FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS ON
ABELIAN VARIETIESDULIP PIYARATNE
Abstract. This article is based on a talk given at the Kinosaki Symposium on Algebraic
Geometry in 2015, about a work in progress [Piy2]. We describe a polarization on a derived
equivalent abelian variety by using Fourier-Mukai theory. We explicitly formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces,
which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes theauthor’s previous joint work [MP1, MP2] with A. Maciocia on principally polarized abelian
3-folds with Picard rank one. Consequently, we see that the strong Bogomolov-Gieseker type
inequalities hold for tilt stable objects on abelian 3-folds.

1. Introduction
The notion of Fourier-Mukai transform was introduced by Mukai in early 1980s (see[Muk2]). In particular, he showed that the Poincaré bundle induces a non-trivial equivalence between the derived categories of an abelian variety and its dual variety. Furthermore,
he studied certain type of vector bundles on abelian varieties called semihomogeneous bundles,
and moduli of them (see [Muk1]). In particular, the moduli space parametrizing simple semihomogeneous bundles on an abelian variety Y with a fixed Chern character is also an abelianvariety, denoted by X. Moreover, the associated universal bundle E on X × Y induces a derived
equivalence ΦX→Y
from X to Y, which is commonly known as the Fourier-Mukai transform.
E
2∗This transform induces a linear isomorphism ΦHE from H2∗alg (X, Q) to Halg (Y, Q), called thecohomological Fourier-Mukai transform. In this article, we realize this linear isomorphism inanti-diagonal form with respect to some twisted Chern characters (see Theorem 3.4). This
generalizes similar work on principally polarized abelian varieties with Picard rank one case(see [MP2]). The following is one of the important consequences of this anti-diagonal representation, which generalizes the known result [BL] for the classical Fourier-Mukai transformwith kernel the Poincaré bundle.
Theorem 1.1 ( = 3.2 ). For a ∈ X, b ∈ Y, let Ξ : Db (X) → Db (Y) be the Fourier-Mukai
functor defined by
Ξ = E∗{a}×Y ◦ ΦX→Y
◦ E∗X×{b} ,E
where E∗{a}×Y denotes the functor E∗{a}×Y ⊗ (−) and similar for E∗X×{b} . If the ample line
bundle L defines a polarization on X, then the line bundle det(Ξ(L))−1 is ample and so it
defines a polarization on Y.
Date: December 18, 2015.
2010 Mathematics Subject Classification. Primary 14F05; Secondary 14J30, 14J32, 14J60, 14K99, 18E10,
18E30, 18E40.
Key words and phrases. Fourier-Mukai transforms, Abelian varieties, Derived category, Polarization, Bridgeland stability conditions, Bogomolov-Gieseker inequality.
1

2

DULIP PIYARATNE

Motivated by Douglas’s work on Π-stability for D-branes, Bridgeland introduced the notionof stability conditions on triangulated categories (see [Bri1]). This categorical stability notion
can be interpreted essentially as an abstraction of the usual slope stability for sheaves on
projective curves. The category of coherent sheaves does not arise as a heart of a Bridgelandstability condition for higher dimensional smooth projective varieties. So more work is needed
to construct the hearts for stability conditions on projective varieties of dimension above
one. In general, when B + iω isR a complexified ample class on a projective variety X, it
is expected that ZXB+iω (−) = − X e−B−iω ch(−) defines a central charge function of somestability condition on X. In this article we conjecturally construct a heart for this central
charge function by using the notion of very weak stability condition (see Conjecture 2.3).
This essentially generalizes the single tilting construction of Bridgeland and Arcara-Bertramfor surfaces, and the conjectural double tilting construction of Bayer-Macrı̀-Toda for 3-folds.
Action of the Fourier-Mukai transform ΦX→Y
induces stability conditions on Db (Y) fromE
the ones on Db (X). This can be defined via the induced the map on Hom(K(Y), C) from
Hom(K(X), C) by the transform. For abelian varieties we view this as ΦX→Y
· ZXΩ = ζ ZYΩ 0 forE
0some ζ ∈ C, where Ω, Ω are complexified ample classes on X, Y respectively. When ζ is real
one can expect that the Fourier-Mukai transform ΦX→Y
gives an equivalence of some heartsEof particular stability conditions on X and Y, whose Ω and Ω 0 are determined by Im ζ = 0.
More precisely we formulate the following:
Conjecture 1.2 ( = 4.2 ). The Fourier-Mukai transform ΦX→Y
: Db (X) → Db (Y) betweenE
the abelian varieties gives the equivalence of stability condition hearts each of which are g-foldtilts as in Conjecture 2.3:

