FourierMukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds II
Abstract.
We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian threefold with Picard rank one by proving that tilt stable objects satisfy the strong BogomolovGieseker type inequality. This is done by showing any FourierMukai transform gives an equivalence of abelian categories which are double tilts of coherent sheaves.
Key words and phrases:
Bridgeland stability conditions, FourierMukai transforms, Abelian threefolds, BogomolovGieseker inequality2010 Mathematics Subject Classification:
Primary 14F05; Secondary 14J30, 14J32, 14J60, 14K99, 18E30, 18E35, 18E40Introduction
There is a growing interest in the study of Bridgeland stability conditions on varieties. This notion was introduced in [Bri1] and some known examples can be found in [Bri2, AB, Oka, Mac1, Mac2, MP, Sch]. See [Huy3] and [BBR, Appendix D] for comprehensive expositions on the subject. Construction of such stability conditions on a given CalabiYau threefold is an important problem but the only known example is on an abelian threefold (see [MP]). For further motivation from, for example, Mathematical Physics, see [Clay, Tod2]. A conjectural construction of such a stability condition for any projective threefold was introduced by Bayer, Bertram, Macrí and Toda in [BMT, BBMT]. Here they introduced the notion of tilt stability for objects in an abelian subcategory of the derived category which is a tilt of coherent sheaves. These have now been studied extensively: [Tod1, LM, BMT, Mac2, MP, Sch]. This conjectural construction has boiled down to the requirement that certain tilt stable objects satisfy a socalled (weak) BogomolovGieseker (BG for short) type inequality. Moreover they went in to propose a stronger version of this inequality which is known to hold for projective 3space (see [Mac2]) and smooth quadric threefold (see [Sch]). In [MP], for a principally polarized abelian threefold, we prove that tilt stable objects satisfy the weak BG type inequality associated to a special complexified ample class. It was achieved by establishing an equivalence of two abelian categories given by the classical FourierMukai transform with kernel the Poincaré bundle. The aim of this paper is to extend those ideas for any nontrivial FourierMukai transform (FMT for short) to establish the strong BG type inequality for the same abelian threefold.
For an abelian variety , the group of FMTs is well understood via the notion of isometric isomorphism (see [Orl2] or [Huy1, Chapter 9]). To any FMT with kernel Orlov constructed an isometric automorphism of the product . He showed that is a surjective map of groups and its kernel consists of trivial FMTs (which send skyscraper sheaves to skyscraper sheaves up to shift) which also preserve up to shift. When is principally polarized, one can canonically identify with an element of and vice versa. The FMT induces an isomorphism and it gives rise to a representation of . In this paper, when is principally polarized with Picard rank one, we obtain an explicit matrix description for this representation in terms of . As a result, any induced nontrivial transform on is an antidiagonal matrix with respect to some suitable twisted Chern characters. This allows us to handle the numerology in the same way as that of the classical FMT. A matrix representation for the induced transform of an abelian surface was also considered in [YY].
When is a complexified ample class, it is expected that defines a central charge function of some stability condition on . The space of all stability conditions carries a natural left action of the group . This can be defined via a natural left action of on . When is a dimensional principally polarized abelian variety with Picard rank one, we can view the action of on explicitly as: for some and (see [MYY, Appendix] for the dimension 2 case). When is real one can expect that the FMT gives an equivalence of some hearts of particular stability conditions of whose and are determined by (see Note 3.2). For example when , following similar ideas in [Yos], one can show that any FMT gives an equivalence of two abelian categories each of which are tilts of (see [Huy2]). Understanding the homological FMT for the case of is the basis of this paper. When the Picard rank is 1, this amounts to understanding the transforms as a numerical matrix. This then allows us, in a similar way to [MP], to show that any nontrivial FMT gives an equivalence of two abelian categories each of which are double tilts of (see Theorem 4.6). Minimal objects are sent to minimal objects again under an FMT. This enables us to obtain an inequality involving the top part of the Chern character of minimal objects in these abelian categories, and this is exactly the strong BG type inequality. Therefore any tilt stable object with zero tilt slope satisfies the strong inequality in [BMT] for our abelian threefold case (see Theorem 4.8).
Toda considered similar ideas in an attempt to construct a “Gepner” type stability condition on a quintic threefold using the spherical twist of the structure sheaf (see [Tod3]). In [Pol], Polishchuk tested the existence of stability conditions for abelian varieties by studying “Lagrangianinvariant” objects of .
Notation
We follow the notation of the first paper [MP] which we summarize and extend as follows.

