The Rank-One Limit of the Fourier-Mukai Transform

Gerard van der Geer and Alexis Kouvidakis

Received: October 15. 2009

Communicated by Gavril Farkas

Abstract. We give a formula for the specialization of the Fourier-Mukai transform on a semi-abelian variety of torus rank 1.

2010 Mathematics Subject Classification: 14C25, 14H40

1. Introduction

Let π : X^{⋆} → S be a semi-abelian variety of relative dimension g over the spectrum S of a discrete valuation ring R with algebraically closed residue field k such that the generic fibre X_{η} is a principally polarized abelian variety. We assume that X^{⋆} is contained in a complete rank-one degeneration X. In particular, the special fibre X_{0} of X is a complete variety over k containing as an open part the total space of the 𝔾_{m}-bundle associated to a line bundle J → B over a (g − 1)-dimensional abelian variety B. The normalization ν : ℙ → X_{0} of X_{0} can be identified with the ℙ^{1}-bundle over B associated to J and X_{0} is obtained by identifying the zero-section of ℙ with the infinity-section of ℙ, both isomorphic to B, by a translation. Moreover, X_{0} is provided with a theta divisor that is the specialization of the polarization divisor on the generic fibre. If c_{η} is an algebraic cycle on X_{η} we can take the Fourier-Mukai transform φ_{η} := F (c_{η}) and consider the limit cycle (specialization) φ_{0} of φ_{η}. A natural question is: What is the limit φ_{0} of φ_{η}? If q : ℙ → B denotes the natural projection of the ℙ^{1}-bundle, the Chow ring A^{∗}(ℙ) of ℙ is the extension A^{∗}(B)[η]/(η^{2} − η · q^{∗}c_{1}(J)) of the Chow ring A^{∗}(B) of B with η = c_{1}(O_{ℙ}(1)). We consider now cycles with rational coefficients. We denote by c_{0} the specialization of the cycle c_{η} on X_{0}. We can write c_{0} as ν_{∗}(γ) with γ = q^{∗}z + q^{∗}w · η.

Theorem 1.1. Let c_{η} be a cycle on X_{η} with c_{0} = ν_{∗}(q^{∗}z + q^{∗}w · η), for z, w ∈ A^{∗}(B). The limit φ_{0} of the Fourier-Mukai transform φ_{η} = F (c_{η}) is given by φ_{0} = ν_{∗}(q^{∗}a + q^{∗}b · η) with

a = F_{B}(w) + ∑_{n=0}^{2g−2} ∑_{m=0}^{n} \frac{(−1)^{m}}{(n + 2)!} F_{B}[(z + w · c_{1}(J)) · c_{1}^{m}(J)] · c_{1}^{n−m+1}(J)

and

b = ∑_{n=0}^{2g−2} ∑_{m=0}^{n} \frac{(−1)^{m}}{(n + 2)!} F_{B}[(((−1)^{n+1} − 1)z − w · c_{1}(J)) · c_{1}^{m}(J)] · c_{1}^{n−m}(J) ,

where F_{B} is the Fourier-Mukai transform on the abelian variety B.

We denote algebraic equivalence by =^{a}. The relation c_{1}(J)=^{a}0 implies the following result.

Theorem 1.2. With the above notation the limit φ_{0} satisfies

φ_{0} =^{a} ν_{∗}(q^{∗}F_{B}(w) − q^{∗}F_{B}(z) · η) .

Note that this is compatible with the fact that for a principally polarized abelian variety A of dimension g the Fourier-Mukai transform satisfies F_{A} ∘ F_{A} = (−1)^{g}(−1_{A})^{∗}. Beauville introduced in [2] a decomposition on the Chow ring with rational coefficients of an abelian variety using the Fourier-Mukai transform. Theorem 1.2 can be used to deduce non-vanishing results for Beauville components of cycles on the generic fibre of a semi-abelian variety of rank 1; we refer to § 7 for examples. We prove the theorem by constructing a smooth model Y of X ×_{S} X to which the addition map X^{⋆} ×_{S} X^{⋆} → X^{⋆} extends and by choosing an appropriate extension of the Poincaré bundle to Y. The proof is then reduced to a calculation in the special fibre. We refer to Fulton’s book [8] for the intersection theory we use. The theory in that book is built for algebraic schemes over a field. In our case we work over the spectrum of a discrete valuation ring. But as is stated in § 20.1 and 20.2 there, most of the theory in Fulton’s book, including in particular the statements we use in this paper, is valid for schemes of finite type and separated over S. However, for us projective space denotes the space of hyperplanes and not lines, which conflicts with Fulton’s book, but is in accordance with [10].

2. Families of abelian varieties with a rank one degeneration

We now assume that R is a complete discrete valuation ring with local parameter t, field of quotients K and algebraically closed residue field k. Suppose that (X^{⋆}, L) is a semi-abelian variety over S = Spec(R) such that the generic fibre X_{η} is abelian and the special fibre X_{0}^{∗} has torus rank 1; moreover, we assume that L is a cubical invertible sheaf (meaning that L satisfies the theorem of the cube, see [7], p. 2, 8) and L_{η} is ample. In particular, the special fibre of X^{⋆} fits in an exact sequence

1 → T_{0} → X_{0}^{∗} → B → 0,

The Rank-One Limit of the Fourier-Mukai Transform 749

where B is an abelian variety over k and T_{0} the multiplicative group 𝔾_{m} over k. The torus T_{0} lifts uniquely to a torus T_{i} of rank 1 over S_{i} = Spec(R/(t^{i+1}) in X_{i}^{∗} = X^{⋆} ×_{S} S_{i}. The quotient X_{i}^{∗}/T_{i} is an abelian variety B_{i} over S_{i}. The system {B_{i}}_{i=1}^{∞} defines a formal abelian variety which is algebraizable, resulting in an abelian scheme B, so that we have an exact sequence of group schemes over S

1 → T → G →^{π} B → 0,

cf. [F-C, p. 34]. We assume now that we are given a line bundle M on B defining a principal polarization λ : B → B^{t} and consider L = π^{∗}(M). This defines a cubical line bundle on G. The extension G is given by a homomorphism c of the character group Z ≅ ℤ of T to B^{t}. The semi-abelian group scheme dual to X^{⋆} defines a similar extension

1 → T^{t} → G^{t} → B^{t} → 0

and the polarization provides an isomorphism ϕ of the character group Z of T with the character group Z^{t} of T^{t}. Now the degenerating abelian variety (i.e. semi-abelian variety) X^{⋆} over S gives rise to the set of degeneration data (cf. [7], p 51, Thm 6.2, or [1], Def. 2.3):

(i) an abelian variety B over S and a rank 1 extension G. This amounts to a S-valued point b of B = B^{t}. (ii) a K-valued point of G lying over b. (iii) a cubical ample sheaf L on G inducing the polarization on B and an

action of Z = Z^{t} on L_{η}. A section s ∈ Γ(G, L) possesses the analogue of a classical Fourier expansion as explained in [7], p. 37. So s can be written uniquely as s = ∑_{χ∈Z} σ_{χ}(s), where σ_{χ} : Γ(G, L) → Γ(B, M_{χ}) is a R-linear homomorphism and M_{χ} is the twist of M by χ: in fact π_{∗}(O_{G}) = ⊕_{χ}O_{χ} with O_{χ} the subsheaf consisting of χ-eigenfunctions. (We refer to [7], p. 43; note also the sign conventions there in the last lines.) We have now by the action

T_{c^{t}(y)}^{∗}M ≅ M_{ϕ(y)} ≅ M ⊗ O_{ϕ(y)}, y ∈ Z^{t}.