X
Y
ΦX→Y
E [k] AΩk = AΩ 0 .
kHere Ωk , Ωk0 are some complexified ample classes on X, Y respectively, 1 6 k 6 (g − 1), andg = dim X = dim Y.
In the last section we establish the above conjecture for abelian surfaces and abelian 3-folds.
In particular, we explicitly give the details for the surface case and our proof is essentiallya generalization of Yoshioka’s in [Yos], and closely follow the one in the author’s PhD thesis[Piy1, Chapter 6]. The equivalences of such stability condition hearts are already known fora principally polarized abelian 3-folds with Picard rank one. Since we have the anti-diagonal
representation of cohomological Fourier-Mukai transform as in Theorem 3.4, essentially thesame arguments work in the general case. In particular, we deduce the following:
Theorem 1.3 ( = 5.2 ). Conjecture 1.2 is true for abelian 3-folds. Moreover, the strong
Bogomolov-Gieseker type inequality introduced in [BMT] holds for abelian 3-folds.
Notation• Unless otherwise stated, throughout this paper, all the varieties are smooth projective anddefined over C. For a variety X, by Coh(X) we denote the category of coherent sheaves on
X, and by Db (X) we denote the bounded derived category of Coh(X).
• For 0 6 i 6 dim X, Coh6i (X) := {E ∈ Coh(X) : dim Supp(E) 6 i}, Coh>i (X) := {E ∈Coh(X) : for 0 6= F ⊂ E, dim Supp(F) > i} and Cohi (X) := Coh6i (X) ∩ Coh>i (X).
• For E ∈ Db (X), E∨ = R Hom(E, OX ). When E is a torsion free sheaf we write its dual byE∗ ; so for a locally free sheaf E, E∨ = E∗ .
• The structure sheaf of a closed subscheme Z ⊂ X as an object in Coh(X) is denoted by OZ
and when Z = {x} for some closed point x ∈ X it is simply denoted by Ox .

FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS

3• When A is the heart of a bounded t-structure on a triangulated category D, by HiA (−) wedenote the corresponding i-th cohomology functor.
• For a set of objects S ⊂ D in a triangulated category D, by hSi ⊂ D we denote its extension
closure, that is the smallest extension closed subcategory of D which contains S.
• We denote the upper half plane {z ∈ C : Im z > 0} by H.
• We will denote an m × m anti-diagonal matrix with entries ak , k = 1, . . . , m by
ak if i = k, j = m + 1 − kAdiag(a1 , . . . , am )ij :=0otherwise.
2. Preliminaries
2.1. Very weak stability conditions. Let us recall the general arguments of very weakstability conditions, which are the variants of weak stability of [Tod2] introduced in [BMS,
Definition B.1]. We closely follow the notions as in [PT].
Let D be a triangulated category, and K(D) its Grothendieck group. We fix a finitely
generated free abelian group Γ and a group homomorphism
cl : K(D) → Γ .
Definition 2.1. A very weak stability condition on D is a pair (Z, A), where A is the heart ofa bounded t-structure on D, and Z : Γ → C is a group homomorphism satisfying the following
conditions:(i) For any E ∈ A, we have Z(E) ∈ H ∪ R60 . Here and elsewhere we write Z(E) =Z(cl(E)), and H is the upper half plane.
(ii) The associated slope function µ : A → R ∪ {+∞} is defined by+∞if Im Z(E) = 0µ(E) =Re Z(E)− Im Z(E) otherwise,
and it satisfies the Harder-Narasimhan property.
We say that E ∈ A is µ-(semi)stable if for any non-zero subobject F ⊂ E in A, we have theinequality: µ(F) < (6) µ(E/F).
The Harder-Narasimhan filtration of an object E ∈ A is a chain of subobjects 0 = E0 ⊂E1 ⊂ · · · ⊂ En = E in A such that each Fi = Ei /Ei−1 is µ-semistable with µ(Fi ) > µ(Fi+1 ).
If such Harder-Narasimhan filtrations exists for all objects in A, we say that µ satisfies theHarder-Narasimhan property.
For a given a very weak stability condition (Z, A), we define its slicing on D (see [Bri1,
Definition 3.3]){P(φ)}φ∈R , P(φ) ⊂ D
as in the case of Bridgeland stability conditions (see [Bri1, Proposition 5.3]). Namely, for
0 < φ 6 1, the category P(φ) is defined to beP(φ) = {E ∈ A : E is µ-semistable with µ(E) = −1/ tan(πφ)} ∪ {0}.
Here we set −1/ tan π = ∞. The other subcategories are defined by settingP(φ + 1) = P(φ)[1].