We will denote an antidiagonal matrix with entries by

For , . and ,

For an interval , . Similarly the subcategory is defined.

For a FourierMukai functor and a heart of a tstructure for which , . For a sequence of integers ,
If is a FourierMukai transform then being WIT is equivalent to .

For a dimensional polarized projective variety with Picard rank one over , the Chern character of any is of the form for some . Here . For simplicity we write . Also we abuse notation to write for the functor .
1. Preliminaries
1.1. Construction of stability conditions for threefolds
Let us quickly recall the conjectural construction of stability conditions for a given projective threefold over as introduced in [BMT]. Let be in with an ample class, i.e. is a complexified ample class. The twisted Chern character with respect to is defined by . The twisted slope of is defined by
We say is (semi)stable, if for any , . The HarderNarasimhan (HN for short) property holds for and so we can define the slopes
of . Then for a given interval , the subcategory is defined by
The subcategories and of are defined by
Now forms a torsion pair on and let the abelian category be the corresponding tilt of . Define the central charge function by .
Following [BMT], the tiltslope of is defined by
In [BMT] the notion of stability for objects in is introduced in a similar way to stability for . Also it is proved that the abelian category satisfies the HN property with respect to stability. Then one can define the slopes for objects in and the subcategory for an interval . The subcategories and of are defined by
Then forms a torsion pair on and let the abelian category be the corresponding tilt.
Conjecture 1.1.
[BMT, Conjecture 3.2.6] The pair is a Bridgeland stability condition on .
Definition 1.2.

Let be the class of stable objects with .

Let be the class of objects with for any .
The objects in are minimal objects in (see [MP, Lemma 2.3]).
Let us assume and is an ample class with is rational. Then similar to the proof of [BMT, Proposition 5.2.2] one can show that the abelian category is Noetherian. Therefore Conjecture 1.1 is equivalent to the following (see [BMT, Corollary 5.2.4]).
Conjecture 1.3.
[BMT, Conjecture 3.2.7] Any satisfies the socalled BogomolovGieseker Type Inequality:
Moreover in [BMT] they proposed the following strong inequality for objects in .
Conjecture 1.4.
[BMT, Conjecture 1.3.1] Any satisfies the socalled Strong BogomolovGieseker Type Inequality:
For any , there exists such that is a short exact sequence (SES for short) in for some (see [MP, Proposition 2.9]). Therefore one only needs to check the BG (respectively, strong BG) type inequality for objects in .
1.2. Autoequivalences of abelian varieties
First of all we briefly introduce FourierMukai theory (see [BBR, Huy1] for further details). Let be smooth projective varieties and let , be the projection maps from to and , respectively. The FourierMukai functor (FM functor for short) with kernel is defined by
When is an equivalence of the derived categories it is called a FourierMukai transform (FMT for short). On the other hand Orlov’s representability theorem (see [Orl1]) says that any equivalence between and is isomorphic to for some . Any FM functor induces a linear map (sometimes called the cohomological FM functor) and it is an isomorphism when is an FMT.
Example 1.5.
Let be a principally polarized abelian variety of dimension . Then the isogeny , is an isomorphism and also . In the rest of the paper let be the FMT with the Poincaré line bundle on as the kernel. In [Muk2] Mukai proved that