This satisfies σ_{χ+1}(s) = ψ(1)τ(χ)T_{b}^{∗}(σ_{χ}(s)), where τ is given by a point of G(K) lying over b and ψ is a cubical trivialization of T_{c^{t}(y)}^{∗}M_{η}^{−1} as in [7], p. 44, Thm. 5.1. We refer to Faltings-Chai’s theorem (6.2) of [7], p. 51 for the degeneration data. The compactification X of X^{⋆} is now constructed as a quotient of the action of Z^{t} on a so-called relatively complete model. Such a relatively complete model P̃ for G can be constructed here in an essentially unique way. If B is trivial (i.e. dim(B) = 0) and if the torus is T = Spec(R[z, z^{−1}]) it is given as the toroidal variety obtained by gluing the affine pieces

U_{n} = Spec(R[x_{n}, y_{n}]),  with  x_{n}y_{n} = t

where G ⊂ P̃ is given by x_{n} = z/t^{n}, y_{n} = t^{n+1}/z, (cf. [13], also in [7], p. 306). By glueing we obtain an infinite chain P̃_{0} of ℙ^{1}’s in the special fibre. We can ‘divide’ by the action of Z^{t}; this is easy in the analytic case, more involved in the algebraic case, but amounts to the same, cf. [13], also [7], p. 55-56. In the special fibre we find a rational curve with one ordinary double point. If instead we divide by the action of nZ^{t} for n > 1 we find a cycle consisting of n copies of ℙ^{1}.

In case the abelian part B is not trivial we take as a relatively complete model the contracted (or smashed) product P̃ ×^{T} G with P̃ the relatively complete model for the case that B is trivial. Call the resulting space P̃. Then P̃ corresponds by Mumford’s [loc. cit., p 29] to a polyhedral decomposition of Z^{t} ⊗ ℝ = ℝ with Z^{t} the cocharacter group of T. Then we essentially quotient by the action of Z^{t} or nZ^{t} as before and obtain a proper X → S. We describe the central fibre X_{0} of X. Let b be the k-valued point of B ≅ B^{t} that determines the above 𝔾_{m}-extension. If M denotes a line bundle defining the principal polarization of B we let M_{b} be the translation of M by b and we set J = M ⊗M_{b}^{−1} and define the projective bundle ℙ = ℙ(J ⊕O_{B}) with projection q : ℙ → B. The bundle ℙ has two natural sections (with images) ℙ_{1} and ℙ_{2} corresponding to the projections J ⊕ O_{B} → J and J ⊕ O_{B} → O_{B}. We have O(ℙ_{1}) ≅ O(ℙ_{2}) ⊗ q^{∗}J and O(1) ≅ O(ℙ_{1}) with O(1) the natural line bundle on ℙ. We denote by \overline{ℙ} the non-normal variety obtained by gluing the sections ℙ_{1} and ℙ_{2} under a translation by the point b. The singular locus of \overline{ℙ} has support isomorphic to B. The line bundle L̃ = O(ℙ_{1}) ⊗ q^{∗}M_{b} ≅ O(ℙ_{2}) ⊗ q^{∗}M descends to a line bundle \overline{L} on \overline{ℙ} with a unique ample divisor D, see [14]. The central fibre X_{0} of the family π : X → S is then equal to ℙ̄. The cubical invertible sheaf L on X^{⋆} extends (uniquely) to X and its restriction to the central fibre ℙ̄ is the line bundle \overline{L}, see [15].

3. Extension of the addition map

The addition map μ : X^{⋆} ×_{S} X^{⋆} → X^{⋆} of the semi-abelian scheme X^{⋆} does not extend to a morphism X ×_{S} X → X, but it does so after a small blow-up of X ×_{S} X as we shall see. The degeneration data of X^{⋆} defines (product) degeneration data for X^{⋆}×_{S} X^{⋆}. Indeed, we can take the fibre product of the relatively complete model P̃′ = P̃ ×_{S} P̃ and this corresponds (e.g. via [13], Corollary (6.6)) to the standard polyhedral decomposition of ℝ^{2} = (Z^{t} ⊗ ℝ)^{2} by the lines x = m and y = n for m, n ∈ ℤ. The special fibre of the model P̃′ is an infinite union of ℙ^{1} × ℙ^{1}- bundles over B × B glued along the fibres over 0 and ∞. The compactified model of X ×_{S} X is obtained by taking the ‘quotient’ of P̃′ under the action of Z^{t} × Z^{t}. This is not regular; for example the criterion of Mumford ([13], p. 29, point (D)]) is not satisfied. We can remedy this by subdividing. For example, by taking the decomposition of ℝ^{2} given by the lines x = m, y = n and x + y = l for m, n, l ∈ ℤ. The special fibre of this model is an infinite union of copies of ℙ^{1} × ℙ^{1}-bundles over B × B blown up in the two anti-diagonal sections (0, ∞) = ℙ_{1} × ℙ_{2} and (∞, 0) = ℙ_{2} × ℙ_{1}. This is regular. Both polyhedral decompositions are invariant under the action of translations (x, y) ↦ (x + a, y + b) for fixed a, b ∈ ℤ. This means that we can form the ‘quotient’ by Z^{t} × Z^{t} ≅ ℤ^{2} (or a subgroup nZ^{t} × nZ^{t}) and obtain a completed semi-abelian variety Y of relative dimension 2g over S. We denote by ϵ : Y → Y′ = X ×_{S} X the natural map. We shall write V for Y_{0} and σ : Ṽ → V for its normalization. Then Ṽ is an irreducible component of the special fibre of P̃′. We denote by τ : Ṽ → ℙ × ℙ the blow up map and by E_{12} and E_{21} the exceptional divisors over the blowing up loci ℙ_{1} × ℙ_{2} and ℙ_{2} × ℙ_{1}, respectively. Now consider the addition map μ : X^{⋆} ×_{S} X^{⋆} → X^{⋆} with X^{⋆} as in § 2. This morphism induces (and is induced by) a map μ̃ : G ×_{S} G → G. However, this map does not extend to a morphism of the relatively complete model P̃′ since the corresponding (covariant) map (Z^{t} ⊗ ℝ)^{2} → (Z^{t} ⊗ ℝ) does not map cells to cells. After subdividing (by adding the lines x+y = l with l ∈ ℤ) this property is satisfied (cf. [11], Thm. 7, p. 25). This means that the map μ extends to μ̃ : P̃′ → P̃ for the polyhedral decomposition given by this subdivision. It is compatible with the action of ℤ and ℤ × ℤ and hence descends to a morphism μ̄ : Y → X. We summarize:

Proposition 3.1. The addition map of group schemes μ : X^{⋆} ×_{S} X^{⋆} → X^{⋆} extends to a morphism μ̄ : Y → X.

