The RankOne Limit of the FourierMukai Transform
Abstract.
We give a formula for the specialization of the FourierMukai transform on a semiabelian variety of torus rank .
1991 Mathematics Subject Classification:
14C25,14H401. Introduction
Let be a semiabelian variety of relative dimension over the spectrum of a discrete valuation ring with algebraically closed residue field such that the generic fibre is a principally polarized abelian variety. We assume that is contained in a complete rankone degeneration . In particular, the special fibre of is a complete variety over containing as an open part the total space of the bundle associated to a line bundle over a dimensional abelian variety . The normalization of can be identified with the bundle over associated to and is obtained by identifying the zerosection of with the infinitysection of by a translation. Moreover, is provided with a theta divisor that is the specialization of the polarization divisor on the generic fibre.
If is an algebraic cycle on we can take the FourierMukai transform and consider the limit cycle (specialization) of . A natural question is: What is the limit of ?
If denotes the natural projection of the bundle, the Chow ring of is the extension with . We consider now cycles with rational coefficients. We denote by the specialization of the cycle on . We can write as with .
Theorem 1.1.
Let be a cycle on with . The limit of the FourierMukai transform is given by with
and
where is the FourierMukai transform of the abelian variety .
We denote algebraic equivalence by . The relation implies the following result.
Theorem 1.2.
With the above notation the limit satisfies
Note that this is compatible with the fact that for a principally polarized abelian variety of dimension the FourierMukai transform satisfies .
Beauville introduced in [2] a decomposition on the Chow ring with rational coefficients of an abelian variety using the FourierMukai transform. Theorem 1.2 can be used to deduce nonvanishing results for Beauville components of cycles on the generic fibre of a semiabelian variety of rank ; we refer to §8 for examples.
We prove the theorem by constructing a smooth model of to which the addition map extends and by choosing an appropriate extension of the Poincaré bundle to . 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 . 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 is a complete discrete valuation ring with local parameter , field of quotients and algebraically closed residue field . Suppose that is a semiabelian variety over such that the generic fibre is abelian and the special fibre has torus rank ; moreover, we assume that is a cubical invertible sheaf (meaning that satisfies the theorem of the cube, see [7], p. 2, 8) and is ample. In particular, the special fibre of fits in an exact sequence
where is an abelian variety over and the multiplicative group over . The torus lifts uniquely to a torus of rank over in . The quotient is an abelian variety over . The system defines a formal abelian variety which is algebraizable, so that we have an exact sequence of group schemes over
cf. [FC, p. 34]. We assume now that we are given a line bundle on defining a principal polarization and consider . This defines a cubical line bundle on . The extension is given by a homomorphism of the character group of to . The semiabelian group scheme dual to defines a similar extension
and the polarization provides an isomorphism of the character group of with the character group of . Now the degenerating abelian variety (i.e. semiabelian variety) over gives rise to the set of degeneration data (cf. [7], p 51, Thm 6.2, or [1], Def. 2.3):

an abelian variety over and a rank extension . This amounts to a valued point of .

a valued point of lying over .

