_{8}  1_{T_{789}}  1_{T_{789}}

We then obtain

p_{1}^{∗}(1_{Y_{123}}) = (1_{T_{189}} + 1_{T_{249}} + 1_{T_{289}} − 1_{T_{379}} − 1_{T_{479}} + 1_{T_{789}})
− 1_{T_{168}} + 1_{T_{357}} + 1_{T_{578}} + 1_{T_{678}}
p_{2}^{∗}(1_{Y_{123}}) = (−1_{T_{179}} + 1_{T_{389}} + 1_{T_{569}} + 1_{T_{589}} − 1_{T_{679}} + 1_{T_{789}})
+ 1_{T_{127}} + 1_{T_{278}} − 1_{T_{348}} + 1_{T_{478}}.

Notice that with the exception of U_{1} and U_{8} that are open sets in A^{5} on which only the exceptional components T_{7}, T_{8} and T_{9} are non empty, all the remaining charts U_{j} are isomorphic to A^{5} and in each of them one has five components T_{k} non empty. On U_{3} ∩ U_{4} the surface T_{378} contains the curve δ_{12}, and on U_{5} ∩ U_{6}, T_{178} contains the curve δ̃_{12}. The curves δ_{12} and δ̃_{12} are different: i.e. T_{1} = ∅ on U_{3} and U_{4}, but their supports map isomorphically onto the same diagonal Δ_{12} : Y_{12} → Y_{12} × Y_{12}. Similarly, U_{2} ∩ U_{4} contains δ_{13} whose support maps isomorphically onto Δ_{13}, whereas U_{5} ∩ U_{7} contains δ̃_{13}, whose support maps still isomorphically onto Δ_{13}: δ_{13} ∩ δ̃_{13} = ∅. Finally, δ_{23} ⊂ U_{2} ∩ U_{6}, δ_{23} ≃ Δ_{23}, while δ̃_{23} ⊂ U_{3} ∩ U_{7}, δ̃_{23} ≃ Δ_{23} and δ_{23} ∩ δ̃_{23} = ∅. The blow-up Z_{1} of X ×_{S} X at z_{1} = 0 = w_{1} is the strict transform of X ×_{S} X in the blow-up of A^{6} along the corresponding linear subvariety. Let (z̃_{1}, w̃_{1}) be a couple of homogeneus coordinates. The exceptional divisor, say E_{1}^{(1)}, is locally a P_{(z̃_{1},w̃_{1})}^{1}–bundle over {z_{1} = 0 = w_{1}}. Then, the intersection E_{1}^{(1)}∩Z_{1} is locally

defined on E_{1}^{(1)} by z_{2}z_{3}z̃_{1} −w_{2}w_{3}w̃_{1} = 0. The blow E_{1}^{(2)} of P^{1} ×{z_{1} = 0 = w_{1}} on P^{1} × {z_{1} = z_{2} = w_{1} = w_{2} = 0} defines the strict transform of E_{1}^{(1)} after the second blow-up along {z_{2} = w_{2}}. Said E_{2}^{(2)} the exceptional divisor of the second blow-up and (z̃_{2}, w̃_{2}) another couple of homogeneus coordinates, one has E_{1}^{(2)} ∩ E_{2}^{(2)} = P_{(z̃_{1},w̃_{1})}^{1} × P_{(z̃_{2},w̃_{2})}^{1} × {z_{1} = z_{2} = w_{1} = w_{2} = 0}. Finally, after

the third blowing at {z_{3} = 0 = w_{3}} the three exceptional divisors E_{1}^{(3)}, E_{2}^{(3)} and E_{3}^{(3)} will intersect the strict transform Z of X ×_{S} X along the exceptional surface T_{789}. This surface is described by the equation z̃_{1}z̃_{2}z̃_{3} − w̃_{1}w̃_{2}w̃_{3} = 0 in E_{1}^{(3)} ∩E_{2}^{(3)} ∩E_{3}^{(3)} = P_{(z̃_{1},w̃_{1})}^{1} ×P_{(z̃_{2},w̃_{2})}^{1} ×P_{(z̃_{3},w̃_{3})}^{1} ×{z_{1} = z_{2} = z_{3} = w_{1} = w_{2} = w_{3} = 0} = (P^{1})^{3}, (z̃_{3}, w̃_{3}) being a third couple of homogeneus coordinates. Let consider the projection T_{789} → P_{(z̃_{2},w̃_{2})}^{1} ×P_{(z̃_{3},w̃_{3})}^{1}. The fiber of this map over a given point in the base (P^{1})^{2} is defined by a linear equation as αz_{1} − βw_{1} = 0. If either α or β (or both) is not zero, then this fiber is reduced to a single point, so the projection map is locally an isomorphism. On the other hand, α = 0 = β happens over the two points (1, 0) × (0, 1) and (0, 1) × (1, 0), where the fiber is a P^{1}. Since T_{789} is non singular, these two copies of P^{1} are Cartier divisors, so by the universal property of blow-ups the map factors through the blow-up Bl of (P^{1})^{2} at the two points (i.e. T_{789} → Bl → (P^{1})^{2}). It is easy to see from this description that T_{789} ≃ Bl. It is straighforward to verify from the second table the description of T̃^{(5)} on each chart U_{j} and the statement concerning its support.

The following result generalizes the description of [N] given in theorem 4.2 for double points degenerations.