We now describe an explicit local construction of the model Y by blowing up the model X ×_{S} X. Let A_{S}^{g+1} = Spec(R[x_{1}, …, x_{g+1}]) denote affine Sspace. In local coordinates, inside A_{S}^{g+1}, we may assume that the g-dimensional fibration π : X^{⋆} → S is given by the equation x_{1}x_{2} = t, where the coordinates x_{3}, …, x_{g+1} are not involved, see [14] p. 361-362. We may assume that the zero section of the family is defined by x_{i} = 1 for i = 1, …, g + 1. We form the fibre product π : Y′ = X ×_{S} X. We denote by Λ the support of the singular locus of X_{0}. The (2g + 1)-dimensional variety Y′ is singular in the special fibre along Σ = Λ ×_{k} Λ ≅ B ×_{k} B of dimension 2g − 2. The generic fibre Y′_{η} is the product X_{η} ×_{K} X_{η} of the abelian variety X_{η}, while the zero fibre Y′_{0} is singular. The local equations of Y′ in a neighborhood of the singular locus of the family are given in our local coordinates by the system x_{1}x_{2} = t, x′_{1}x′_{2} = t. The singular locus Σ of Y′ is given by the equations x_{1} = x_{2} = x′_{1} = x′_{2} = t = 0. The above blow up ϵ : Y → Y′ is a small blow up and can be described directly as follows: we blow up Y′ along its subvariety Π defined by x_{1} = x′_{2} = 0 (a 2-plane contained in the central fibre of Y′). The proper transform Y of Y′ is smooth. In local coordinates, the blow-up is given by the graph Γ_{ϕ} ⊆ Y′ × ℙ^{1} of the rational map ϕ : Y′ ⟶ ℙ^{1} given by ϕ(x_{1}, …, x′_{g+1}, t) = (x_{1} : x′_{2}). The equations of the graph Γ_{ϕ} ⊆ Y′ × ℙ^{1} ⊆ A_{S}^{2(g+1)} ×_{S} ℙ_{S}^{1} are given by the system

x_{1}x_{2} = t, ux′_{2} − vx_{1} = 0, ux_{2} − vx′_{1} = 0 ,

where u, v are homogeneous coordinates on ℙ^{1}. For later calculations we write down the morphism μ̄ explicitly on the special fibre. We start with g = 1; then B is trivial and we may restrict the map to an irreducible component of the special fibre of the relatively complete model P̃ ×_{S} P̃ and get the map m : ℙ^{1} × ℙ^{1} → ℙ^{1} given by ((a : b), (a′ : b′)) ↦ (aa′ : bb′). This is not defined in the points (0, ∞) and (∞, 0). After blowing up these points (which is the blow up Y → Y′) the rational map becomes a regular map m̃ : Ṽ → ℙ^{1}. It is defined by the two sections prop(p_{1}^{∗}{0}) + prop(p_{2}^{∗}{0}) and prop(p_{1}^{∗}{∞})+prop(p_{2}^{∗}{∞}) of the linear system |τ^{∗}(F_{1} +F_{2})−E_{12} −E_{21}| with F_{1} and F_{2} the horizontal and vertical fibre (with prop( ) meaning the proper transform). The map m̃ descends to a map m̄ : V → ℙ̄ which is the restriction of the morphism μ̄ : Y → X to the central fibre. For the case g > 1, note that we have the addition map μ_{X^{⋆}}. Its restriction to the special fibre extends to a map of the relatively complete model and then restricts to a morphism m̃ : Ṽ → ℙ that lifts the addition map μ_{B} of B. That means that it comes from a surjective bundle map (cf. [10], Ch. II, Prop. 7.12)

δ : m_{1}^{∗}(J ⊕ O_{B}) ≅ (p_{1}^{∗}q^{∗}J ⊗ p_{2}^{∗}q^{∗}J) ⊕ OṼ → N

with m_{1} := μ_{B} ∘ (q × q) ∘ τ : Ṽ → B and N = τ^{∗}(p_{1}^{∗}O(ℙ_{1}) ⊗ p_{2}^{∗}O(ℙ_{1})) ⊗ O(−E_{12} −E_{21})) with p_{i} : ℙ×ℙ → ℙ the ith projection. Then m_{1}^{∗}(J ⊕O_{B})^{∨} ⊗N is isomorphic to the direct sum of

τ^{∗}p_{1}^{∗}O(ℙ_{i}) ⊗ τ^{∗}p_{2}^{∗}O(ℙ_{i}) ⊗ O(−E_{12} − E_{21}) (i = 1, 2).

The map δ is then given by the two sections prop(p_{1}^{∗}ℙ_{i}) + prop(p_{2}^{∗}ℙ_{i}) of τ^{∗}p_{1}^{∗}O(ℙ_{i}) ⊗ τ^{∗}p_{2}^{∗}O(ℙ_{i}) ⊗ O(−E_{12} − E_{21}) for i = 1, 2. The map m̃ descends to a map m̄ : V → ℙ̄ which is the restriction of the morphism μ̄ : Y → X to the central fibre.

4. Extension of the Poincaré bundle

We denote by j_{0} : X_{0} ↪ X and i_{0} : Y_{0} ↪ Y the inclusions of the special fibre. Recall that we write V for Y_{0} and Ṽ for its normalization. We denote by P_{η} the Poincaré bundle on Y′_{η} and by P_{B} the Poincaré bundle on B.

Theorem 4.1. The Poincaré bundle P_{η} has an extension P such that the pull back of P_{0} := i_{0}^{∗}P to Ṽ satisfies σ^{∗}P_{0} ≅ τ^{∗}(q × q)^{∗}P_{B} ⊗ O(−E_{12} − E_{21}).

Proof. We have the following commutative diagram of maps

V ^{m̄}  \overline{ℙ} Ṽ_{σ}_{τ} _{m̃} ℙ_{ν} ℙ_{q}  ℙ × ℙ_{q×q}_{p_{i}}  ℙ_{q} B  B × B_{q_{i}}  _{μ_{B}} B

Let L be the theta line bundle on the family X introduced in § 2. We define the extension of P_{η} by

P := μ̄^{∗}L ⊗ ρ_{1}^{∗}L^{−1} ⊗ ρ_{2}^{∗}L^{−1},

where we denote by ρ_{1}, ρ_{2} : Y → X the compositions of the natural projections ρ′_{i} : Y′ → X with the blowing up map ϵ : Y → Y′ of § 3. We then have σ^{∗}P_{0} = σ^{∗}(m̄^{∗}j_{0}^{∗}L)⊗σ^{∗}i_{0}^{∗}ρ_{1}^{∗}L^{−1}⊗σ^{∗}i_{0}^{∗}ρ_{2}^{∗}L^{−1}. Now m̄^{∗}j_{0}^{∗}L = m̄^{∗}L̄ and by using the description of L̄ in § 2 we have σ^{∗}(m̄^{∗}j_{0}^{∗}L) = m̃^{∗}ν^{∗}L̄ = m̃^{∗}(O(ℙ_{1}) ⊗ q^{∗}M_{b}). In view of O(ℙ_{1}) = O(1) we have by the discussion at the end of § 3 that

m̃^{∗}O(ℙ_{1}) = τ^{∗}p_{1}^{∗}O(ℙ_{1}) ⊗ τ^{∗}p_{2}^{∗}O(ℙ_{1}) ⊗ O(−E_{12} − E_{21})