a cubical ample sheaf on inducing the polarization on and an action of on .
A section can be written uniquely as , where is a linear homomorphism and is the twist of by : in fact with the subsheaf consisting of eigenfunctions. (We refer to [7], p. 43; note also the sign conventions there in the last lines.) We have now by the action
This satisfies , where is given by a point of lying over and is as in [7], p. 44. We refer to FaltingsChai’s theorem (6.2) of [7], p. 51 for the degeneration data.
The compactification of is now constructed as a quotient of the action of on a socalled relatively complete model. Such a relatively complete model for can be constructed here in an essentially unique way. If is trivial (i.e. ) and if the torus is it is given as the toroidal variety obtained by gluing the affine pieces
where is given by , cf. [13], also in [7], p. 306]. By glueing we obtain an infinite chain of ’s in the special fibre. We can ‘divide’ by the action of ; this is easy in the analytic case, more involved in the algebraic case, but amounts to the same, cf. [13], also [7], p. 5556.
In the special fibre we find a rational curve with one ordinary double point. If instead we divide by the action of for we find a cycle consisting of copies of .
In case the abelian part is not trivial we take as a relatively complete model the contracted (or smashed) product with the relatively complete model for the case that is trivial. Call the resulting space . Then corresponds by Mumford’s [loc. cit., p 29] to a polyhedral decomposition of with the cocharacter group of . Then we essentially divide through the action of or as before and obtain a proper .
We describe the central fibre of . Let be the valued point of that determines the above extension. If denotes a line bundle defining the principal polarization of we let be the translation of by and we set and define the projective bundle with projection . The bundle has two natural sections (with images) and corresponding to the projections and . We have and with the natural line bundle on . We denote by the nonnormal variety obtained by gluing the sections and under a translation by the point . The singular locus of has support isomorphic to . The line bundle descends to a line bundle on with a unique ample divisor , see [14]. The central family of the family is then equal to . The cubical invertible sheaf on extends (uniquely) to and its restriction to the central fiber is the line bundle , see [15].
3. Extension of the addition map
The addition map of the semiabelian scheme does not extend to a morphism , but it does so after a small blowup of as we shall see.
The degeneration data of defines (product) degeneration data for . Indeed, we can take the fibre product of the relatively complete model and this corresponds (e.g. via [13], Corollary (6.6)) to the standard polyhedral decomposition of by the lines and for . The special fibre of the model is an infinite union of bundles over glued along the fibres over and . The compactified model of is obtained by taking the ‘quotient’ of under the action of . 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 given by the lines and for .
The special fibre of this model is an infinite union of copies of bundles over blown up in the two antidiagonal sections and . This is regular.
Both the polyhedral decompositions are invariant under the action of translations for fixed . This means that we can form the ‘quotient’ by (or a subgroup ) and obtain a completed semiabelian abelian variety of relative dimension over . We denote by the natural map. We shall write for and for its normalization. Then is an irreducible component of the special fibre of . We denote by the blow up map and by and the exceptional divisors over the blowing up loci and , respectively.
Now consider the addition map with as in the preceding section. This morphism is induces (and is induced by) by a map . However, this map does not extend to a morphism of the relatively complete model since the corresponding (covariant) map does not have the property that it maps cells to cells. After subdividing (by adding the lines with ) this property is satisfied (cf. [11], Thm. 7, p. 25). This means that the map extends to for the polyhedral decomposition given by this subdivision. It is compatible with the action of and and hence descends to a morphism . We summarize:
Proposition 3.1.
The addition map of group schemes extends to a morphism .
In the next section we shall see that the change from the model to is a small blowup.
For later calculations we write down this map explicitly on the special fibre. We start with ; then is trivial and we may restrict the map to an irreducible component of the special fibre of the relatively complete model and get the map given by . This is not defined in the points and . After blowing up these points (which corresponds exactly to the change from to ) the rational map becomes a regular map . It is defined by the two sections and of the linear system with and the horizontal and vertical fibre (with meaning the proper transform). The map descends to a map which is the restriction of the morphism to the central fiber.
For the case that , note that we have the addition map . Its restriction to the special fibre extends to a map of the relatively complete model and then restricts to a morphism that lifts the addition map of . That means that it comes from a surjective bundle map (cf. [10], Ch. II, Prop. 7.12)
with and with the th projection. Then is isomorphic to the direct sum of
The map is then given by the two sections of for . The map descends to a map which is the restriction of the morphism to the central fiber.
4. An explicit model of
We now describe an explicit local construction of the model by blowing up the model . Let denote affine space. In local coordinates, inside , we may assume that the dimensional fibration is given by the equation , where the coordinates are not involved, see [14] p. 361362. We may assume that the zero section of the family is defined by for .