Theorem 5.3. Let f : X → S be a semistable degeneration of surfaces as we have considered above. With the same notations as in lemma 5.2, let π : Bl → P^{1} × P^{1} be the morphism definying the blow-up of P^{1} × P^{1} at the points {(0, 1) × (1, 0)} and {(1, 0) × (0, 1)}, being Bl ≃ T_{789}. Let F_{1} = π^{∗}({pt} × P^{1}) and F_{2} = π^{∗}(P^{1} × {pt}) be the two fundamental fibers and let E_{1} and E_{2} be the two exceptional divisors of π. The following description of [N] ∈ Ker ρ^{(4)} (cf. (5.1)) holds:

[N] = a_{178}δ̃_{12} + a_{279}δ̃_{13} + a_{378}δ_{12} + a_{489}δ̃_{23} + a_{579}δ_{13} + a_{689}δ_{23} + Γ.

The 1-cycle Γ ⊂ Bl and the (rational) numbers a’s are subject to the following requirements:

Γ = xF_{1} + yF_{2} + zE_{1} + wE_{2}, with w = z − 1, x, y, z, w ∈ Q

a_{178} − a_{378} = a_{279} − a_{579} = a_{489} − a_{689} = 1 and the relations among them are given by the following set of equalities

\begin{matrix}a_{178} = −w,  a_{279} = −(y + w),  a_{378} = −z, \\ a_{489} = x + z,  a_{579} = −(y + z),  a_{689} = x + w. \end{matrix}

Furthermore, for those degenerations with N^{2} ≠ 0, the class [N^{2}] ∈ E_{1}^{0,4}(Z) = H^{0}(T̃^{(5)}, Q) (cf. (5.2)) can be exhibited as:

\begin{aligned}[N^{2}] = & b_{12789}T_{12789} + b_{16789}T_{16789} + b_{24789}T_{24789} \\ & + b_{34789}T_{34789} + b_{35789}T_{35789} + b_{56789}T_{56789}. \end{aligned}

The (rational) numbers b’s must satisfy the following equation:

−b_{12789} + b_{16789} − b_{24789} + b_{34789} − b_{35789} − b_{56789} = 1.

Hence, the induced classes of [N] in gr_{2}^{L}H^{4}(T, Q)(1) and of [N^{2}] in gr_{0}^{L}H^{4}(T, Q) (i.e. modulo boundary relations via the restriction maps ρ^{(3)} and ρ^{(5)} cf. (1.6)) determine algebraic cocycles of dimension one and zero respectively.

Proof. We will determine [N] as a cocycle making the following square commute (i.e. this is the one one has to study for a degeneration of K-3 surfaces of the type mentioned above)

gr_{3}^{L}H^{2}(X̃^{∗} × X̃^{∗}, Q) ⟶^{[N] ·} gr_{5}^{L}H^{6}(X̃^{∗} × X̃^{∗}, Q)(1) = (E_{2}^{1,5})(1)
(p_{1})^{∗}↑  ↓^{(p_{2})_{∗}}
E_{2}^{−1,3} = gr_{3}^{L}H^{2}(X̃^{∗}, Q) ⟶^{N} (gr_{1}^{L}H^{2}(X̃^{∗}, Q))(−1) = (E_{2}^{1,1})(−1)

Note that besides the commutativity of the square, one has to impose another condition on [N] in order for it to represent the operator N. That arises from (5.1). Namely, the representative of N in (E_{1}^{2,2})(1) = H^{2}(T̃^{(3)}, Q)(1) must belong to the kernel of the related restriction map ρ^{(4)}. This condition was automatically satisfied for double point degenerations since T^{(4)} = ∅ always in that case. We will explicitly describe a representative [N] of N in (E_{1}^{2,2})(1) that satisfies the commutativity of the following square

\begin{matrix}H^{1}(T̃^{(2)}, Q)(−1) & ⟶^{[N] ·} & H^{5}(T̃^{(2)}, Q)(1) \\ ^{p_{1}^{∗}}↑ & & ↓^{(p_{2})_{∗}} \\ H^{1}(Ỹ^{(2)}, Q)(−1) & & H^{1}(Ỹ^{(2)}, Q)(−1). \end{matrix} (5.3)

With the notations introduced in lemma 5.2 we first remark that the cocycles [δ_{ij}] = (Δ_{ij})_{∗}(1_{Y_{ij}}) (i, j = 1, 2, 3, i ≠ j), Δ_{ij} : Y_{ij} → Y_{ij} × Y_{ij} being the diagonal embedding, evidently satisfy the cohomological equality

(p_{2})_{∗}(Δ_{∗}(1_{Y_{ij}}) · (p_{1})^{∗}(v)) = (p_{2})_{∗}(Δ_{∗}Δ^{∗}p_{1}^{∗}(v)) = (p_{2})_{∗}(Δ_{∗}(v)) = v

for 1_{Y_{ij}} ∈ H^{0}(Y_{ij}, Q) and any element v ∈ H^{1}(Ỹ^{(2)}, Q)(−1). However, since a simple linear combination as a_{178}δ̃_{12} + a_{279}δ̃_{13} + a_{378}δ_{12} + a_{489}δ̃_{23} + a_{579}δ_{13} + a_{689}δ_{23} (the coefficients a’s are integers) does not satisfy the requirement of being in the kernel of the restriction map ρ^{(4)} (cf. (5.1) and (1.6)), we have to add to the above “diagonal” definition a 1-cocycle Γ ⊂ T_{789}, so that the completed linear combination defines an element in (E_{2}^{2,2})(1) representing N. Notice that since the exceptional surface T_{789} projects down via p_{2}, onto the triple point P, this modification by Γ does not spoil the commutativity of (5.3), once we have checked it for the partial representative of [N] given in terms of the above diagonals. The 1-cycle Γ will be described as a combination of the generators F_{1}, F_{2}, E_{1}, E_{2} of the Neron-Severi group NS(T_{789}). First of all, let consider the six curves T_{k789} for k = 1, …, 6. They are elements of T̃^{(4)}. We describe them using the generators of NS(T_{789}). Because π(T_{1789}) = {(0, 1) × (1, 0)}, T_{1789} = E_{2}. Similarly, we have T_{3789} = E_{1}, as π(T_{3789}) = {(1, 0) × (0, 1)}. The remaining four curves are described using the projection formula. For example, we know that π(T_{2789}) = (0, 1)×P^{1} and that π^{∗}((0, 1)×P^{1}) = F_{1} = E_{2}+T_{2789}. Hence we have T_{2789} = F_{1} − E_{2}. With a similar procedure we obtain T_{4789} = F_{2} − E_{1}, T_{5789} = F_{1} − E_{1} and T_{6789} = F_{2} − E_{2}. The geometry of the intersections among the generators of NS(T_{789}) is well known, namely E_{1} · E_{2} = E_{1} · F_{2} = E_{1} · F_{1} = E_{2} · F_{1} = E_{2} · F_{2} = F_{1} · F_{1} = F_{2} · F_{2} = 0, E_{1} · E_{1} = −1 = E_{2} · E_{2} and F_{1} · F_{2} = 1. Let Γ = xF_{1} +yF_{2} +zE_{1} +wE_{2} be an element of NS(T_{789}), with x, y, z, w ∈ Q. Then, we must solve

