FOURIER-MUKAI TRANSFORMS FOR QUOTIENT
VARIETIES
TOM BRIDGELAND AND ANTONY MACIOCIA
Abstract. We study Fourier-Mukai transforms for non-singular projective varieties whose canonical bundles have finite order. Our results
lead to new transforms for Enriques and bielliptic surfaces.
1. Introduction
Fourier-Mukai transforms are now well-established as a useful tool for
computing moduli spaces of sheaves on non-singular projective varieties [2],
[9]. Further interest in these transforms has been generated by the appearance of derived categories of coherent sheaves in the homological mirror
symmetry conjecture [8]. Despite this, comparatively few Fourier-Mukai
transforms are known, and it is therefore of considerable interest to construct new examples.
In this paper we study Fourier-Mukai transforms for complex projective
varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Applying our results
to the case of Enriques and bielliptic surfaces, we obtain new examples of
transforms for complex surfaces. These results are used in [4], where we find
all pairs of minimal complex surfaces with equivalent derived categories.
A Fourier-Mukai (FM) transform is an exact equivalence
Φ : D(Y ) −→ D(X)
between the bounded derived categories of coherent sheaves on two nonsingular projective varieties X and Y . A result of D. Orlov [13] states that
for any such equivalence Φ there is an object P of D(Y × X) such that
L
Φ(−) ∼
= RπX,∗ (P ⊗ πY∗ (−)),
π
π
Y
X
where Y ←−
Y × X −→
X are the projection maps.
Suppose X is a non-singular complex projective variety whose canonical
bundle ωX has finite order n. There is a finite unbranched cover of X by
e with trivial canonical bundle, and X is
a non-singular projective variety X
e
the quotient of X by an action of the cyclic group of order n. We shall refer
e → X as the canonical cover of X.
to the quotient morphism pX : X
If Y is another non-singular projective variety, and
Φ : D(Y ) −→ D(X)
is a FM transform, we shall show that ωY also has order n, and that if
pY : Ye → Y is the canonical cover of Y , then there is a Zn -equivariant FM
transform
e : D(Ye ) −→ D(X),
e
Φ
1
2
such that the following two squares of functors commute
e
Φ
e
D(
xX)


p∗X 
ypX,∗
Φ
D(X).
e
D(
−→
xY )


p∗Y 
ypY,∗
D(Y )
−→
Conversely, if Ye is a non-singular projective variety, with trivial canonical
bundle, and a free Zn -action with quotient Y , and
e : D(Ye ) −→ D(X)
e
Φ
is an equivariant FM transform, we shall show that the quotient map pY :
Ye → Y is a canonical cover, and that there is a FM transform Φ such that
the diagram above commutes.
Notation. All varieties will be over the complex numbers C. By a sheaf
on a variety X we mean a coherent OX -module, and a point of X always
means a closed (or geometric) point. The structure sheaf of such a point
x ∈ X will be denoted Ox .
The bounded derived category of coherent sheaves on a variety X is denoted D(X). Its objects are complexes of OX -modules with bounded and
coherent cohomology sheaves. The translation (or shift) functor is written
[1], so that the symbol E[m] means the object E of D(X) shifted to the left
by m places.
The canonical bundle of a non-singular projective variety X is denoted
ωX . An object E of D(X) will be called special if E ⊗ ωX ∼
= E and simple
if HomD(X) (E, E) = C.
2. Fourier-Mukai transforms
Throughout this section X and Y will be non-singular projective varieties.
A Fourier-Mukai (FM) transform relating X and Y is an exact equivalence
of categories
Φ : D(Y ) −→ D(X).
Here exact means commuting with the translation functors and taking triangles to triangles. In this section we list some basic properties of FM transforms. We start with the following simple consequence of Serre duality. For
a proof see [4, Lemma 2.1].
Lemma 2.1. If there is a FM transform D(Y ) → D(X) then the canonical
bundles of X and Y have the same order.
¤
Given an object P of D(Y × X) there is an exact functor
ΦP
Y →X : D(Y ) −→ D(X)
defined by the formula
L
∗
ΦP
Y →X (−) = RπX,∗ (P ⊗ πY (−)),
as in the introduction. Functors of this type will be called integral functors.
It is easily checked [10, Proposition 1.3], that the composite of two integral
functors is again an integral functor.
3
It was proved by Orlov [13, Theorem 2.2] that any FM transform is isomorphic to an integral functor. Moreover, the object P of D(Y ×X) defining
the functor is unique up to isomorphism. We refer to it as the kernel of the
transform.
Note that P is always a simple object of D(Y × X). This is because there
is a relative transform
ΦY : D(Y × Y ) −→ D(Y × X)
which takes the structure sheaf of the diagonal ∆ ⊂ Y × Y to the object P.
Thus
HomD(Y ×X) (P, P) = HomD(Y ×Y ) (O∆ , O∆ ) = C.
Note also that there is an isomorphism
(1)
P ⊗ π ∗ (ωX ) ∼
= P ⊗ π ∗ (ωY )
X
Y
because, up to shifts, these objects are the kernels of the left and right adjoint
functors of Φ respectively (see e.g. [3, Lemma 4.5]), which are isomorphic
because Φ is an equivalence.
Suppose one has a FM transform Φ : D(Y ) → D(X) such that for each
point y ∈ Y there is a point f (y) ∈ X with
Φ(Oy ) = Of (y) .
Then f defines a morphism Y → X, and for some line bundle L on Y , there
is an isomorphism of functors
Φ(−) ∼
= f∗ (L ⊗ −).