and m̃^{∗}q^{∗}M_{b} = τ^{∗}(q × q)^{∗}μ_{B}^{∗}M_{b}. On the other hand we have

σ^{∗}(i_{0}^{∗}ρ_{i}^{∗}L) = τ^{∗}p_{i}^{∗}ν^{∗}L̄ = τ^{∗}p_{i}^{∗}O(ℙ_{1}) ⊗ τ^{∗}(q × q)^{∗}q_{i}^{∗}M_{b}

and putting this together we get the result. □

5. The basic construction

The fibration π : Y → S is a flat map since Y is irreducible and S is smooth 1dimensional, see [10], Ch. III, Proposition 9.7. The maps ρ_{i} = Y → X, i = 1, 2, defined in the proof of Theorem 4.1, are flat maps too since they are maps of smooth irreducible varieties with fibres of constant dimension g, see e.g. [12], Corollary of Thm. 23.1. We denote by Y_{0} (resp. Y_{η}) the special fibre (resp. the generic fibre) and by i_{0} : Y_{0} → Y (resp. i_{η} : Y_{η} → Y) the corresponding embedding. According to [8], Example 10.1.2, i_{0} is a regular embedding. Similarly, j_{0} : X_{0} → X is a regular embedding. We consider the diagram

Y_{0}  ^{i_{0}}_{π_{0}}  Y_{π} Spec(k) _{s}  S

Let i_{0}^{∗} : A_{k}(Y) → A_{k−1}(Y_{0}) be the Gysin map (see [8], Example 5.2.1). Since Y_{0} is an effective Cartier divisor in Y the Gysin map i_{0}^{∗} coincides with the Gysin map for divisors (see [8], Example 5.2.1 (a) and § 2.6). We now consider specialization of cycles, see [8], § 20.3. Note that according to [8], Remark 6.2.1., in our case we have s^{!}a = i_{0}^{∗}a, a ∈ A_{∗}(Y). If Z is a flat scheme over the spectrum of a discrete valuation ring S the specialization homomorphism σ_{Z} : A_{k}(Z_{η}) → A_{k}(Z_{0}) is defined as follows, see [8], pg. 399: If β_{η} is a cycle on Z_{η} we denote by β an extension of β_{η} in Z (e.g. the Zariski closure of β_{η} in Z) and then σ_{Z}(β_{η}) = i_{0}^{∗}(β), where i_{0} : Z_{0} → Z is the natural embedding. Let c_{η} be a cycle on X_{η} and let φ_{η} = F (c_{η}) be the Fourier-Mukai transform. It is defined by F (c_{η}) = ρ_{2∗}(e^{c_{1}(P_{η})} ·ρ_{1}^{∗}c_{η}) ∈ A_{∗}(X_{η}). Let σ_{X} : A_{k}(X_{η}) → A_{k}(X_{0}) be the specialization map. We have to determine σ_{X}(F (c_{η})). If β_{η} is a cycle on Y_{η}, we have ρ_{2∗}σ_{Y} (β_{η}) = σ_{X}ρ_{2∗}(β_{η}) by applying [8] Proposition 20.3 (a) to the proper map ρ_{2} : Y → X. By choosing β_{η} = e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η} we have

(1)  σ_{X}(F (c_{η})) = ρ_{2∗}σ_{Y} (e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η}) .

Therefore, in order to compute σ_{X}(F(c_{η})) we have to identify σ_{Y} (e^{c_{1}(P_{η})}·ρ_{1}^{∗}c_{η}). We take the extension e^{c_{1}(P)} of e^{c_{1}(P_{η})} and the extension of ρ_{1}^{∗}c_{η} given by ρ_{1}^{∗}c, where c is the Zariski closure of c_{η} in X. Since i_{η} : Y_{η} → Y is an open embedding and hence a flat map of dimension 0, we have i_{η}^{∗}(e^{c_{1}(P)} · ρ_{1}^{∗}c) = e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η}, see [8], Proposition 2.3 (d). In other words, the cycle e^{c_{1}(P)} · ρ_{1}^{∗}c extends the cycle e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η} and hence σ_{Y} (e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η}) = i_{0}^{∗}(e^{c_{1}(P)} · ρ_{1}^{∗}c). Now, for any k-cycle a on Y we have the identity

i_{0}^{∗}(c_{1}(P) · a) = c_{1}(P_{0}) · i_{0}^{∗}(a)

in A_{k−2}(Y_{0}), where P_{0} = i_{0}^{∗}P is the pull back of the line bundle and i_{0}^{∗}a the Gysin pull back to the divisor Y_{0}. This follows from applying the formula in [8], Proposition 2.6 (e) to i_{0} : Y_{0} → Y, with D = Y_{0}, X = Y and L = P the Poincaré bundle. Hence

(2)  σ_{Y} (e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η}) = e^{c_{1}(P_{0})} · i_{0}^{∗}(ρ_{1}^{∗}c) .

By the Moving Lemma (see [8], § 11.4), we may choose the cycle c on the regular X such that it intersects the singular locus Λ of the central fibre properly. Since Λ ⊆ X_{0} the cycle c_{0} = j_{0}^{∗}(c) meets Λ properly by the following dimension argument. We have dim(c ∩ Λ) = dim(c_{0} ∩ Λ), hence

dim(c_{0} ∩ Λ) = dim(c) + dim(Λ) − dim(X)

= (dim(c) − 1) + dim(Λ) − (dim(X) − 1)

= dim(c_{0}) + dim(Λ) − dim(X_{0}).

Since Λ is of codimension 1 in X_{0} = ℙ̄, saying that c_{0} meets Λ properly, is equivalent to saying that no component of c_{0} is contained in Λ.

Lemma 5.1. There exists a cycle γ on ℙ with c_{0} = ν_{∗}γ that meets the sections ℙ_{i} for i = 1, 2 properly.

Proof. If Λ is the singular locus of ℙ̄ and A = ℙ_{1} ∪ ℙ_{2} its preimage in ℙ, then ℙ̄\Λ ≅ ℙ\A. We may assume that the cycle c_{0} is irreducible and we consider the support of c_{0} ∩ (ℙ̄\Λ) as a subset W of ℙ\A. Its Zariski closure γ = W̄ is an irreducible cycle on ℙ. Then ν_{∗}γ is an irreducible cycle on ℙ̄ since the map ν is a projective map. Also, ν_{∗}γ ∩ (ℙ̄\Λ) = c_{0} ∩ (ℙ̄\Λ), hence ν_{∗}γ is the Zariski closure of c_{0} ∩ (ℙ̄\Λ) and so, by the irreducibility, we have ν_{∗}γ = c_{0}. □

Lemma 5.2. If c_{0} = ν_{∗}γ, then we have i_{0}^{∗}ρ_{1}^{∗}c = σ_{∗}(τ^{∗}(p_{1}^{∗}γ)).