[N] = a_{178}δ̃_{12} + a_{279}δ̃_{13} + a_{378}δ_{12} + a_{489}δ̃_{23} + a_{579}δ_{13} + a_{689}δ_{23} + Γ

for Γ subject to the condition that [N] is in ker ρ^{(4)}, for ρ^{(4)} = ∑_{u=1}^{4}(−1)^{u−1}ρ_{u}^{(4)} (cf. (1.6)). For example we have ρ^{(4)}(a_{178}δ̃_{12}) = −a_{178}(δ̃_{12} · T_{9}), while ρ^{(4)}(a_{279}δ̃_{13}) = a_{279}(δ̃_{13} · T_{8}). Following these rules we obtain the system

\begin{matrix}a_{178} = Γ · T_{1789} = −w,  a_{279} = −Γ · T_{2789} = −(y + w) \\ a_{378} = Γ · T_{3789} = −z,  a_{489} = Γ · T_{4789} = x + z \\ a_{579} = −Γ · T_{5789} = −(y + z),  a_{689} = Γ · T_{6789} = x + w. \end{matrix} (5.4)

For the standard choice of the orientations of {z_{1}, z_{2}, z_{3}} and {w_{1}, w_{2}, w_{3}} and the numbering of the T_{i}’s setted in lemma 5.2, the local description of the pullbacks \frac{dz_{i}}{z_{i}} ∧ \frac{dz_{j}}{z_{j}} and \frac{dw_{i}}{w_{i}} ∧ \frac{dw_{j}}{w_{j}} (i ≠ j, i, j = 1, 2, 3) in terms of cohomology classes v_{T_{ij}} and v_{T_{ijk}}, is given following the tables shown in the proof of lemma 5.2. Let v_{ij} ∈ H^{1}(Ỹ^{(2)}, Q)(−1), then via the multiplicative rule described in the Appendix (cf. the similar calculation done in the proof of theorem 4.2) we obtain

[N] · p_{1}^{∗}(v_{12} + v_{13} + v_{23}) =
= [N] · (v_{T_{18}} + v_{T_{78}} + v_{T_{29}} + v_{T_{79}} + v_{T_{49}} + v_{T_{89}}) =

= a_{178}g_{1}(δ̃_{12} · v_{T_{18}}) − a_{378}g_{7}(δ_{12} · v_{T_{78}}) + a_{279}g_{2}(δ̃_{13} · v_{T_{29}}) − a_{579}g_{7}(δ_{13} · v_{T_{79}})+

+a_{489}(g_{4}(δ̃_{23} · v_{T_{49}}) − a_{689}(g_{8}(δ_{23} · v_{T_{89}}) =

= a_{178}v_{78}(1) − a_{378}v_{38}(1) + a_{279}v_{79}(1) − a_{579}v_{59}(1) + a_{489}v_{89}(1) − a_{689}v_{69}(1) where g_{j} are the pushforward maps defined in the Appendix. Applying the map (p_{2})_{∗} we have

(p_{2})_{∗}([N] · p_{1}^{∗}(v_{12} + v_{13} + v_{23}))

= (a_{178} − a_{378})v_{12} + (a_{279} − a_{579})v_{13} + (a_{489} − a_{689})v_{23}.

The commutativity of the diagram (5.3) is then equivalent to the requirement

a_{178} − a_{378} = a_{279} − a_{579} = a_{489} − a_{689} = 1 (5.5)

The linear system (5.4) may be then read as z − w = 1. Therefore, any curve Γ = xF_{1} + yF_{2} + zE_{1} + wE_{2} satisfying the condition z − w = 1 can be used in the description of [N] ∈ (E_{1}^{2,2})(1). The description of [N^{2}] is similar. For instance, from proposition 2.1 we have

[N^{2}] ∈ gr_{0}^{L}H^{4}(T, Q) ≃ \frac{H^{0}(T̃^{(5)}, Q)}{Im(ρ^{(5)} : H^{0}(T̃^{(4)}, Q) → H^{0}(T̃^{(5)}, Q))}

Via the procedure described in (2.1), [N^{2}] is then determined in terms of the commutativity of the following square

gr_{4}^{L}H^{2}(X̃^{∗} × X̃^{∗}, Q) ⟶^{[N^{2}] ·}  gr_{4}^{L}H^{6}(X̃^{∗} × X̃^{∗}, Q) = E_{2}^{2,4}
(p_{1})^{∗}↑  ↓^{(p_{2})_{∗}}
E_{2}^{−2,4} = gr_{4}^{L}H^{2}(X̃^{∗}, Q) ⟶^{N^{2}} (gr_{0}^{L}H^{2}(X̃^{∗}, Q))(−2) = (E_{2}^{2,0})(−2).