is an autoequivalence of the derived category ,

, and

.
If we assume the Picard rank of is and write , then we have (see [Huy1, Lemma 9.23]).
Following work of Orlov the group of FMTs from to can be described explicitly as follows (see [Orl2] and [Huy1, Chapter 9] for further details). Let be two abelian varieties. Then one can write any morphism as a matrix , , and . The morphism of is defined by is said to be isometric if it is an isomorphism and its inverse . When , we denote the group of all isometric automorphisms of by . . Then for some morphisms
Let be an FMT between two abelian varieties and with kernel . Let us define the map by . Let , where are the projection maps from , is the structure sheaf on the diagonal , and is the Poincaré bundle on .
Let be an object such that and let be the FMT from to with kernel . Then it satisfies
for any (see [Orl2]). Now define the equivalence by
so that fits into the following commutative diagram (see [Huy1], [Orl2]).
The equivalence can also be expressed in a simple form as follows.
Lemma 1.6.
Example 1.7.
Let be a principally polarized abelian variety. The following examples are important in this paper (see [Huy1, Ex. 9.38]). Here is the diagonal embedding.
Let be abelian varieties and let , , be FMTs such that . Then one can show that and . So we have a well defined group homomorphism
Lemma 1.8.
[Huy1, Prop. 9.55] The map is an epimorphism and its kernel consists of autoequivalences where , and . So .
Let be a central extension of the group . The equivalences , and generate a subgroup of which is isomorphic to . Therefore is a subgroup of . Also can be realized as a subgroup of and so . Moreover when is principally polarized, any isometric automorphism in is of the form satisfying . In the rest of the paper we abuse notation by dropping from this matrix. In this way we canonically identify the isomorphism . Therefore, for a principally polarized abelian variety we have for some
2. Matrix Representations of and Induced FMTs on
2.1. Finite dimensional matrix representations of
Following [Kna], we explicitly construct a variant of the symmetric power representation of all dimensions () of .
For , let be the vector space of homogeneous polynomials over in variables of degree . Then . So the set
is a basis of . Here
We have . Let us define the map by
for and . Then one can easily check that is a dimensional linear representation of . We now explicitly compute the matrix representation of with respect to the basis . Let
By setting , we have the following.
Proposition 2.1.
The entry of is
Here are polynomials of , , , with coefficients from . Therefore .
2.2. Induced cohomological FMTs
We now recall some important notions from finite continued fraction theory (see [HW] for further details). Let be a sequence of integers. Define for by
The key result for us is the following standard fact which we reproduce for the reader’s convenience:
Proposition 2.2.
If we write the finite continued fraction by
then , , and .
Let be a dimensional principally polarized abelian variety. The transform is defined by
Here is the FMT from to with the Poincaré line bundle on as its kernel and is .
Proposition 2.3.
The isometric automorphism associated to the FMT is
Proof.
By induction on . ∎
Assume the Picard rank of is one and let be . As usual, we write and so the induced transform on can be expressed as a invertible matrix.
Example 2.4.
The following examples of induced FMTs on are important in this paper. We identify them in matrix form as images of the corresponding isometric automorphisms under as given by Proposition 2.1.





Since is principally polarized, any FMT in is isomorphic to for some sequence of integers , , and . The induced transform on gives a well defined group homomorphism
Therefore, we have and . Also .
For , let , , and . By Proposition 2.3, the induced transform on is and its entry is given explicitly in Proposition 2.1.
Up to shift, any nontrivial FMT in is isomorphic to some FMT with a universal bundle on as the kernel. Therefore for some such that and for any .
Example 2.5.
For the case
Example 2.6.
For the case
Let us simply denote the twisted Chern character by . Since , we have
Since , we obtain the following presentation.
Theorem 2.7.
Remark 2.8.
As a result of this theorem, we can see that the induced transform on of any nontrivial FMT with respect to the appropriate twisted Chern characters looks somewhat similar to the induced transform of on with usual Chern characters.
3. Action of FMTs on Stability Conditions
A Bridgeland stability condition on a triangulated category consists of a stability function together with a slicing of satisfying certain axioms. Equivalently, one can define by giving a bounded tstructure on together with a stability function on the corresponding heart satisfying the HN property. Then is usually written as the pair or . See [Bri1], [Huy3] or [BBR, Appendix D] for further details. Let and let be a group homomorphism. Then
defines a left action of the group on . Moreover this can be extended to the natural left action of on the space of all stability conditions on by defining .
Let be a principally polarized dimensional abelian variety with Picard rank one and let be . Then the Todd class of is trivial and so for any object in the Mukai vector is the Chern character. Any complexified class in is of the form for some , where . Assume . Define the function by
If we denote the Mukai pairing on by then . It is expected that is a central charge of some stability condition on (see [BMT, Conjecture 2.1.2], [Pol]). This is already known to be true for completely, and the authors proved the case of , for in [MP, Theorem 3.3].
Let be a nontrivial FMT with kernel the universal bundle on . From the previous section, the induced transform on is for some satisfying and . Also