Proof. We denote the restriction of ρ_{i} to the special fibre again by ρ_{i}. Then we have i_{0}^{∗}ρ_{1}^{∗}c = ρ_{1}^{∗}c_{0} since ρ_{1} is a flat map and i_{0}, j_{0} are regular embeddings (see [8], Theorem 6.2 (b) and Remark 6.2.1). We will use the following commutative diagram

Ṽ_{τ}  ^{σ} V  _{ϵ}_{ρ_{i}} ℙ × ℙ_{p_{i}}  \overline{ℙ} × \overline{ℙ}_{ρ′_{i}} ℙ _{ν}  \overline{ℙ}

We may assume that c_{0} and γ are irreducible k-cycles. We claim that ρ_{1}^{∗}c_{0} is irreducible. Indeed, the map ρ_{1} is a flat map of relative dimension g. The cycle ρ_{1}^{∗}c_{0} is then a cycle of pure dimension k + g and contains the proper transform of (ρ′_{1})^{∗}c_{0} and that is an irreducible cycle. Any other irreducible component of ρ_{1}^{∗}c_{0} must have support on the preimage of Λ. But since the cycle c_{0} intersects Λ along a k − 1-cycle, there is no irreducible component of ρ_{1}^{∗}c_{0} on the preimage of Λ. On the other hand, since γ meets the sections ℙ_{i} properly, the cycle τ^{∗}p_{1}^{∗}γ is an irreducible cycle, and hence so is σ_{∗}(τ^{∗}p_{1}^{∗}γ). But as ρ_{1}^{∗}c_{0} and σ_{∗}(τ^{∗}p_{1}^{∗}γ) coincide outside the exceptional divisor of V, they have to coincide everywhere. □

Proposition 5.3. We have σ_{X}(F(c_{η})) = ρ_{2∗}(e^{c_{1}(P_{0})} · σ_{∗}(τ^{∗}p_{1}^{∗}γ)).

Proof. By equation (2) and Lemma 5.2 we have

(3)  σ_{Y} (e^{c_{1}(P_{η})} · ρ_{1}^{∗}c_{η}) = e^{c_{1}(P_{0})} · σ_{∗}τ^{∗}(p_{1}^{∗}γ) .

The result follows from equation (1). □

In order to calculate the limit of the Fourier-Mukai transform we are thus reduced to a calculation in the special fibre.

6. A calculation in the special fibre - Proof of the main theorem

Recall the normalization map σ : Ṽ → V. Suppose we have a cycle ρ on Ṽ with σ_{∗}ρ = c_{0}. We can consider the intersection c_{1}(P_{0})^{k} · c_{0}, that is a successive intersection of a cycle with a Cartier divisor on the singular variety V. On the other hand we have the cycle σ_{∗}(c_{1}(σ^{∗}P_{0})^{k} · ρ) and the projection formula ([8], Proposition 2.5 (c)) implies that

c_{1}(P_{0})^{k} · c_{0} = σ_{∗}(c_{1}(σ^{∗}P_{0})^{k} · ρ).

Now we will use the following diagram of maps.

Ṽ  ^{σ}_{τ}  V ℙ × ℙ_{p_{1}}  _{α_{2}}  _{q×q} _{α_{1}}  _{p_{2}} \overline{ℙ}  ℙ _{q}_{ν}  ℙ × B _{β_{1}}_{κ_{1}}  B × ℙ_{β_{2}}  _{κ_{2}} ℙ_{q}  _{ν} \overline{ℙ} B  B × B_{q_{1}}  _{q_{2}}  B

Lemma 6.1. Let x be a cycle on B × B. Then the following holds. (1) p_{2∗}((q × q)^{∗}x) = 0.

(2) p_{2∗}((q × q)^{∗}x · p_{1}^{∗}η) = q^{∗}q_{2∗}x.

Proof. For (1) we observe that p_{2∗} = κ_{2∗}α_{1∗}, and (q ×q)^{∗} = α_{1}^{∗}β_{2}^{∗} and α_{1∗}α_{1}^{∗} = 0. For (2) we use the identities

p_{2∗}((q × q)^{∗}x · p_{1}^{∗}η) = p_{2∗}(α_{2}^{∗}β_{1}^{∗}x · α_{2}^{∗}κ_{1}^{∗}η) = p_{2∗}α_{2}^{∗}(β_{1}^{∗}x · κ_{1}^{∗}η)
= κ_{2∗}α_{1∗}α_{2}^{∗}(β_{1}^{∗}x · κ_{1}^{∗}η) = κ_{2∗}β_{2}^{∗}β_{1∗}(β_{1}^{∗}x · κ_{1}^{∗}η)
= κ_{2∗}β_{2}^{∗}(x · β_{1∗}κ_{1}^{∗}η) = q^{∗}q_{2∗}(x · q_{1}^{∗}q_{∗}η) = q^{∗}q_{2∗}x.
□

Consider the following diagram of maps

ℙ_{i}_{λ_{i}}  ℙ_{i} × ℙ_{j}_{λ_{ij}}  E_{ij}_{ϵ_{ij}}^{π_{ij}} ℙ_{q}  ℙ × ℙ_{p_{1}} _{q×q}  Ṽ_{τ}  _{σ} B  B × B_{q_{1}}  _{q_{2}}  V B

where p_{i}, q_{i} are the projections to the ith factor, π_{ij} the canonical map of the projective bundle E_{ij} and the maps λ_{i}, λ_{ij} and ϵ_{ij} the natural inclusions. The map (q × q) ∘ λ_{ij} is an isomorphism. By the adjunction formula, the normal bundles to ℙ_{1}, ℙ_{2} are N_{ℙ_{1}}(ℙ) = J and N_{ℙ_{2}}(ℙ) = J^{−1}. The exceptional divisors E_{12} and E_{21} are projective bundles over the blowing up loci ℙ_{i} × ℙ_{j}. By identifying ℙ_{i} × ℙ_{j} with B × B, via the map (q × q) ∘ λ_{ij}, we have E_{12} = ℙ(q_{1}^{∗}J^{−1} ⊕ q_{2}^{∗}J) and E_{21} = ℙ(q_{1}^{∗}J ⊕ q_{2}^{∗}J^{−1}). We set ξ_{ij} = c_{1}(O(1)) on E_{ij}. By standard theory [[10], ch. II, Theorem 8.24 (c)] we have ϵ_{ij}^{∗}E_{ij} = −ξ_{ij}. We now introduce the notation

γ := c_{1}(J), γ_{i} = q_{i}^{∗}γ, η_{i} = p_{i}^{∗}η, i = 1, 2.

Note that γ is algebraically equivalent to 0, but not rationally equivalent to 0. We have the quadratic relations

(ξ_{ij} − π′_{ij}^{∗}γ_{j})(ξ_{ij} + π′_{ij}^{∗}γ_{i}) = 0

where π′_{ij} : E_{ij} → B × B is the natural map, showing that ξ_{ij}^{2} is expressible in lower powers.

Lemma 6.2. Suppose that ξ satisfies the relation ξ^{2} + (a − b)ξ − ab = 0. Then, with ϕ_{k} = (b^{k} −(−a)^{k})/(b+a) we have ξ^{k} = ϕ_{k}ξ +abϕ_{k−1} for any k ≥ 1 (where we put ϕ_{0} = 0).