The related E_{1} description is

H^{0}(T̃^{(3)}, Q)(−2) ⟶^{[N^{2}] ·} H^{4}(T̃^{(3)}, Q)
p_{1}^{∗}↑  ↓^{(p_{2})_{∗}}
H^{0}(Ỹ^{(3)}, Q)(−2)  H^{0}(Ỹ^{(3)}, Q)(−2).

The scheme T̃^{(5)} is the disjoint union of the zero dimensional schemes T_{12789}, T_{16789}, T_{35789} and T_{56789}. Their support map all isomorphically onto the diagonal Δ_{123} : Y_{123} → Y_{123} × Y_{123}. With a similar procedure as the one used above to describe [N], we write

\begin{aligned}[N^{2}] = & b_{12789}T_{12789} + b_{16789}T_{16789} + b_{24789}T_{24789} \\ & + b_{34789}T_{34789} + b_{35789}T_{35789} + b_{56789}T_{56789} \end{aligned}

for some integers b’s. Imposing the commutativity of the above diagram, by means of the description of the pullbacks p_{1}^{∗}(1_{Y_{123}}) and p_{2}^{∗}(1_{Y_{123}}) as shown in the last table appearing in the proof of lemma 5.2, we finally get the condition

−b_{12789} + b_{16789} − b_{24789} + b_{34789} − b_{35789} − b_{56789} = 1.

It is straightforward to verify that both [N] and [N^{2}] make diagrams like (2.1) commute, for any choice of the indices ∗ and r.

Remark 5.4.

It is easy to verify that the description of [N] and [N^{2}] given in theorem 5.3 holds also for a normal–crossings degeneration (not semistable) like f(z_{1}, …, z_{n}) = z_{i}^{2}z_{j}, i, j ∈ [1, n], i ≠ j. This applies in particular to the case of normal–crossings degenerations of curves with triple points as described above. The desingularization process of the threefold X ×_{S} X is obtained via two blow-ups along z_{i} = 0 = w_{i} and z_{j} = 0 = w_{j} by analogy to what we have done in Remark 4.3. For the description of [N] we also refer to the same Remark.

6. An arithmetic interpretation of the monodromy operator in mixed characteristic

The calculations on the geometric description of [N^{i}] that we have done in the previous sections only involve the (local) geometry of the special fiber of a degeneration. Hence they equally hold in mixed characteristic also, i.e. for a degeneration f : X → Spec(Λ) = S, where Λ is a Henselian discrete valuation ring with η and v as its generic and closed points respectively. In analogy with the classical case, the model X is assumed to be proper and the map f is supposed to be flat, smooth over the generic point η and with a normal– crossings special fiber Y defined over the finite field k(v) of characteristic p > 0. Locally, for the étale topology X is S-isomorphic to S[x_{1}, …, x_{n}]/(x_{1}^{e_{1}} ⋯ x_{k}^{e_{k}} − π), where π is a uniformizing parameter in Λ and e_{i} ∈ Z, ∀i = 1, …, k. For simplicity, we also assume that Λ is a finite extension of Z_{ℓ} or Q_{ℓ}, where l ≠ p is a prime number. The complex of nearby cycles is then defined as RΨ(Λ) := ī^{−1}Rj̄_{∗}Λ. Here i : Y → X (resp. j : X_{η} → X) is the natural closed (resp. open) embedding that one “extends” to the algebraic closure k(v̄) of k(v) (resp. a separable closure k(η̄) of k(η)). Assume that the multiplicities e_{i} are prime to ℓ and g.c.d.(e_{i}, p) = 1. Then, the wild inertia acts trivially on RΨ(Λ) and the theory exposed in [16] shows that the nearby cycle complex has an abstract description in the derived category D^{+}(Y, Λ[Z_{ℓ}(1)]) of the abelian category of complexes of sheaves of Λ[Z_{ℓ}(1)]–modules on Y, by a complex A_{X ,Λ}^{•}, supported on Y. A_{X ,Λ}^{•} can be interpreted as the analogue of the Steenbrink resolution in the classical case. Therefore, the related study of it goes in parallel with the classical one in equal characteristic zero. We refer to op.cit. and [7] (e.g. Théorème 3.2) for further detail. The power maps (n ∈ [0, 2d], i ≥ 0, d = dim X_{η}) N^{i} : H^{n}(Xη̄, Λ) → H^{n}(Xη̄, Λ)(−i) define elements

N^{i} ∈ ⨁_{n≥0}[H^{2d−n}(Xη̄, Λ)(d) ⊗ H^{n}(Xη̄, Λ)(−i)]^{G} = [H^{2d}(Xη̄ × Xη̄, Λ)(d − i)]^{G}

invariant for the action of the Galois group G = Gal(η̄/η) on the cohomology of the product Xη̄ ×Xη̄. Assume that f : X → S has at worst triple points. Then, the singularities of both X ×_{S} X and Y × Y can be resolved locally around each singular point by a sequence of at most three blows-up, as we described in details in §§ 2,4,5. The resulting degeneration h : Z → S is normal–crossings with special fiber T = T_{1} ∪ … ∪ T_{N}. Let Xη̄ × Xη̄ = Zη̄ be its geometric generic fiber. Denote by Ñ = 1 ⊗ N + N ⊗ 1 the logarithm of the local monodromy on the product degeneration h. Then, the analogue of proposition 2.1 is the following

Proposition 6.1. Assume the monodromy-weight conjecture on H^{∗}(Zη̄, Λ) and the semisimplicity of the Frobenius on the inertia invariants. Then

N^{i} ∈ [Ker(Ñ) ∩ H^{2d}(Zη̄, Λ(d − i))]^{F =1}

≃ [Ker(Ñ) ∩ (gr_{2(d−i)}^{L}H^{2d}(Zη̄, Λ))(d − i)]^{F =1}
≃ ((gr_{2(d−i)}^{L}H^{2d}(T, Λ))(d − i))^{F =1}
≃
[\frac{Ker(ρ^{(2(i+1)} : H^{2(d−i)}(T̃^{(2i+1)}, Λ)(d−i) → H^{2(d−i)}(T̃^{(2(i+1))}, Λ)(d−i))}{Image ρ}]^{F =1}

where F is the geometric Frobenius.

