The rank-one limit of the Fourier-Mukai transform

We give a formula for the specialization of the Fourier-Mukai transform on a semi-abelian variety of torus rank 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 ∼ = B with the infinity-section of P 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 of P is the extension CH * (B)[η]/(η 2 − η • q * c 1 (J)) 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.
We denote algebraic equivalence by a =.The relation c 1 (J) a =0 implies the following result.
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 §8 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].

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, 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 i=1 defines a formal abelian variety which is algebraizable, so that we have an exact sequence of group schemes over S We assume now that we are given a line bundle M on B defining a principal polarization λ : B → B t and consider 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.semiabelian 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 .
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 P 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 where G ⊂ P 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 P0 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 P × T G with P the relatively complete model for the case that B is trivial.Call the resulting space P .Then P 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 divide through 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 −1 b 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 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 P has support isomorphic to B. The line bundle L = O(P 1 ) ⊗ q * M b ∼ = O(P 2 ) ⊗ q * M descends to a line bundle L on P with a unique ample divisor D, see [14].The central family X 0 of the family π : X → S is then equal to P. The cubical invertible sheaf L on X ⋆ extends (uniquely) to X and its restriction to the central fiber P is the line bundle L, see [15].

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 P ′ = P × S P 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 P ′ 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 P ′ 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 the 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 abelian variety Y of relative dimension 2g over S. We denote by ǫ : 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 P ′ .We denote by τ : Ṽ → P 1 × P 1 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 the preceding section.This morphism is induces (and is induced by) by a map μ : G × S G → G.However, this map does not extend to a morphism of the relatively complete model P ′ since the corresponding (covariant) map (Z t ⊗ R) 2 → (Z t ⊗ R) does not have the property that it maps 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 μ : P ′ → P 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 μ : Y → X .We summarize: Proposition 3.1.The addition map of group schemes µ : In the next section we shall see that the change from the model X × S X to Y is a small blow-up.
For later calculations we write down this map 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 P × S P 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 corresponds exactly to the change from X × S X to Y) the rational map becomes a regular map m : Ṽ → P .The map m descends to a map m : V → P which is the restriction of the morphism μ : Y → X to the central fiber.
For the case that 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 m : 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) The map δ is then given by the two sections prop(p The map m descends to a map m : V → P which is the restriction of the morphism μ : Y → X to the central fiber.

An explicit model of Y
We now describe an explicit local construction of the model Y by blowing up the model X × S X .Let A g+1 S = Spec(R[x 1 , . . ., x g+1 ]) denote affine S-space.In local coordinates, inside A g+1 S , 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 fiber product π : Y ′ = X × S X .We denote by T the support of the singular locus of X 0 .The 2g + 1 dimensional variety Y ′ is singular in the special fiber along Σ = T × k T ∼ = B × k B of dimension 2g − 2. The generic fiber Y ′ η is the product X η × K X η of the abelian variety X η , while the zero fiber 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 The above blow up ǫ : 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 fiber 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 : S are given by the system , where u, v are homogeneous coordinates on P 1 .

Extension of the Poincaré bundle
We denote by j 0 : X 0 ֒→ X and i 0 : Y 0 ֒→ Y the inclusions of the special fiber.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 5.1.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

B
Let L be the theta line bundle on the family X introduced in section 2. We define the extension of P 0 by 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 ǫ : Y → Y ′ of section 4. We then have and m * q * M b = τ * (q × q) * µ * B M b .On the other hand using the description of L in §2 we see and putting this together we find