Proof. Immediate by checking the relation with ξ = b or ξ = −a. □

Applying the above for the classes ξ_{ij} of the bundles E_{ij}, considered as bundles over B × B via the isomorphism (q × q) ∘ λ_{ij}, we get, by choosing

ϕ_{k} = ∑_{m=0}^{k−1}(−1)^{m}γ_{1}^{m}γ_{2}^{k−1−m},

that

\begin{matrix}ξ_{12}^{k} = & π′_{12}^{∗}ϕ_{k} · ξ_{12} + π′_{12}^{∗}(γ_{1}γ_{2}ϕ_{k−1}), \\ ξ_{21}^{k} = & (−1)^{k+1}π′_{21}^{∗}ϕ_{k} · ξ_{21} + (−1)^{k}π′_{21}^{∗}(γ_{1}γ_{2}ϕ_{k−1}) . \end{matrix}

We view now the bundles E_{ij} as bundles over ℙ_{i} × ℙ_{j} and, for any k ≥ 0, we write ξ_{ij}^{k} = π_{ij}^{∗}A_{ij}(k)ξ_{ij} + π_{ij}^{∗}B_{ij}(k), for some cycles A_{ij}(k), B_{ij}(k) on ℙ_{i} × ℙ_{j}. By the above relations we have

(q × q)_{∗}λ_{ij∗}A_{ij}(k) = (−1)^{(k+1)j}ϕ_{k} .

Lemma 6.3. We have

λ_{ij∗}A_{ij}(k) = (−1)^{(k+1)j}[(q × q)^{∗}ϕ_{k} · η_{1}η_{2} − (q × q)^{∗}(ϕ_{k} γ_{j}) · η_{i}] .

Proof. We let ψ_{ij} = (q ×q)∘λ_{ij} : ℙ_{i} ×ℙ_{j} → B ×B be the natural isomorphism. We then have the identity

λ_{ij∗}A_{ij}(k) = λ_{ij∗}(ψ_{ij}^{∗}ψ_{ij∗}A_{ij}(k)) = (q × q)^{∗}ψ_{ij∗}A_{ij}(k) · λ_{ij∗}1_{ℙ_{i}×ℙ_{j}}.

But λ_{ij∗}1_{ℙ_{i}×ℙ_{j}} = p_{1}^{∗}ℙ_{i} · p_{2}^{∗}ℙ_{j} = η_{i}(η_{j} − p_{j}^{∗}q^{∗}γ) = η_{1}η_{2} − η_{i} · (q × q)^{∗}γ_{j} and the result follows. □

Lemma 6.4. For a cycle class x = q^{∗}z + q^{∗}w · η on ℙ the cycle class τ_{∗}(τ^{∗}p_{1}^{∗}x · (E_{12}^{k} + E_{21}^{k})) for k ≥ 1 is given by

∑_{m=0}^{k−2}(−1)^{m}{(q × q)^{∗}q_{1}^{∗}[(((−1)^{k+1} − 1)z + (−1)^{k+1}wγ) γ^{m}] · η_{1}η_{2}

+ (−1)^{k}(q × q)^{∗}q_{1}^{∗}[(z + wγ) γ^{m}] · η_{1} · p_{2}^{∗}q^{∗}γ

+ (q × q)^{∗}q_{1}^{∗}(zγ^{m+1}) · η_{2}} · p_{2}^{∗}q^{∗}γ^{k−2−m}.

Note that for k = 1 the above sum is zero.

Proof. Since ϵ_{ij}^{∗}E_{ij} = −ξ_{ij} we have E_{ij}^{k} = (−1)^{k−1}ϵ_{ij∗}ξ_{ij}^{k−1}. Therefore

\begin{matrix}τ_{∗}(τ^{∗}p_{1}^{∗}x · E_{ij}^{k}) = & (−1)^{k−1}p_{1}^{∗}x · τ_{∗}ϵ_{ij∗}ξ_{ij}^{k−1} \\ = & (−1)^{k−1}p_{1}^{∗}x · λ_{ij∗}π_{ij∗}(π_{ij}^{∗}A_{ij}(k − 1)ξ_{ij} + π_{ij}^{∗}B_{ij}(k − 1)) \\ = & (−1)^{k−1}p_{1}^{∗}x · λ_{ij∗}A_{ij}(k − 1) \end{matrix}

since π_{ij∗}ξ_{ij} = 1_{ℙ_{i}×ℙ_{j}}. Note that since A_{ij}(0) = 0 the above calculation shows that τ_{∗}(τ^{∗}p_{1}^{∗}x · E_{ij}) = 0. By Lemma 6.3 and by using the relation

p_{1}^{∗}x = (q × q)^{∗}q_{1}^{∗}z + (q × q)^{∗}q_{1}^{∗}w · η_{1},

we have

\begin{matrix}τ_{∗}(τ^{∗}p_{1}^{∗}x · E_{ij}^{k}) = & (−1)^{k(j+1)+1}((q × q)^{∗}q_{1}^{∗}z + (q × q)^{∗}q_{1}^{∗}w · η_{1}) \\ · [(q × q)^{∗}ϕ_{k−1} · η_{1}η_{2} − (q × q)^{∗}(ϕ_{k−1}γ_{j}) · η_{i}] \end{matrix}

758 Gerard van der Geer and Alexis Kouvidakis and this equals

\begin{matrix}(−1)^{k(j+1)+1}[(q × q)^{∗}(q_{1}^{∗}z · ϕ_{k−1}) · η_{1}η_{2} − (q × q)^{∗}(q_{1}^{∗}z · ϕ_{k−1}γ_{j}) · η_{i} \\ + (q × q)^{∗}(q_{1}^{∗}w · ϕ_{k−1}) · η_{1}^{2}η_{2} − (q × q)^{∗}(q_{1}^{∗}w · ϕ_{k−1}γ_{j}) · η_{1}η_{i}] \end{matrix}

We then have, by using the formula η^{2} = q^{∗}γ · η, that

\begin{matrix}τ_{∗}(τ^{∗}p_{1}^{∗}x · E_{12}^{k}) = (−1)^{k+1}[(q × q)^{∗}(q_{1}^{∗}(z + wγ) · ϕ_{k−1}) · η_{1}η_{2} \\ −(q × q)^{∗}(q_{1}^{∗}(z + wγ) · ϕ_{k−1}) · η_{1} · p_{2}^{∗}q^{∗}γ] \end{matrix}

and

τ_{∗}(τ^{∗}p_{1}^{∗}x · E_{21}^{k}) = −(q × q)^{∗}(q_{1}^{∗}z · ϕ_{k−1}) · η_{1}η_{2} + (q × q)^{∗}(q_{1}^{∗}(z γ) · ϕ_{k−1}) · η_{2}.

Using ϕ_{k−1} = ∑_{m=0}^{k−2}(−1)^{m}γ_{1}^{m} · γ_{2}^{k−2−m} we deduce the proposition. □ We state now the basic result of this section. Proposition 6.5. Let z, w be cycles on B. Then we have

p_{2∗}τ_{∗}(e^{c_{1}(σ^{∗}P_{0})} · τ^{∗}(p_{1}^{∗}(q^{∗}z + q^{∗}w · η)) = q^{∗}a + q^{∗}b · η,

with a and b as in Theorem 1.1. Proof. We put x = q^{∗}z + q^{∗}w · η. We want to calculate

p_{2∗}τ_{∗}(e^{τ^{∗}(q×q)^{∗}c_{1}(P_{B})−E_{12}−E_{21}} · τ^{∗}(p_{1}^{∗}x))

which equals

p_{2∗}(e^{(q×q)^{∗}c_{1}(P_{B})} · τ_{∗}(e^{−E_{12}−E_{21}} · τ^{∗}p_{1}^{∗}x)).