The following result shows the relation of proposition 6.1 with the arithmetic of the degeneration h

Theorem 6.2. Assume the monodromy-weight conjecture on Zη̄ and the semisimplicity of the action of the frobenius F on H^{∗}(Zη̄, Λ)^{I}. Then, for i > 0 and d = dim Xη̄

ord_{s=d−i} det(Id − F N(v)^{−s}|H^{2d}(Zη̄, Λ)^{I}) =

rk[\frac{Ker(ρ^{2(i+1)}:H^{2(d−i)}(T̃^{(2i+1)}, Λ)(d−i) → H^{2(d−i)}(T̃^{(2(i+1))}, Λ)(d−i))}{Image ρ}]^{F =1}.

N(v) is the number of elements of the finite residue field k(v).

Proof. cf. [2], theorem 3.5.

This result explains geometrically the pole of the local factor at v of the Lfunction L(H^{2d}(Zη̄, Q_{ℓ}), s) at the points s = d − 1 and s = d − 2, with the presence of the “diagonal” cycles representing the monodromy powers on the strata of T as we previously described.

7. Appendix (by Spencer Bloch)

Our objective in this appendix is to define a multiplication between the total complex of E_{1}-terms of the Steenbrink spectral sequence and the graded complex

(7.1) H^{∗}(Y^{(•)}), ρ = restriction which is the E_{1} complex converging to the cohomology of the special fiber Y. We order the components Y = Y_{1} ∪ … ∪ Y_{N} and write a_{i_{0},...,i_{m}} ∈ H^{∗}(Y_{i_{0},...,i_{m}}, Q). The E_{1}-terms of the Steenbrink spectral sequence can be arrayed in a triangular diagram (compare [7], (2.3.8.1)) where each • denotes some H^{∗}(Y^{(m)}, Q(n)).

\begin{matrix}• \\ ↑ & ↖ \\ • & ← & • \\ ↑ & ↖ & ↑ & ↖ \\ • & ← & • & ← & • \\ ⋮ & ⧸^{wt0} & ⋮ & ⧸^{wt1} & ⋮ & ... \\ • & ← & • & ← & • & ← • \end{matrix} (7.2)

Here the horizontal arrows are Gysin maps and the vertical arrows are restriction maps. The diagonal arrows are (upto twist) the maps N which, on the level of E_{1} are either the identity or 0. The Steenbrink E_{1}-terms, i.e. the H^{∗}(Y, gr_{r}^{L}RΨ(Q)), are direct sums of terms on a NE-SW diagonal, with weight r meeting the ”x-axis” at x = r. The complex H^{∗}(Y^{(•)}) is embedded as the left hand column, and the resulting multiplication on it is the usual (associative) product

a_{i_{0},...,i_{m}} ⊗ b_{j_{0},...,j_{n}} ↦ \begin{cases}0 & i_{m} ≠ j_{0} \\ (a · b)_{i_{0},...,i_{m},j_{1},...,n_{n}} & i_{m} = j_{0} \end{cases} (7.3)

The bottom row is a quotient complex calculating the homology of the closed fiber H_{∗}(Y) (with appropriate twist). Our multiplication induces an action of the left hand column on the bottom row, which we will show induces the cap product ([14], p. 254)

H^{q}(Y) ⊗ H_{n}(Y) → H_{n−q}(Y). (7.4)

This module structure, unifying and extending the classical cocycle calculations for cup and cap product, is of independent interest. Quite possibly it can be extended to a product on the whole E_{1}-complex, but the daunting sign calculations involved have prevented us from working it out. We will apply this construction to calculate the product

[N^{i}] · : H^{∗}(T, gr_{r}^{L}A_{Z,Q}^{•}) → H^{∗+2d}(T, gr_{r−2i}^{L}A_{Z,Q}^{•}(d − i)) (7.5)

from (2.1). We return to the situation in section 2. In particular, Z → X ×_{S} X is a resolution, and T ⊂ Z is the special fiber, which we assume is a normal crossings divisor. We write E_{1}(Z) for the Steenbrink spectral sequence associated to the degeneration Z/S.

Lemma 7.1. There exists a class [N^{i}] in E_{1}(Z) satisfying 1. d_{1}[N^{i}] = 0, and the induced class in E_{2} is the i-th power of the mon-

odromy operator

N^{i} ∈ gr_{2(d−i)}^{L}H^{2d}(X̃^{∗} × X̃^{∗}, Q(d − i))

2. N([N^{i}]) = 0, i.e. in the diagram (7.2), [N^{i}] lies in the left hand vertical column.

Proof. We see from proposition (2.1) that the class of N^{i} is killed by N in E_{2}(Z). Let M denote the map on E_{1} which is inverse to N insofar as possible, i.e. M maps down and to the right in diagram (7.2). M is zero on the bottom line. Let x ∈ E_{1} represent N^{i} in E_{2}. Then Nx = d_{1}y. (Here d_{1} = d′ + d′′ is the total differential.) Since N commutes with d′ and d′′, and Nx has no term on the bottom row, it follows that [N^{i}] := x − d_{1}My is supported on the left hand column, i.e. killed by N.