4

DULIP PIYARATNE

For an interval I ⊂ R, we define P(I) to be the smallest extension closed subcategory of
D which contains P(φ) for each φ ∈ I. For 0 6 s 6 1, the pair (P((s, 1]), P((0, s])) of
subcategories of A = P((0, 1]) is a torsion pair, and the corresponding tilt is P((s, s + 1]).
Note that the category P(1) contains the following category
C := {E ∈ A : Z(E) = 0}.
It is easy to check that C is closed under subobjects and quotients in A. In particular, C isan abelian subcategory of A. Moreover, the pair (Z, A) gives a Bridgeland stability condition
on D if C = {0}.
2.2. Bridgeland stability on varieties. Let X be a smooth projective variety and let Db (X)be the bounded derived category of coherent sheaves on X. A stability condition σ on Db (X)is numerical if the central charge function Z : K(X) → C factors through the Chern character
map ch : K(X) → H2∗
alg (X, Q).
Usually, a stability condition σ on Db (X) is called geometric if all the skyscraper sheaves
Ox of x ∈ X are σ-stable of the same phase. The following result gives some properties ofgeometric stability conditions on varieties.
Proposition 2.2. Let X be a smooth projective variety of dimension n. Let σ = (Z, A) be ageometric stability condition on Db (X) with all skyscraper sheaves Ox of x ∈ X are σ-stablewith phase one. If E ∈ A then HiCoh(X) (E) = 0 for i ∈/ {−n + 1, −n + 2, . . . , 0}.
Proof. The following proof is adapted from [Bri2, Lemma 10.1]. Let P be the corresponding
slicing of σ. Since A = P((0, 1]) and Coh0 (X) ⊂ P(1), from the Harder-Narasimhan property,
we only need to consider E ∈ A such that HomX (Coh0 (X), E) = 0. For any skyscraper sheaf
Ox of x ∈ X we have Ox [i] ∈ P(1 + i) and E[i] ∈ P((i, 1 + i]). Therefore, for all i < 0,
∼ HomX (E, Ox [n − 1 − i])∗ = 0. So by [BM,HomX (E, Ox [i]) = 0, and HomX (Ox , E[1 + i]) =Proposition 5.4], E is quasi-isomorphic to a complex of locally free sheaves of length n. This
completes the proof as required.
When X is a smooth projective curve, the central charge function Z(E) = − deg(E) +i rk(E) together with the heart Coh(X) of the standard t-structure defines a geometric stability
condition on Db (X). However, for a smooth projective variety X with dim X > 2, there isno numerical stability condition on Db (X) with Coh(X) as the heart of a stability condition(see [Tod1, Lemma 2.7] for a proof). Motivated by the constructions for smooth projective
surfaces (see [Bri2, AB]) together with some observations in Mathematical Physics, for any
dimensional smooth projective variety X, it is expected that the function defined byZ
ZB+iω (−) = − e−B−iω ch(−)
X

is a central charge function of some geometric stability condition on Db (X) (see [BMT, Conjecture 2.1.2]). Here B + iω ∈ NSC (X) is a complexified ample class on X, that is by definitionB, ω ∈ NSR (X) with ω an ample class. By using the notion of very weak stability, let us
conjecturally construct a heart for this central charge function.
Let n = dim X. For 0 6 k 6 n, we define the k-truncated Chern character by
ch6k (E) = (ch0 (E), ch1 (E), . . . , chk (E), 0, . . . , 0),FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS

5(k)and the function ZB+iω : K(X) → C by(k)
ZB+iω (E) = −in−kZX

e−B−iω ch6k (E).
(1)

The usual slope stability on sheaves gives the very weak stability condition (ZB+iω , Coh(X)).
Moreover, we formulate the following:
(k)

(k)

Conjecture 2.3 ([Piy2]). For each 1 6 k < n, the pair σk = (ZB+iω , AB+iω ) gives a very(k)

weak stability condition on Db (X), where the hearts AB+iω , 1 6 k 6 n are defined by
(1)
AB+iω = Coh(X)
.
(k+1)
A
= P ((1/2, 3/2]) 
B+iω

σk

(n)

Moreover, the pair σn = (ZB+iω , AB+iω ) is a Bridgeland stability condition on Db (X).
(1)

(1)

Remark 2.4. As mentioned before, (Zω,B , AB+iω = Coh(X)) is a very weak stability
condition for any variety and a Bridgeland stability condition for curves. By [Bri2, AB],
(2)
(2)
(Zω,B , AB+iω ) is a Bridgeland stability condition for surfaces. In [BMT], the authors proved
(2)

(2)

that the pair (ZB+iω , AB+iω ) is again a very weak stability condition for 3-folds. Here the(1)stability was called tilt slope stability. The usual Bogomolov-Gieseker inequality for ZB+iωstable sheaves plays a crucial role in these proofs. Clearly the same arguments work for(3)
any higher dimensional varieties. Therefore, we can always construct the category AB+iω
when dim X > 2. In [BMT], the authors conjectured that this category is a heart of a
Bridgeland stability condition with the central charge ZB+iω . Moreover, they reduced it(2)(2)
to prove Bogomolov-Gieseker type inequalities for ZB+iω stable objects E ∈ AB+iω with
(2)

Re ZB+iω (E) = 0:
ω2 B
ch (E).
18 1
So far this conjectural inequality is known to hold for some Fano 3-folds [BMT, Mac, Sch, Li],abelian 3-folds [MP1, MP2, Piy1, BMS] and the étale quotients of abelian 3-folds [BMS].
chB
3 (E) 6(1)

2.3. Fourier-Mukai theory. Let us quickly recall some of the important notions in FourierMukai theory. Further details can be found in [Huy2].
Let X, Y be smooth projective varieties and let pi , i = 1, 2 be the projection maps fromX × Y to X and Y, respectively. The Fourier-Mukai functor ΦX→Y
: Db (X) → Db (Y) withE
kernel E ∈ Db (X × Y) is defined by
L

X→Y
ΦE
(−) = Rp2∗ (E ⊗ p∗1 (−)).