Since E_{12} · E_{21} = 0 we have

e^{−E_{12}−E_{21}} = 1 + ∑_{k=1}^{2g} \frac{(−1)^{k}}{k!} (E_{12}^{k} + E_{21}^{k})

and so τ_{∗}(e^{−E_{12}−E_{21}} · τ^{∗}p_{1}^{∗}x) equals

p_{1}^{∗}x + ∑_{k=1}^{2g} \frac{(−1)^{k}}{k!} τ_{∗}[τ^{∗}p_{1}^{∗}x · (E_{12}^{k} + E_{21}^{k})].

We have

p_{2∗}((q × q)^{∗}e^{c_{1}(P_{B})} · p_{1}^{∗}x) =
= p_{2∗}(e^{(q×q)^{∗}c_{1}(P_{B})} · p_{1}^{∗}(q^{∗}z + q^{∗}w η))
= p_{2∗}((q × q)^{∗}(e^{c_{1}(P_{B})}q_{1}^{∗}z) + (q × q)^{∗}(e^{c_{1}(P_{B})}q_{1}^{∗}w) p_{1}^{∗}η)
= 0 + q^{∗}q_{2∗}(e^{c_{1}(P_{B})}q_{1}^{∗}w) = q^{∗}F_{B}(w)

by Lemma 6.1. Combining the above with Lemma 6.4 we find that p_{2∗}τ_{∗}(e^{τ^{∗}(q×q)^{∗}c_{1}(P_{B})−E_{12}−E_{21}} · τ^{∗}(p_{1}^{∗}x))

Documenta Mathematica 15 (2010) 747–763

is the sum of the four terms: the first is q^{∗}F_{B}(w), the second is

\begin{matrix}∑_{k=2}^{2g} ∑_{m=0}^{k−2} \frac{(−1)^{k+m}}{k!} \{p_{2∗}[(q × q)^{∗}[e^{c_{1}(P_{B})}q_{1}^{∗}[(((−1)^{k+1} − 1)z+ \\ (−1)^{k+1}wγ) γ^{m}]] · η_{1}]\} · η · q^{∗}γ^{k−2−m}, \end{matrix}

the third term is

∑_{k=2}^{2g} ∑_{m=0}^{k−2} \frac{(−1)^{m}}{k!} \{p_{2∗}[(q × q)^{∗}[e^{c_{1}(P_{B})}q_{1}^{∗}[(z + wγ) γ^{m}]] · η_{1}]\} · q^{∗}γ^{k−1−m},

and finally the fourth is

∑_{k=2}^{2g} ∑_{m=0}^{k−2} \frac{(−1)^{k+m}}{k!} \{p_{2∗}[(q × q)^{∗}[e^{c_{1}(P_{B})}q_{1}^{∗}(zγ^{m+1})]]\} · η · q^{∗}γ^{k−2−m} .

By applying now Lemma 6.1 and by making the substitution n = k − 2 we get the desired expression. □

Corollary 6.6. Let z, w be cycles on B. Then modulo algebraic equivalence we have

p_{2∗}τ_{∗}(e^{c_{1}(σ^{∗}P_{0})} · τ^{∗}(p_{1}^{∗}(q^{∗}z + q^{∗}w · η))=^{a} q^{∗}F_{B}(w) − q^{∗}F_{B}(z) · η.

Proof. Indeed, since c_{1}(J)=^{a}0 it is clear that a=^{a} F_{B}(w) and b=^{a} −q^{∗}F_{B}(z) since the only non zero term of the sum corresponds to m = 0, n = 0. □

We conclude now with the proof of the basic Theorem 1.1 and Theorem 1.2:

Proof. By Proposition 5.3 we have φ_{0} = σ_{X}F (c_{η}) = ρ_{2∗}(e^{c_{1}(P_{0})} · σ_{∗}(τ^{∗}p_{1}^{∗}γ)). By the projection formula we have e^{c_{1}(P_{0})} · σ_{∗}(τ^{∗}p_{1}^{∗}γ) = σ_{∗}(e^{c_{1}(σ^{∗}P_{0})} · τ^{∗}p_{1}^{∗}γ). Observe now that ρ_{2} ∘ σ = ν ∘ (p_{2} ∘ τ) : Ṽ → ℙ̄, see the diagram in the proof of Lemma 5.2. The proof then follows from Proposition 6.5 and Corollary 6.6. □

7. Applications

Let X → S be a completed rank-one degeneration as described in § 2. According to Beauville [2] we have a decomposition of A_{ℚ}^{i}(X_{η}) into subspaces which are eigenspaces for the action by multiplication by an integer on X_{η}:

A_{ℚ}^{i}(X_{η}) = ⊕_{j}A_{(j)}^{i}(X_{η})

such that n^{∗}(x) = n^{2i−j} x for x ∈ A^{i}(X_{η}). (Beauville works over ℂ, but his proof does not use more than the Fourier-Mukai transform which works over the residue field of η.) The multiplication map n acts as multiplication by n^{2i} on homology and therefore all cycles in A_{(j)}^{i}(X_{η}) are homologically trivial for j ≠ 0. Since under the Fourier-Mukai transform we have F (A_{(j)}^{i}(X_{η})) = A_{(j)}^{g−i+j}(X_{η}), the elements of A^{i} that lie in A_{(j)}^{i} are characterized by the codimension of their Fourier transform (namely g − i + j).

Suppose now that c = ∑ c^{(j)} ∈ A^{i}(X_{η}) with c^{(j)} ∈ A_{(j)}^{i}(X_{η}), where the decomposition corresponds to φ := F (c) = ∑ φ^{(j)} with φ^{(j)} ∈ A^{g−i+j}(X_{η}).

Theorem 7.1. Let c = c_{η} = ∑ c^{(j)} ∈ A^{i}(X_{η}) with c^{(j)} ∈ A_{(j)}^{i}(X_{η}). Assume

for some j′ that φ_{0}^{(j′)} ≠ 0, where φ_{0} is the specialization of φ and φ_{0}^{(j′)} the codimension g − i + j′-part of φ_{0}. Then c^{(j′)} ≠ 0.

Proof. The specialization map preserves the codimension of cycles. Therefore, if c^{(j′)} = 0 then φ^{(j′)} = 0, hence φ_{0}^{(j′)} = 0 and this contradicts our assumption.