Here is some notation. The special fiber will be Y = ⋃ Y_{i}, with 0 ≤ i ≤ N. Write H^{∗}(Y) for cohomology in some fixed constant ring like Z or C.

I = {i_{0}, …, i_{m}}; J = {j_{0}, …, j_{n}} (strictly ordered); Y_{I} = ⋂_{i_{k}∈I} Y_{i_{k}}

We will say the pair I, J is admissible if

∃p such that i_{m} = max(I) = j_{p} and {j_{0}, …, j_{p}} ⊂ I.

In this case, write j_{0} = i_{b_{0}}, …, j_{p−1} = i_{b_{p−1}}. Define

a(I, J) := b_{0} + … + b_{p−1} + mp.

With I, J admissible as above, write

J′ = {j_{0}, …, j_{p}}; J′′ = {j_{p}, …, j_{n}}; J = J′ ∪ J′′; J′ ∩ J′′ = {j_{p}} = {i_{m}}

Write

I′ = J′; I′′ = (I − J′) ∪ {i_{m}}; I = I′ ∪ I′′; {i_{m}} = I′ ∩ I′′

Let K = I′′ ∪ J′′, and define

θ(I, J) : H^{α}(Y_{I}) ⊗ H^{β}(Y_{J}) → H^{α+β+2p}(Y_{K}) (7.6)

θ(I, J)(x ⊗ y) := (−1)^{a(I,J)}g_{j_{0}} ∘ ⋯ g_{j_{p−1}}(x · y). (7.7)

Here x · y ∈ H^{α+β}(Y_{I∪J}), the g_{j} are Gysin maps, and

g_{j_{0}} ∘ ⋯ g_{j_{p−1}} : H^{∗}(Y_{I∪J}) → H^{∗+2p}(Y_{I′′∪J′′}).

If the pair I, J is not admissible, define θ(I, J) = 0. Define for I as above and 0 ≤ k ≤ N

σ(I, k) := #{i ∈ I | i < k}

For k ∉ I we have the restriction rest_{k} : H^{∗}(Y_{I}) → H^{∗}(Y_{I∪{k}}). Define

d′ := ∑_{k∉I}(−1)^{σ(I,k)}rest_{k} : H^{∗}(Y_{I}) → ⨁_{k∉I}H^{∗}(Y_{I∪{k}})

Similarly, for k ∈ I we have the Gysin g_{k} : H^{∗}(Y_{I}) → H^{∗+2}(Y_{I−{k}}). We define

d′′ = ∑_{k∈I}(−1)^{σ(I,k)}g_{k} : H^{∗}(Y_{I}) → ⨁_{k∈I}H^{∗+2}(Y_{I−{k}}).

Theorem 7.2. With notation as above (I, J not necessarily admissible) the following diagram is commutative:

H^{∗}(Y_{I}) ⊗ H^{∗}(Y_{J}) ⟶^{θ(I,J)}  H^{∗}(Y_{K})
↓^{d′⊗1+(−1)^{m}1⊗(d′+d′′)}  ↓^{d′+d′′}
⨁_{Ĩ,J̃}H^{∗}(YĨ) ⊗ H^{∗}(YJ̃) ⟶^{θ(Ĩ,J̃)} ⨁_{K̃=Ĩ′′∪J̃′′}H^{∗}(YK̃)

Remark 7.3.

A priori the theorem does not suffice to determine the desired mapping

H^{∗}(Y^{•}) ⊗ E_{1} → E_{1} a ⊗ b ↦ a ∗ b

because a given H^{∗}(Y_{K}) occurs many times in the diagram (2.1) (at every point along a NW pointing diagonal). However, if we add the condition that the weights (SW-NE diagonals in (7.2)) should be added, the mapping is defined. It has the property that

a ∗ Nb = N(a ∗ b)

In particular, there is an induced action on E_{1}/NE_{1} which we identify with the bottom row in (7.2). This simple complex calculates H_{∗}(Y), and the product coincides with the cap product. To see this, one notes that the product is correct for two elements in weight 0, and that if each H^{∗}(Y_{I}) is replaced by Z, the acyclic model theorem ([14], p. 165) can be applied.

proof of theorem. The proof consists of many separate cases. In each case we will check the sign carefully (this is the delicate part) and omit checking that the maps coincide set-theoretically (which is straightforward). case: i_{m} ∉ J. In this case, the pair I, J is not admissible, so θ(I, J) = 0. We must show

⊕_{Ĩ,J̃}θ(Ĩ, J̃) ∘ (d′ ⊗ 1 + (−1)^{m}1 ⊗ (d′ + d′′)) = 0. (7.8)

We may ignore non-admissible Ĩ, J̃. The only way admissible Ĩ, J̃ can occur in this situation is if for some p ≥ 0 we have j_{p−1} < i_{m} < j_{p} and {j_{0}, …, j_{p−1}} ⊂ I. (If a subscript for j doesn’t fall in {0, …, n}, ignore it, i.e. take j_{−1} = −∞, j_{n+1} = +∞.) Assume these conditions hold. Then the pair I ∪ {j_{p}}, J is admissible and occurs in the image of d′ ⊗ 1. Also the pair I, J ∪ {i_{m}} is admissible and occurs in the image of (−1)^{m}(1 ⊗ d′). We must show these two contributions cancel. Suppose j_{0} = i_{b_{0}}, …, j_{p−1} = i_{b_{p−1}}. Then the sign condition we need to verify is

σ(I, j_{p}) + b_{0} + ⋯ + b_{p−1} + p(m + 1) ≡

1 + m + σ(J, i_{m}) + b_{0} + ⋯ + b_{p−1} + pm mod (2)

This is correct because σ(I, j_{p}) = m + 1 and σ(J, i_{m}) = p. case: i_{m} = j_{p} ∈ J, {j_{0}, …, j_{p−1}} ⊄ J. This is the other case where I, J is not admissible, so θ(I, J) = 0. To get admissible Ĩ, J̃ we must have

∃k, 0 ≤ k ≤ p − 1 such that j_{k} ∉ I, {j_{0}, …, ĵ_{k}, …, j_{p−1}} ⊂ I.