The map Σ : Y × X → X × Y is defined by Σ(y, x) = (x, y) for any x ∈ X and y ∈ Y.
L

L

Let EL = Σ∗ (E∨ ⊗ p∗2 ωY ) [dim Y], and ER = Σ∗ (E∨ ⊗ p∗1 ωX ) [dim X]. Then we have the
X→Y
Y→X
.
adjunctions (see [Huy2, Proposition 5.9]): ΦY→Xa ΦE
EL a ΦE
R
X→Y
When ΦE is an equivalence of the derived categories, usually it is called a Fourier-Mukai
transform. On the other hand by Orlov’s Representability Theorem (see [Huy2, Theorem

6

DULIP PIYARATNE

5.14]), any equivalence between Db (X) and Db (Y) is isomorphic to a Fourier-Mukai transform
X→YΦE
for some E ∈ Db (X × Y).
X→Y
Any Fourier-Mukai functor ΦE: Db (X) → Db (Y) induces a linear map ΦHE : H2∗
alg (X, Q) →2∗
Halg (Y, Q), usually called the cohomological Fourier-Mukai functor, and it is an isomorphism
X→Y
when ΦEis a Fourier-Mukai transform. The induced transform fits into the following
commutative diagram, due to the Grothendieck-Riemann-Roch theorem.
Db (X)ΦX→YE/ Db (Y)

[−]

[−]



K(X)
vX (−)

H2∗alg (X, Q)

ΦK
E

ΦH
E



/ K(Y)vY (−)/ H2∗ (Y, Q)
alg√
Here vZ (−) = ch(−) tdZ is the Mukai vector map, where ch : K(Z) → H2∗
alg (Z, Q) is the
Chern character map and tdZ is the Todd class of Z.
Pdim X
2i
Let v ∈ H2∗
i=0 vi for vi ∈ Halg (X, Q) and thealg (X, Q) be a Mukai vector. Then v =PXi
Mukai dual of v is defined by v∗ = dim
i=0 (−1) vi . A symmetric bilinear form h−, −iX called
Mukai pairing is defined by the formulaZhv, wiX = − v∗ · w · ec1 (X)/2 .
X

Note that for an abelian variety X, tdX = 1 and c1 (X) = 0. Hence the Mukai vector v(E)of E ∈ Db (X) is the same as its Chern character ch(E).
2∗Due to Mukai and Căldăraru-Willerton, for any u ∈ H2∗
alg (Y, Q) and v ∈ Halg (X, Q) we have
(2)

ΦHEL (u) , v X = hu , ΦHE (v)iY(see [Huy2, Proposition 5.44], [CW]).
2.4. Abelian varieties. Over any field, an abelian variety X is a complete group variety,that is X is an algebraic variety equipped with the maps m : X × X → X, (x, y) 7→ x + y (thegroup law), and (−1) : X → X, x 7→ −x (the inverse map), together with the identity elemente ∈ X. For a ∈ X, the morphism ta : X → X is defined by ta = m(−, a) : x 7→ x + a. Overthe field of complex numbers, an abelian variety is a complex torus with the structure of a
projective algebraic variety.
Let Pic0 (X) be the subgroup of the abelian group Pic(X) consisting of elements represented
by line bundles which are algebraically equivalent to zero, and the corresponding quotient
Pic(X)/ Pic0 (X) is the Néron-Severi group NS(X). The group Pic0 (X) is naturally isomorphicb
to an abelian variety called the dual abelian variety of X, denoted by X.
b is the uniquely determined line bundle
The Poincaré line bundle P on the product X × X∼ b . In [Muk2],b and (ii) P
satisfying (i) PX×{bx} ∈ Pic(X) is represented by bx ∈ X,
b = OX{e}×Xb
Xb is an equivalence of: Db (X) → Db (X)Mukai proved that the Fourier-Mukai functor ΦX→P
the derived categories, that is a Fourier-Mukai transform.

FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS

7∼ E for all
A vector bundle E on an abelian variety X is called homogeneous if we have t∗x E =x ∈ X. A vector bundle E on X is homogeneous if and only if E can be filtered by line bundles
from Pic0 (X) (see [Muk1]). We call a vector bundle E is semihomogeneous if for every x ∈ X∼ E ⊗ PX×{bx} . A vector bundle E isthere exists a flat line bundle PX×{bx} on X such that t∗x E =∼ C. A rank r simple semihomogeneous bundle E has thecalled simple if we have EndX (E) =Chern character
ch(E) = r ec1 (E)/r .

(3)

Furthermore, due to Mukai, for any DX ∈ NSQ (X), there exists simple semihomogeneous
bundles E on X with ch(E) = r eDX for some r ∈ Z>0 . See [Orl] for further details.
b
X
The image of an ample line bundle L on X under the Fourier-Mukai transform ΦX→
is
P
b
X
∼b
ΦX→
P (L) = L

b Moreover, −c1 (L)b is an ample
for some rank χ(L) = c1 (L)g /g! semihomogeneous bundle L.
bdivisor class on X (see [BL]). Therefore, we have the following:
Lemma 2.5. Let `X ∈ NSQ (X) be an ample class on X. Under the induced cohomologicalb
X→Xtransform ΦHP of ΦP
we have−`Xb
ΦHP (e`X ) = (`g
X /g!) eb satisfying
for some ample class `Xb ∈ NSQ (X),g(`g
X /g!)(`Xb /g!) = 1.