We form the fiber product . We denote by the support of the singular locus of . The dimensional variety is singular in the special fiber along of dimension . The generic fiber is the product of the abelian variety , while the zero fiber is singular. The local equations of in a neighborhood of the singular locus of the family are given in our local coordinates by the system . The singular locus of is given by the equations .
The above blow up is a small blow up and can be described directly as follows: we blow up along its subvariety defined by (a 2plane contained in the central fiber of ). The proper transform of is smooth. In local coordinates, the blowup is given by the graph of the rational map given by . The equations of the graph are given by the system
where are homogeneous coordinates on .
5. Extension of the Poincaré bundle
We denote by and the inclusions of the special fiber. Recall that we write for and for its normalization. We denote by the Poincaré bundle on and by the Poincaré bundle on .
Theorem 5.1.
The Poincaré bundle has an extension such that the pull back of to satisfies .
Proof.
We have the following commutative diagram of maps
Let be the theta line bundle on the family introduced in section 2. We define the extension of by
where we denote by the compositions of the natural projections with the blowing up map of section 4. We then have . Now , so . In view of we have , where is the line bundle introduced at the end of section 3. We thus get
and . On the other hand using the description of in §2 we see
and putting this together we find
∎
6. The basic construction
The fibration is a flat map since is irreducible and is smooth dimensional, see [10], Ch. III, Proposition 9.7. The maps , , 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 , see e.g. [12], Corollary of Thm. 23.1.
We denote by (resp. ) the special fibre (resp. the generic fibre) and by (resp. ) the corresponding embedding. According to [8], Example 10.1.2., is a regular embedding. Similarly, is a regular embedding. We consider the diagram
Let be the Gysin map (see [8], Example 5.2.1). Since is an effective Cartier divisor in the Gysin map coincides with the Gysin map for divisors (see [8], Example 5.2.1 (a) and ).
We now consider specialization of cycles, see [8], . Note that according to [8], Remark 6.2.1., in our case we have . If is a flat scheme over the spectrum of a discrete valuation ring the specialization homomorphism is defined as follows, see [8], pg. 399: If is a cycle on we denote by an extension of in (e.g. the Zariski closure of in ) and then , where is the natural embedding.
Let be a cycle on and let be the FourierMukai transform. It is defined by . Let be the specialization map. We have to determine .
If is a cycle on we have by applying [8] Proposition 20.3 (a) to the proper map . By choosing we have
(1) 
Therefore, in order to compute we have to identify . We take the extension of and the extension of given by , where is the Zariski closure of in . Since is an open embedding and hence a flat map of dimension , we have , see [8], Proposition 2.3 (d). In other words, the cycle extends the cycle and hence .
Now, for any cycle on we have the identity
in , where is the pull back of the line bundle and the Gysin pull back to the divisor . This follows from applying the formula in [8], Proposition 2.6 (e) to , with , and the Poincaré bundle. Hence
(2) 
By the Moving Lemma (see [8], ), we may choose the cycle on the regular such that it intersects the singular locus of the central fiber properly. Since the cycle meets properly by the following dimension argument. We have , hence
Since is of codimension in , saying that meets properly, is equivalent to saying that no component of is contained in .
Lemma 6.1.
There exists a cycle on with that meets the sections for properly.
Proof.
If is the singular locus of and its preimage in , then . We may assume that the cycle is irreducible and we consider the support of as a subset of . Its Zariski closure is an irreducible cycle on . Then is an irreducible cycle on since the map is a projective map. Also, , hence is the Zariski closure of and so, by the irreducibility, we have . ∎
Lemma 6.2.
If , then we have .
Proof.
We denote the restriction of to the special fibre again by . Then we have since is a flat map and are regular embeddings (see [8], Theorem 6.2 (b) and Remark 6.2.1). We will use the following commutative diagram
We may assume that and are irreducible cycles. We claim that is irreducible. Indeed, the map is a flat map of relative dimension . The cycle is then a cycle of pure dimension and contains the proper transform of and that is an irreducible cycle. Any other irreducible component of must have support on the preimage of . But since the cycle intersects along a cycle, there is no irreducible component of on the preimage of . On the other hand, since meets the sections properly, the cycle is an irreducible cycle, and hence so is . But as and coincide outside the exceptional divisor of , they have to coincide everywhere. ∎
Proposition 6.3.
We have .
In order to calculate the limit of the FourierMukai transform we are thus reduced to a calculation in the special fibre.
7. A calculation in the special fibre  Proof of the main theorem
Recall the normalization map . Suppose we have a cycle on with . We can consider the intersection , that is a successive intersection of a cycle with a Cartier divisor on the singular variety . On the other hand we have the cycle and the projection formula ([8], Proposition 2.5 (c)) implies that
Now we will use the following diagram of maps.
Lemma 7.1.
Let be a cycle on . Then the following holds.

.

.
Proof.
For (1) we observe that , and and . For (2) we use the identities
∎
Consider the following diagram of maps
where are the projections to the th factor, the canonical map of the projective bundle and the maps , and the natural inclusions. The map is an isomorphism.
By the adjunction formula, the normal bundles to are and . The exceptional divisors and are projective bundles over the blowing up loci . By identifying with , via the map , we have and . We set on . By standard theory [[10], ch. II, Theorem 8.24 (c)] we have .
We now introduce the notation
Note that is algebraically equivalent to , but not rationally equivalent to . We have the quadratic relations