Assume this. Then the pairs (I ∪{j_{k}}, J) and (I, J −{k_{k}}) are admissible. The first occurs in θ(I∪{j_{k}}, J)∘(d′⊗1) and the second in (−1)^{m}θ(I, J−{j_{k}})∘1⊗d′′. The necessary sign condition for cancellation is

σ(I, j_{k}) + a(I ∪ {j_{k}}, J) ≡^{?} m + 1 + k + a(I, J − {j_{k}}) mod (2).

To check this sign condition write j_{r} = i_{b_{r}} for 0 ≤ r ≤ p − 1, r ≠ k. Then

a(I, J − {j_{k}}) = b_{0} + ⋯ + b_{k−1} + b_{k+1} + ⋯ + b_{p−1} + (p − 1)m
a(I ∪ {j_{k}}, J) = b_{0} + ⋯ + b_{k−1} + σ(I, j_{k}) + (b_{k+1} + 1) +
+ ⋯ + (b_{p−1} + 1) + p(m + 1).

This yields the necessary congruence. For the rest of the proof we assume I, J is admissible. We examine the various terms in (7.8) and show they occur with the same signs in (d′ + d′′) ∘ θ(I, J). We first consider terms coming from d′ ⊗ 1, so the target is labelled by Ĩ = I ∪ {k}, J̃ = J. case: k < i_{m} = j_{p}. In this case, since j_{p} = min J′′ and k ∉ I ⊃ J′, we have k ∉ J. The pair Ĩ = I ∪ {k}, J is admissible with Ĩ′′ = I′′ ∪ {k} and the same decomposition J = J′∪J′′. Let K̃ = Ĩ′′∪J′′ = K∪{k}. Since k < j_{p} = min J′′, we have

σ(K, k) = σ(I′′, k) = σ(I, k) − σ(J′, k)

What we must show, therefore, is that

a(I, J) − a(Ĩ, J) ≡ σ(J′, k) mod (2)

Write

Ĩ = {ĩ_{0}, …, ĩ_{m+1}}; j_{0} = ĩb̃_{0}, …, j_{p−1} = ĩb̃_{p−1};
a(Ĩ, J) = b̃_{0} + ⋯ + b̃_{p−1} + (m + 1)p
I = {i_{0}, …, i_{m}}; j_{r} = i_{b_{r}}, 0 ≤ r ≤ p − 1
a(I, J) = b_{0} + ⋯ + b_{p−1} + mp

where

b̃_{ℓ} = \begin{cases}b_{ℓ} & i_{b_{ℓ}} < k \\ b_{ℓ} + 1 & i_{b_{ℓ}} > k \end{cases}

Thus

a(Ĩ, J) − a(I, J) = p − #{j ∈ J′ − {j_{p}} | j > k} =

#{j ∈ J′ | j < k} = σ(J′, k).

This is the desired congruence. We continue to consider the contribution of d′ ⊗ 1 with I, J admissible. case: k > i_{m}, k ≠ j_{p+1}. In this case I ∪ {k}, J is not admissible so θ(Ĩ, J) = 0. case: k = j_{p+1}. Here Ĩ:= I ∪ {k}, J̃:= J is admissible with

J̃′ = {j_{0}, …, j_{p+1}} = J′ ∪ {k} = J′ ∪ {j_{p+1}}
J̃′′ = {j_{p+1}, …, j_{n}} = J′′ − {j_{p}}; K̃ = Ĩ′′ ∪ J̃′′ = K − {j_{p}}

Note in this case k > i_{m} so σ(I, k) = m+1. The claim is here that the diagram

H^{∗}(Y_{I}) ⊗ H^{∗}(Y_{J}) ⟶^{θ(I,J)} H^{∗}(Y_{K})
↓^{(−1)^{m+1}rest.⊗1}  ↓^{(−1)^{σ(K,jp)}Gysin_{jp}}
H^{∗}(YĨ) ⊗ H^{∗}(Y_{J}) ⟶^{θ(Ĩ,J)} H^{∗}(YK̃)

commutes. Note that the right hand vertical arrow (with the sign) is part of 1 ⊗ d′′. To verify the signs we need

a(I, J) + σ(K, j_{p}) ≡ m + 1 + a(Ĩ, J).

Since K = I′′ ∪ J′′ and k = max(I′′) = min(J′′) it is clear that

σ(K, j_{p}) = #I′′ − 1 = m − p.

Also j_{p} = i_{m} so with the usual notation j_{r} = i_{b_{r}} we get

a(Ĩ, J) = b_{0} + ⋯ + b_{p−1} + m + (m + 1)(p + 1).

Now the desired congruence becomes

b_{0} + ⋯ + b_{p−1} + pm + m − p ≡ b_{0} + ⋯ + b_{p−1} + m + (m + 1)(p + 1) + m + 1

This is correct. We now consider terms occurring in (−1)^{m}(1 ⊗ d′) on the left of the diagram in the statement of the theorem. We assume given k ∉ J. case: k > j_{p}. Note in this case k ∉ I. Taking J̃ = J ∪{k}, K̃ = K ∪{k}, I claim the diagram below is commutative:

H^{∗}(Y_{I}) ⊗ H^{∗}(Y_{J}) ⟶^{θ(I,J)} H^{∗}(Y_{K})
↓_{(−1)^{m+σ(J,k)}1⊗rest}  ↓_{(−1)^{σ(K,k)}rest}
H^{∗}(Y_{I}) ⊗ H^{∗}(YJ̃) ⟶^{θ(I,J̃)} H^{∗}(KK̃)

(In other words, the contribution in this case is to d′ on the right.) Set

J̃ = J′ ∪ J̃′′; J̃′′ = J′′ ∪ {k}; K = I′′ ∪ J′′.