□

This theorem, which holds as well for cycles modulo algebraic equivalence, can be used to prove non-vanishing results for cycles. For the rest of this section we work modulo algebraic equivalence. For example, consider a threefold Z/S such that Z_{η} is a smooth cubic threefold and Z_{0} is a generic nodal cubic threefold. We shall consider the Picard variety of the Fano surface of this degenerating cubic threefold and this will give us a degenerating abelian variety of dimension 5, cf. [5]. As is well-known the nodal cubic threefold Z_{0}, and hence its Fano surface, corresponds to a canonical genus 4 curve C in ℙ^{3}, see e.g. [9] Section 2. The genericity assumption means that the curve C is a generic curve and hence we may assume by Ceresa’s result [4] that the class C^{(1)} does not vanish in the Jacobian B of the curve C. Since C is a trigonal curve we have by [6] that C^{(j)}=^{a} 0 for j ≥ 2. Hence the Beauville decomposition of C is [C]=^{a} C^{(0)} + C^{(1)} with F_{B}(C^{(0)}) ∈ A_{(0)}^{1}(B) and F_{B}(C^{(1)}) ∈ A_{(1)}^{2}(B). The Picard variety X /S of the Fano surface of Z/S defines a principally polarized semi-abelian variety with central fibre a rank-one extension of the Jacobian B of the curve C, see [9], Corollary 6.3 and Section 10. The principal polarization on X_{η} is induced by a geometrically defined divisor Θ. Let Σ be the Fano surface of lines in Z_{η}. If s ∈ Σ we denote by l_{s} the corresponding line in Z_{η}. For each s ∈ Σ we have the divisor

D_{s} = {s′ ∈ Σ, l_{s′} ∩ l_{s} ≠ ∅}

on Σ as defined in [5]. We then have a natural map

Σ → Pic^{0}(Σ), s ↦ D_{s} − D_{s_{0}},

with s_{0} ∈ Σ a base point. It is well known that the cohomology class of Σ in Pic^{0}(Σ) is equal to that of the cycle Θ^{3}/3!, see [5]. By [2], Propositions 3 and 4, we have that A_{(j)}^{3}(X_{η}) = 0 for j < 0 and A_{(j)}^{5}(X_{η}) = 0 for j ≠ 0. We have therefore the decomposition

[Σ]=^{a} Σ^{(0)} + Σ^{(1)} + Σ^{(2)} with Σ^{(j)} ∈ A_{(j)}^{3}.

Indeed, Σ^{(j)} ∈ A_{(j)}^{3}(X_{η}), hence F (Σ^{(j)}) ∈ A_{(j)}^{2+j}(X_{η}) which is zero for j ≥ 3.

Now we show that Σ^{(1)}≠^{a} 0, and we thus obtain a cycle which is homologically but not algebraically equivalent to zero. Since Θ ∈ A_{(0)}^{1}(X_{η}) this implies that Σ is homologically, but not algebraically equivalent to Θ^{3}/3! . We denote by X the completed rank one degeneration of X_{η}. The class [Σ] degenerates to a cycle [Σ_{0}] = ν_{∗}(γ) on the central fibre X_{0} of class

γ=^{a} q^{∗}[C] + \frac{1}{2} q^{∗}[C ∗ C] · η,

where C ∗ C is the Pontryagin product, see [9], Propositions 10.1 and 8.1. In

order to see that Σ^{(1)}≠^{a} 0 it suffices by Theorem 7.1 to show that φ_{0}^{(1)}≠^{a} 0 with φ_{0} the limit of the Fourier-Mukai transform. By Theorem 1.2, we have

φ_{0}=^{a} ν_{∗}(\frac{1}{2} q^{∗}[F_{B}(C) · F_{B}(C)] − q^{∗}F_{B}(C) · η),

hence

φ_{0}^{(1)}=^{a} ν_{∗}(q^{∗}[F_{B}(C^{(0)}) · F_{B}(C^{(1)})] − q^{∗}F_{B}(C^{(1)}) · η).

Since C^{(1)}≠^{a} 0 we conclude that φ_{0}^{(1)}≠^{a} 0, and this implies the result. By using the specialization of the Fourier-Mukai transform we can deduce the specialization of the Beauville decomposition. We do this working modulo algebraic equivalence.

Proposition 7.2. Let c = c_{η} ∈ A^{i}(X_{η}) with specialization c_{0} = ν_{∗}(q^{∗}z + q^{∗}w · η), where z ∈ A^{i}(B) and w ∈ A^{i−1}(B). Let c = ∑ c^{(j)} with c^{(j)} ∈ A_{(j)}^{i}(X_{η}), and let z = ∑ z^{(j)} with z^{(j)} ∈ A_{(j)}^{i}(B) and w = ∑ w^{(j)} with w^{(j)} ∈ A_{(j)}^{i−1}(B)

be the Beauville decompositions. If c_{0}^{(j)} is the specialization of c^{(j)}, then

c_{0}^{(j)}=^{a} ν_{∗}(q^{∗}z^{(j)} + q^{∗}w^{(j)} · η) .

Proof. By the proof of the main theorem in [2], the component c^{(j)} is defined as (−1)^{g}F ((−1)^{∗}ϕ^{(j)}) with ϕ^{(j)} ∈ A^{g−i+j}(X_{η}) (notation as above). The inversion on X_{η} leaves the cell decomposition of the toroidal compactification invariant and hence extends naturally to X_{0}. So c_{0}^{(j)} equals (−1)^{g}F ((−1)^{∗}ϕ_{0}^{(j)}) with ϕ_{0}^{(j)} ∈ A^{g−i+j}(X_{0}). Therefore, by Theorem 1.2, we have

\begin{aligned}c_{0}^{(j)} & =^{a} (−1)^{g}F ((−1)^{∗}ν_{∗}(q^{∗}F_{B}(w^{(j)}) − q^{∗}F_{B}(z^{(j)}) · η)) \\ & =^{a} (−1)^{g+j}(−1)^{g−1+j}ν_{∗}(−q^{∗}z^{(j)} − q^{∗}w^{(j)} · η) = ν_{∗}(q^{∗}z^{(j)} + q^{∗}w^{(j)} · η) . \end{aligned}

□

For example, let C → S be a genus g curve with C_{η} a smooth curve and C_{0} a one-nodal curve with normalization C̃_{0}. Let p be the node of C_{0} and x_{1}, x_{2} the points of C̃_{0} lying over p. The compactified Jacobian X = \overline{P_{C/S}} is then a complete rank one degeneration with central fibre the ℙ^{1}-bundle over Pic^{0}(C̃_{0}) associated to the line bundle J = O(x_{1} − x_{2}). Let ū : C → X be the compactified Abel-Jacobi map and let c_{η} = [ū(C_{η})]. The cycle c_{η} specializes then to the cycle c_{0} = [ū(C_{0})] with c_{0}=^{a} ν_{∗}(q^{∗}[pt] + q^{∗}c̃_{0} · η), where [pt] is the class of a point and c̃_{0} is the class of the Abel-Jacobi image of the smooth curve C̃_{0} in Pic^{0}(C̃_{0}), see e.g. [9], Proposition 7.1. By Proposition 7.2 we have then

c_{0}^{(j)} =^{a} \{ \begin{matrix}q^{∗}c̃_{0}^{(j)} · η, j ≠ 0 , \\ q^{∗}[pt] + q^{∗}c̃_{0}^{(0)} · η, j = 0 . \end{matrix}

Acknowledgement The second author thanks the Korteweg-de Vries Instituut van de Universiteit van Amsterdam, where part of this work was done, for its support and hospitality.

References

[1] V. Alexeev, I. Nakamura: On Mumford’s construction of degenerating abelian varieties. Tohoku Math. J. 51 (1999), p. 399–420.

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Gerard van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Postbus 94248 1090 GE Amsterdam The Netherlands G.B.M.vanderGeer@uva.nl Alexis Kouvidakis Department of Mathematics University of Crete GR-71409 Heraklion Greece kouvid@math.uoc.gr 764

Documenta Mathematica 15 (2010)