Moreover, for each 0 6 i 6 g,
 i`(−1)g−i `g
X g−iΦP X =`.
i!
g!(g − i)! Xb
H

3. Cohomological Fourier-Mukai Transforms
Let Y be a g-dimensional abelian variety and let DY ∈ NSQ (Y). Let X be the fine moduli
space of rank r simple semihomogeneous bundles E on Y with c1 (E)/r = DY . Due to Mukai
X is a g-dimensional abelian variety. Let E be the associated universal bundle on X × Y; soby (3) we havech(E{x}×Y ) = r eDY .
Let ΦX→Y: Db (X) → Db (Y) be the corresponding Fourier-Mukai transform from Db (X) toEb
Y→X
D (Y) with kernel E. See [Orl] for further details. Then its quasi inverse is given by ΦΣ∗ E∨ [g].
Again, by (3) we havech(EX×{y} ) = r eDX
for some DX ∈ NSQ (X).
A polarization on X is by definition the first Chern class c1 (L) of an ample line bundle L
on X. However, it is usual to say the line bundle L itself a polarization.
b defined
Let a ∈ X and b ∈ Y. Consider the Fourier-Mukai functor Γ from Db (X) to Db (Y)bybYΓ = ΦY→◦ E∗{a}×Y ◦ ΦX→Y
◦ E∗X×{b} [g],PE

8

DULIP PIYARATNEb →Γ : Db (Y)where E∗{a}×Y denotes the functor E{a}×Y ⊗∗ (−) and similar for E∗X×{b} . Let b
Db (X) be the Fourier-Mukai functor defined byb
Y→YbΓ = EX×{b} ◦ ΦY→XΣ∗ E∨ ◦ E{a}×Y ◦ ΦΣ∗ P∨ [g].

Then bΓ and Γ are adjoint functors to each other. By direct computation, Γ (Ox ) = OZxb and Γ (Oyb ) = OZ for some 0-subscheme Zyb ⊂ X; where
for some 0-subscheme Zx ⊂ Y,bythe lengths of Zx and Zyb are r3 and r respectively. Therefore, the Fourier-Mukai kernel∼ Extg (F, O b )[−g]. So Γ (Cohi (X)) ⊂ Cohi (Y)b with F∨ =b
F of Γ is F ∈ Cohg (X × Y),
X×Yb ⊂ Cohi (X) for all i. Also by direct computation, Γ (OX ) and b
and b
Γ (Cohi (Y))
Γ (O b ) areYhomogeneous bundles of rank r and r3 respectively. Let `X ∈ NSQ (X) be an ample class.
Under the induced cohomological map Γ H we have
Γ H (e`X ) = r e`Yb ,gb satisfying r2 `g
for some class `Yb ∈ NSQ (Y)X = `Yb . So we haveb
Γ H (e`Yb ) = r3 e`X .
Moreover, for each 0 6 i 6 g,
Γ H (`iX ) = r `iYb ,b
Γ H (`iYb ) = r3 `iX .

b By Theorem
By the Nakai–Moishezon criterion, one can show `Yb is an ample class on Y.
b
Y→Y
2.5, under the induced cohomological map of ΦΣ∗ P∨ we have−`YΦHΣ∗ P∨ (e`Yb ) = (`g,b /g!) eYfor some ample class `Y ∈ NSQ (Y).
Let Ξ : Db (X) → Db (Y) be the Fourier-Mukai functor defined by
Y→YΞ = E∗{a}×Y ◦ ΦX→Y
◦ E∗X×{b} = ΦΣ∗ P∨ ◦ Γ .
Eb−`Y . Therefore,
The image of e`X under its induced cohomological transform ΞH is (r3 `g
X /g!) e
we deduce the following.

Theorem 3.1 ([Piy2]). If `X ∈ NSQ (X) is an ample class then−`Ye−DY ΦHE e−DX (e`X ) = (r `g,X /g!) eg2
for some ample class `Y ∈ NSQ (Y), satisfying (`gMoreover, for each
X /g!)(`Y /g!) = 1/r .
0 6 i 6 g,
 i(−1)g−i r `g
`X
X g−i
−DY
H −DX
=
`
.
e
ΦE e
i!
g!(g − i)! Y

Theorem 3.2 ([Piy2]). If the ample line bundle L defines a polarization on X, then the line
bundle det(Ξ(L))−1 is ample and so it defines a polarization on Y.
Let us introduce the following notation:
X
Notation 3.3. Let B, `X ∈ NSQ (X). For E ∈ Db (X), the entries vB,`
(E), i = 0, . . . , g areidefined by
XvB,`(E) = i! `g−i· chB
Xi (E).
i

FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS

9B
Here chBi (E) is the i-th component of the B-twisted Chern character ch (E) defined bychB (E) = e−B ch(E). The vector vB,`X (E) is defined by


B,`X
X
vB,`X (E) = vB,`
(E),
.
.
.
,
v
(E)
.
g
0

Theorem 3.4 ([Piy2]). If we consider v−DX ,`X , vDY ,`Y as column vectors, thenvDY ,`Y (ΦX→Y
E (E)) =g!
g−1, (−1)g v−DX ,`X (E).
g Adiag 1, −1, . . . , (−1)
r `X