We have

a(I, J) = a(I, J̃)
σ(J, k) = σ(J′′, k) + p + 1
σ(K, k) = σ(J′′, k) + #I′′ = σ(J′′, k) + m + 1 − p

It follows that

m + σ(J, k) + a(I, J̃) ≡ σ(K, k) + a(I, J) mod (2)

which is the desired sign relation in this case. case: k < j_{p}, k ∉ I. In this case, the pair I, J ∪ {k} is not admissible, so the contribution is zero. case: k < j_{p}, k ∈ I.

In this case the pair I, J̃ is admissible with

J̃:= J ∪ {k} = J̃′ ∪ J′′; J̃′ = J′ ∪ {k}
I = Ĩ = J̃′ ∪ Ĩ′′; Ĩ′′ = I′′ − {k}; K̃ = K − {k} = Ĩ′′ ∪ J′′

The term in question contributes to d′′ on the right, and the diagram which commutes is:

H^{∗}(Y_{I}) ⊗ H^{∗}(Y_{J}) ⟶^{θ(I,J)} H^{∗}(Y_{K})
↓_{(−1)^{m+σ(J,k)}rest}  ↓_{(−1)^{σ(K,k)}Gysin_{k}}
H^{∗}(Y_{I}) ⊗ H^{∗}(YJ̃) ⟶^{θ(I,J̃)} H^{∗}(YK̃)

The signs will be correct if

a(I, J) + σ(K, k) ≡ m + σ(J, k) + θ(I, J̃) mod (2)

Write J̃ = {j̃_{0}, …, j̃_{m+1}} and j̃_{r} = ib̃_{r}, r ≤ p. The desired congruence reads

b_{0} + ⋯ + b_{p−1} + mp + σ(K, k) ≡^{?} m + σ(J, k) + b̃_{0} + ⋯ + b̃_{p} + (p + 1)m

We have

b̃_{ℓ} = \begin{cases}b_{ℓ} & ℓ < σ(J′, k) \\ σ(I, k) & ℓ = σ(J′, k) \\ b_{ℓ−1} & ℓ > σ(J′, k) \end{cases}

The condition becomes

σ(K, k) ≡^{?} σ(J, k) + σ(I, k) = σ(J′, k) + σ(J′, k) + σ(I′′, k),

which is true. Finally we consider terms coming from (−1)^{m}(1 ⊗ d′′) in the lefthand vertical arrow in the diagram of the theorem. In what follows j ∈ J. case: j ∈ J′′, j ≠ j_{p}. Define

J̃ = J − {j}; K = I′′ ∪ J′′; K̃ = K − {j} = I′′ ∪ J̃′′.

The diagram which commutes is:

H^{∗}(Y_{I}) ⊗ H^{∗}(Y_{J}) ⟶^{θ(I,J)} H^{∗}(Y_{K})
↓^{1⊗(−1)^{m+σ(J,j)}Gysin_{j}} ↓^{(−1)^{σ(K,j)}Gysin_{j}}
H^{∗}(Y_{I}) ⊗ H^{∗}(YJ̃) ⟶^{θ(I,J̃)} H^{∗}(YK̃)

The sign condition to be checked is

m + σ(J, j) + a(I, J̃) ≡^{?} a(I, J) + σ(K, j) mod (2).

Our conditions imply j > j_{p} so a(I, J) = a(I, J̃). Also,

#I′′ + #J′ = m + 2 ≡ m mod (2),

so

σ(K, j) = #I′′ + σ(J′′, j) − 1 ≡ m + #J′ + σ(J′′, j) − 1
σ(J, j) = σ(J′, j) + σ(J′′, j) − 1 = #J′ − 1 + σ(J′′, j).

This is the desired condition. case: j = j_{p}. In this case, I, J − {j} is not admissible, so we get no contribution. case: j ∈ J, j < j_{p}. In this case, j ∈ J′, j ≠ j_{p}. Set

J̃ = J − {j}; J̃′ = J′ − {j}; J̃′′ = J′′
Ĩ = I; Ĩ′′ = I′′ ∪ {j}; I = Ĩ = J̃′ ∪ Ĩ′′
K = I′′ ∪ J′′; K̃ = Ĩ′′ ∪ J̃′′ = K − {j}.

The sign condition to show we gat a contribution to d′′ on the right is

a(I, J) + σ(K, j) ≡^{?} m + σ(J, j) + a(Ĩ, J̃) mod (2).

Writing j = j_{ℓ} = i_{b_{ℓ}} the condition becomes

b_{0} + ⋯ + b_{p−1} + mp + σ(K, j) ≡^{?}

b_{0} + ⋯ + b̂_{ℓ} + b_{ℓ+1} + ⋯ + b_{p−1} + m(p − 1) + m + σ(J, j)

This is true because

b_{ℓ} = σ(I, j) = σ(I′′, j) + σ(J′, j)
σ(K, j) = σ(I′′, j); σ(J, j) = σ(J′, j).

The proof is completed by checking that all the terms on the right in the theorem (i.e. in d′ + d′′) are accounted for precisely once in the above enumeration of cases.

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C. Consani

Department of Mathematics Massachusetts Institute of Technology Cambridge, Ma. 02139 USA katia@math.mit.edu S. Bloch Department of Mathematics The University of Chicago Chicago, Il. 606139 USA bloch@math.uchicago.edu