The basic construction
The fibration π : Y → S is a flat map since Y is irreducible and S is smooth 1-dimensional, see [10], Ch.III, Proposition 9.7.The maps ρ i = Y → X , i = 1, 2, defined in the proof of Theorem 5.1, are flat maps too since they are maps of smooth irreducible varieties with fibers 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 ) 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 . Therefore, in order to compute σ X (F (c η )) we have to identify σ Y (e c1(Pη ) • ρ * 1 c η ).We take the extension e c1(P) of e c1(Pη ) and the extension of ρ * , 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 : ) .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 T of the central fiber properly.Since T ⊆ X 0 the cycle c 0 = j * 0 (c) meets T properly by the following dimension argument.We have dim(c 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 (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 Consider the following diagram of maps 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 ǫ 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 P1 (P) = J and N P2 (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 ) on E ij .By standard theory [ [10], ch.II, Theorem 8.24 (c)] we have We now introduce the notation Note that γ is algebraically equivalent to 0, but not rationally equivalent to 0. We have the quadratic relations Proof.Assuming by induction that ξ k = φ k ξ + abφ k−1 we find so the result follows by induction from the recurrence φ k+1 = (b − a)φ k + abφ k−1 that can be left to the reader.
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 . We view now the bundles E ij as bundles over P i × P j and, for any k ≥ 0, we write By the above relations we have Proof.We let ψ ij = (q × q) • λ ij : P i × P j → B × B be the natural isomorphism.We then have the identity We have p 2 * ((q × q) * e c1(PB ) • p * 1 x) = p 2 * (e (q×q) * c1(PB ) • p * 1 (q * z + q * w η)) = p 2 * ((q × q) * (e c1(PB ) q * 1 z) + (q × q) * (e c1(PB ) q * 1 w) p * 1 η) = 0 + q * q 2 * (e c1(PB ) q * 1 w) = q * F B (w) by Lemma 7.1.Combining the above with Lemma 7.4 we find that )) is the sum of the four terms: the first is q * F B (w), the second is and finally the fourth is By applying now Lemma 7.1 and by making the substitution n = k − 2 we get the desired expression.
Corollary 7.6.Let z, w be cycles on B. Then in algebraic equivalence we have Proof.Indeed, since c 1 (J)

Applications
Let X → S be a completed rank-one degeneration as described in §2.According to Beauville [2] we have a decomposition of CH i Q (X η ) into subspaces which are eigenspaces for the action of the integers on 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 i (j) (X η ) are homologically trivial for j = 0. Since under the Fourier-Mukai transform we have (X η ), the elements of A i that lie in A i (j) can be characterized by their codimension (namely g − i + j).Suppose now that c = c (j) ∈ A i (X η ) with c (j) ∈ A i (j) (X η ), where the decomposition corresponds to ϕ := F (c) = ϕ (j) with ϕ (j) ∈ A g−i+j (X η ).
Proof.The specialization map preserves the codimension of cycles.Therefore, if c (j) = 0 then ϕ (j) = 0, hence ϕ (j) 0 = 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.The genericity assumption means that the corresponding canonical genus 4 curve C in P 3 which is used to construct the Fano threefold, see e.g.[9] Section 2, 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 (B).The Picard variety X /S of Z 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 ∈ S we have the divisor D s = {s ′ ∈ S, l s ′ ∩ l s = ∅} on S as defined in [5].We then have a natural map 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 3 (j) (X η ) = 0 for j < 0 and A 5 (j) (X η ) = 0 for j = 0 in algebraic equivalence.We have therefore the decomposition [Σ] a = Σ (0) + Σ (1) + Σ (2)  with Σ (j) ∈ A 3 (j) .
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 fiber X 0 of class 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 8.1 to show that ϕ Since C (1)   a = 0 we conclude that ϕ (1) 0 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.
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 C0 .Let p be the node of C 0 and x 1 , x 2 the points of C0 lying over p.The compactified Jacobian X = P C/S is then a complete rank one degeneration with central fiber the P 1 -bundle over Pic q * c(j) 0 • η, j = 0 , q * [pt] + q * c(0) 0 • η, j = 0 .

BLemma 7 . 1 .
Now we will use the following diagram of maps.Let x be a cycle on B × B. Then the following holds.