4. Stability Conditions Under Fourier-Mukai Transforms
This section generalizes some of the similar results in [MP2, Piy1]. Recall that a Bridgelandstability condition σ on a triangulated category D consists of a stability function Z together
with a slicing P of D satisfying certain axioms. Equivalently, one can define σ by giving abounded t-structure on D together with a stability function Z on the corresponding heart A
satisfying the Harder-Narasimhan property. Then σ is usually written as the pair (Z, P) or(Z, A).
Let Υ : D → D 0 be an equivalence of triangulated categories, and let W : K(D) → C be agroup homomorphism. Then(Υ · W) ([E]) = W [Υ−1 (E)]defines an induced group morphism Υ · W in Hom(K(D 0 ), C) by the equivalence Υ. Moreover,this can be extended to a natural induced stability condition on D 0 by defining Υ · (Z, A) =(Υ · Z, Υ(A)).
Let X, Y be two derived equivalent g-dimensional abelian varieties as in Section 3, which is
given by the Fourier-Mukai transform ΦX→Y
: Db (X) → Db (Y). Let `X ∈ NSQ (X) be an ample
Eclass on X and let `Y ∈ NSQ (Y) be the induced ample class on Y as in Theorem 3.1.
Let u = λeiα be a complex number. Consider the function ZX−DX +u`X : K(X) → C definedbyZ
ZX−DX +u`X (E) = − e−(−DX +u`X ) ch(E).
X

For E ∈ Db (Y) we have
D

E

−1
X→Y
)
ΦE
· ZX−DX +u`X (E) = e−DX +u`X , ch (ΦX→Y
(E)
E
X
D
E
−DX +u`X
H −1= e, (ΦE ) (ch(E))
XH−DX +u`X= ΦE e, ch(E) Y u`
DY−DY H −DX= e
ΦE e(e X ), ch(E) YeDEg= (r `geDY −`Y /u , ch(E) ,X u /g!)Yg−`Y /u .
since by Theorem 3.1, e−DY ΦHE e−DX (eu`X ) = (r `g
X u /g!) e

So we have the following

relation:

X→Yg
Lemma 4.1. We have ΦE· ZX−DX +u`X = ζ ZYDY −`Y /u , for ζ = r `g
X u /g!.

10

DULIP PIYARATNE

Assume there exist a stability condition for any complexified ample class −DX + u`X with aheart AX−DX +u`X and a slicing PX−DX +u`X associated to the central charge function ZX−DX +u`X .



X
P
(φ)
⊂ R>0 eiπφ ; that is
From Lemma 4.1 for any φ ∈ R, ζ ZYDY −`Y /u ΦX→Y
−DX +u`XEX→YZYDY −`Y /u ΦE
PX−DX +u`X (φ) ⊂ R>0 ei(πφ−arg(ζ)) .
So we would expect
X→Y

ΦE



arg(ζ)P−DX +u`X (φ) = PDY −`Y /u φ −
,
π


X

Y

and soarg(ζ)
arg(ζ)−
ΦE
P−DX +u`X ((0, 1]) = PDY −`Y /u
,−
+1 .
π
π
E
D
For 0 6 α < 1, PYDY −`Y /u ((α, α + 1]) = PYDY −`Y /u ((0, α]) [1], PYDY −`Y /u ((α, 1]) is a tilt of
AYDY −`Y /u = PYDY −`Y /u ((0, 1]) associated to a torsion theory coming from ZYDY −`Y /u stability.

X
Y
Therefore, one would expect ΦX→YA
−DX +u`X is a tilt of ADY −`Y /u associated to a torsion
Etheory coming from ZYDY −`Y /u stability, up to shift.
Moreover, for the Fourier-Mukai transform ΦX→Y
when ζ is real, that is, ug ∈ R, we wouldE
expect the following equivalence:
X→Y

X



Y

Conjecture 4.2 ([Piy2]). The Fourier-Mukai transform ΦX→Y
: Db (X) → Db (Y) gives theEequivalence of stability condition hearts conjecturally constructed in Conjecture 2.3:

X
Y
ΦX→Y
E [k] AΩk = AΩ 0 .
k0−ikπ/g ` /λ are complexified ample classes on
Here Ωk = −DX
X and Ωk = DY − eY
X and Y respectively, for k ∈ {1, 2, . . . , (g − 1)}, and some λ ∈ R>0 .

+ λeikπ/g `5. Equivalences of Stability Condition Hearts
The main aim of this section is to show that Conjecture 4.2 is true for abelian surfaces andabelian 3-folds. In order to achieve this we study various stabilities of sheaves and complexesof them under the Fourier-Mukai transforms. This generalizes author’s joint work [MP1, MP2]
with A. Maciocia.
5.1. Abelian surfaces. Let X, Y be derived equivalent abelian surfaces and let `X , `Y be
ample classes on them respectively as in Theorem 3.1. Let Ψ be the Fourier-Mukai transformX→Yb = ΦY→X
ΦEfrom X to Y with kernel E, and let Ψ. We haveΣ∗ E∨
Ψ(Coh(X)) ⊂ hCoh(Y), Coh(Y)[−1], Coh(Y)[−2]i,∼ [−2] and Ψ ◦ Ψ∼ [−2], we have the followingb Since Ψb◦Ψ =b =and similar relation for Ψ.
convergence of the spectral sequences.
(4)

(5)


b p Ψq (E) =⇒ Hp+q−2 (E) 
Ep,q
=Ψ
2
Coh(X)b q (E) =⇒ Hp+q−2 (E) Ep,q= Ψp Ψ
2
Coh(Y).

qqb p (E) = Hpb
Here and elsewhere we write ΨCoh(X) (Ψ(E)) and Ψ (E) = HCoh(Y) (Ψ(E)). Immediately from the convergence of this spectral sequence for E ∈ Coh(X), we deduce that

FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS

11b 0 (E) ∈ Coh(X)[−2], and ΨΨb 2 (E) ∈ Coh(X);• ΨΨ01b Ψ (E) ,→ Ψ
b 2 Ψ0 (E), and a surjection Ψ
b 0 Ψ2 (E)  Ψ
b 2 Ψ1 (E).
• there is an injection Ψ
Let us recall the notation in Conjecture 4.2 for our derived equivalent abelian surfaces.
Consider the complexified ample classes Ω = −DX + iλ`X , Ω 0 = DY + i`Y /λ on X, Y reR(1)
spectively. The function defined by ZΩ = −i X e−Ω (ch0 , ch1 , 0) together with the standard
heart Coh(X) defines a very weak stability condition σ1 on Db (X). Define the subcategories
FX = PXσ1 ((0, 1/2]), T X = PXσ1 ((1/2, 1])of Coh(X) in terms of the associated slicing PXσ1 . Then the Bridgeland stability condition
heart in Conjecture 4.2 isAX = hFX [1], T X i = PXσ1 ((1/2, 3/2]).
We consider similar subcategories associated to Ω 0 on Y.
Let us sketch the proof of the equivalence in Conjecture 4.2. This is essentially a generalization of Yoshioka’s in [Yos], and closely follow the one in the author’s PhD thesis [Piy1,
Section 6].
We need the following results about cohomology sheaves of the images under the FourierMukai transforms.
Proposition 5.1 ([Piy2]). Let E ∈ Coh(X). Then we have the following:
1. (i) If E ∈ T X then Ψ2 (E) = 0, and (ii) if E ∈ FX then Ψ0 (E) = 0.
2. (i) Ψ2 (E) ∈ T Y , and (ii) Ψ0 (E) ∈ FY .
3. (i) if E ∈ T X then Ψ1 (E) ∈ T Y , and (ii) if E ∈ FX then Ψ1 (E) ∈ FY .
In other words, the results of the above proposition say
Ψ(T X ) ⊂ hFY , T Y [−1]iΨ(FX ) ⊂ hFY [−1], T Y [−2]i.

b Since AX = hFX [1], T X i and AY = hFY [1], T Y i, we have Ψ[1](AX ) =Similar results hold for Ψ.
Y
A as required.
5.2. Abelian 3-folds. Let X, Y be derived equivalent abelian 3-folds and let `X , `Y be ample
classes on them respectively as in Theorem 3.1. Let Ψ be the Fourier-Mukai transform ΦX→YE∼∼
b = ΦY→Xbbfrom X to Y with kernel E, and let Ψ
.
Then
Ψ
◦
Ψ
[−3]
and
Ψ
◦
Ψ
[−3].
=
=
Σ∗ E∨
Let us recall the notation in Conjecture 4.2 for our derived equivalent abelian 3-folds.
Consider the complexified ample classes√Ω = (−DX + λ`X /2) + i 3λ`X /2√Ω 0 = (DY − `Y /(2λ)) + i 3`Y /(2λ)R(1)
on X, Y respectively. The function defined by ZΩ = X e−Ω (ch0 , ch1 , 0, 0) together withthe standard heart Coh(X) defines a very weak stability condition σ1 on Db (X). Define thesubcategories
F1X = PXσ1 ((0, 1/2]), T1X = PXσ1 ((1/2, 1])of Coh(X) in terms of the associated slicing PXσ1 ; and the corresponding tilt be
BX = hF1X [1], T1X i = PXσ1 ((1/2, 3/2]).

12

DULIP PIYARATNE

Then as mentioned in Remark 2.4, due to [BMT] the function defined byZ(2)ZΩ = −i e−Ω (ch0 , ch1 , ch2 , 0)X

together with the heart B defines a very weak stability condition σ2 on Db (X). Define thesubcategories
F2X = PXσ2 ((0, 1/2]), T2X = PXσ2 ((1/2, 1])X

of BX in terms of the associated slicing PXσ2 . Then the expected Bridgeland stability condition
heart in Conjecture 2.3 isAX = hF2X [1], T2X i = PXσ2 ((1/2, 3/2]).
We consider similar subcategories on Y associated to Ω 0 .
Under the Fourier-Mukai transform Ψ, for any E ∈ Coh(X), HiCoh(Y) (Ψ(E)) = 0 wheni 6∈ {0, 1, 2, 3}.
We can visualize BX and BY as follows:A

BX = hF1X [1], T1X i :A∈T1X , B∈F1X

B

−1

0
C

BY = hF1Y [1], T1Y i :C∈T1Y , D∈F1Y

D

−1

0

Following similar arguments in [MP1, MP2, Piy1], one can prove
H0Coh(Y) (Ψ(F1X )) ⊂ F1Y , H3Coh(Y) (Ψ(F1X )) ⊂ T1Y.

H3Coh(Y) (Ψ(T1X )) = {0}, H2Coh(Y) (Ψ(T1X )) ⊂ T1Y
That is Ψ (F1X ) [1] ⊂ hBY , BY [−1], BY [−2]i,
=

Ψ
B
−1

B0
Ψ
−1

0

B1
Ψ

B2
Ψ

0

1

B3
Ψ

2

3

BiΨ =HiCoh(Y) (Ψ(B))

and Ψ (T1X ) ⊂ hBY , BY [−1], BY [−2]i.
A

Ψ
−1

=
0

−1

A0
Ψ

A1
Ψ

0

1

A2
Ψ

2

AiΨ =HiCoh(Y) (Ψ(A))

Hence,
Ψ (BX ) ⊂ hBY , BY [−1], BY [−2]i.

3FOURIER-MUKAI TRANSFORMS AND STABILITY CONDITIONS

13

Similarly, one can prove
Ψ[1](BY ) ⊂ hBX , BX [−1], BX [−2]i.
bTherefore, under the Fourier-Mukai transforms Ψ and Ψ[1],the images of the complexes inXY
the subcategories B and B behave somewhat like the images of sheaves on abelian surfaces
under the Fourier-Mukai transform (see (4)). Also as similar to [MP1, MP2, Piy1], one canproveΨ(T2X ) ⊂ hF2Y , T2Y [−1]i,Ψ(F2X ) ⊂ hF2Y [−1], T2Y [−2]iband similar relations for Ψ[1].
Since AX = hF2X [1], T2X i and AY = hF2Y [1], T2Y i we have Ψ[1](AX ) =Y
A as required.
Consequently, by similar arguments as in [MP1, MP2, Piy1] for principally polarized abelian
3-folds with Picard rank one, one can prove the strong Bogomolov-Gieseker type inequality
introduced in [BMT] holds for both X and Y.
Theorem 5.2 ([Piy2]). Conjecture 4.2 is true for abelian 3-folds. Moreover, the strong
Bogomolov-Gieseker type inequality introduced in [BMT] holds for abelian 3-folds.
Acknowledgements
The author is grateful to Arend Bayer, Tom Bridgeland, Antony Maciocia and Yukinobu
Toda for very useful discussions relating to this work, and also to the organizers of theKinosaki Symposium on Algebraic Geometry 2015. Special thanks go to Antony Maciociafor his guidance given to the author’s doctoral studies. This work is supported by the World
Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
References[AB]

D. Arcara and A. Bertram, Bridgeland-stable moduli spaces for K-trivial surfaces. With an appendix
by M. Lieblich, J. Eur. Math. Soc. 15 (2013), 1–38.
[BMS] A. Bayer, E. Macrı̀, and P. Stellari, The space of stability conditions on abelian threefolds, and on someCalabi-Yau threefolds, preprint, arXiv:1410.1585.
[BMT] A. Bayer, E. Macrı̀, and Y. Toda, Bridgeland stability conditions on 3-folds I: Bogomolov-Gieseker type
inequalities, J. Algebraic Geom. 23 (2014), 117–163.
[BL]
C. Birkenhake, H. Lange, The dual polarization of an abelian variety, Arch. Math. (Basel) 73 (1999),
no. 5, 380 – 389.
[Bri1] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math 166 (2007), 317–345.
[Bri2], Stability conditions on K3 surfaces, Duke Math. J. 141 (2008), 241–291.
[BM] T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for K3 and elliptic fibrations, J. Algebraic
Geom. 11 (2002), no. 4, 629 – 657.
[CW] A. Căldăraru and S. Willerton, The Mukai pairing, I: A categorical approach, New York J. Math. 16(2010), 61 – 98.
[Huy1] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs,
The Clarendon Press, Oxford University Press, Oxford, 2006.
[Huy2], Derived and abelian equivalence of K3 surfaces, J. Algebraic Geom. 17 (2008), no. 2, 375–400.
[Li]
C. Li, Stability conditions on Fano threefolds of Picard number one, preprint, arXiv:1510.04089.
[Muk1] S. Mukai, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ. 18 (1978),no. 2, 239 – 272.
b with its application to Picard sheaves, Nagoya Math. J. 81[Muk2] S. Mukai, Duality between D(X) and D(X)(1981), 153 – 175.
[Mac] E. Macrı̀, A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space, Algebra Number Theory 8 (2014), 173–190.

14

DULIP PIYARATNE[MP1] A. Maciocia and D. Piyaratne, Fourier-Mukai Transforms and Bridgeland Stability Conditions on
Abelian Threefolds, Algebr. Geom. 2 (2015), no. 3, 270 – 297.
[MP2]
, Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds II,
preprint, arXiv:1310.0299v2.
[Orl] D. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them,(Russian) Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 131 – 158; translation in Izv. Math. 66(2002), no. 3, 569 – 594.
[Piy1] D. Piyaratne, Fourier-Mukai Transforms and Stability Conditions on Abelian Threefolds, PhD thesis,
University of Edinburgh (2014), http://hdl.handle.net/1842/9635.
[Piy2], TBA, in preparation.
[PT] D. Piyaratne and Y. Toda, Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas
invariants, preprint, arXiv:1504.01177.
[Sch] B. Schmidt, A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold,
Bull. Lond. Math. Soc. 46 (2014), 915–923.
[Tod1] Y. Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157 – 208.
[Tod2], Curve counting theories via stable objects I: DT/PT correspondence, J. Amer. Math. Soc. 23(2010), 1119–1157.
[Yos] K. Yoshioka, Stability and the Fourier-Mukai transform II, Compos. Math. 145 (2009), no. 1, 112 – 142.
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The Universityof Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583,
Japan.
E-mail address: dulip.piyaratne@ipmu.jp