To see this note that by [3, Lemma 4.3], the kernel P of Φ is a sheaf on
Y × X, flat over Y , such that for each y ∈ Y , Py = Of (y) . But if ∆ ⊂ X × X
denotes the diagonal, then the sheaf O∆ is a universal sheaf parameterising
structure sheaves of points of X. It follows that f is a morphism of varieties,
and
P = (f × idX )∗ (O∆ ) ⊗ πY∗ (L)
for some line bundle L on Y . The claim follows.
Many examples of FM transforms for surfaces are constructed using the
following theorem, which is a simple consequence of the results of [3]. See
[4, Corollary 2.8] for a proof.
Theorem 2.2. Let X be a non-singular projective surface with a fixed polarisation and let Y be a two-dimensional, complete, non-singular, fine moduli
space of stable, special sheaves on X. Then there is a universal sheaf P on
Y × X and the resulting functor Φ = ΦP
¤
Y →X is a FM transform.
We shall need the following well-known observation. Suppose we are in
the situation of Theorem 2.2, and suppose that E is a stable sheaf on X
with the same Chern character as the sheaves Py . We claim that E must
be isomorphic to one of the Py . If not, for each y ∈ Y we must have
HomX (E, Py ) = HomX (Py , E) = 0.
Since Py is special, Serre duality implies that Ext2X (E, Py ) = 0, and since
E has the same Chern character as Py , and Φ is an equivalence,
χ(E, Py ) = χ(Py , Py ) = χ(Φ(Oy ), Φ(Oy )) = χ(Oy , Oy ) = 0,
4
so this is enough to show that HomiX (E, Py ) = 0 for all i. This is impossible
by [3, Example 2.2], because if Ψ is the inverse of Φ,
HomiX (E, Py ) = HomiY (Ψ(E), Oy ).
We conclude this section with a couple of well-known examples of FM
transforms which will be useful later.
Example 2.3. The first example of a FM transform for a K3 surface was
the reflection functor of [11], although Mukai never explicitly mentions the
fact that it is an equivalence of derived categories. To construct it, take a
K3 surface X and let P be the ideal sheaf I∆ of the diagonal in X × X. For
any x ∈ X one has Px ∼
= Ix . Theorem 2.2 shows that the functor ΦP
X→X is
a FM transform.
Example 2.4. Let (X, `) be a principally polarised abelian surface, and let
Y be the moduli space of stable sheaves on X of Chern character (4, 2`, 1).
This moduli space is fine, complete and two-dimensional, so there is a universal sheaf P on Y ×X, and the resulting functor ΦP
Y →X is a FM transform.
In fact Y is isomorphic to X. See [9, Proposition 7.1] for details.
3. Equivariant sheaves and cyclic covers
Let G be a finite group acting freely on a non-singular projective variety
e The quotient variety X = X/G
e
X.
is non-singular and projective and there
e
is a quotient morphism p : X → X.
e is a sheaf E
e on X
e together with
A G-equivariant sheaf (or G-sheaf) on X
∗
e
e
isomorphisms λg : E → g (E) for each g ∈ G, such that λ1 = idEe and such
that for any pair g, h ∈ G,
(2)
λhg = g ∗ (λh ) ◦ λg .
e → Fe of the underlying sheaves
A morphism of G-sheaves is a morphism f : E
commuting with the given isomorphisms:
e
e
λFg ◦ f = g ∗ (f ) ◦ λE
g.
It is well-known [5, Chap. V] that the functor p∗ defines an equivalence
e
between the category of sheaves on X and the category of G-sheaves on X.
What we want to do now is to show that a corresponding result holds for
objects of the derived category.
e to be an object E
e together with isoLet us define a G-object of D(X)
∗
e → g (E)
e in D(X)
e for each g ∈ G satisfying the relations
morphisms λg : E
(2).
e of D(X)
e there is an object E of
Proposition 3.1. For every G-object E
e∼
D(X) such that E
= p∗ (E).
Proof. We proceed by induction on the number r of non-zero cohomology
e If r = 1 then up to shifts E
e is a G-sheaf on X
e as above, so we
sheaves of E.
e
may assume r > 1. Shifting E to the left or right we may as well assume
e = 0 for i > 0 and that H 0 (E)
e is non-zero.
that H i (E)
5
Let τ<0 and τ≥0 be the truncation functors of the standard t-structure on
e There is a triangle
D(X).
e −→ E
e −→ τ≥0 (E)
e −→ τ<0 (E)[1],
e
τ<0 (E)
e
e
E
e = τ≥0 (E)
e = H 0 (E)
e is a sheaf. Putting λA
e
where A
g = τ≥0 (λg ) makes A into
∗
e = p A. Similarly τ<0 (E)
e is a
a G-sheaf, so there is a sheaf A on X with A
G-object, and since it has r − 1 non-zero cohomology sheaves, we may apply
our induction hypothesis and conclude that it is the pull-back of an object
F of D(X). Thus there is a triangle
˜
f
e −→ p∗ A −→
p∗ F [1].
p∗ F −→ E
We claim that there is a morphism f : A → F [1] in D(X) such that f˜ =
e∼
p∗ (f ). Assuming this, note that the cone E of f then satisfies E
= p∗ E and
we are done.
To prove the claim replace F by an injective resolution
d−1
d0
· · · −−−−→ I −1 −−−−→ I 0 −−−−→ I 1 −−−−→ · · · ,
as in [6, Lemma I.4.6] and consider the map
α : HomXe (p∗ A, p∗ I 0 ) −→ HomXe (p∗ A, p∗ I 1 ).
The group G acts on both of these vector spaces and α is equivariant.
Thus each space decomposes as a direct sum over the irreducible representations of G and α respects these decompositions.
The complex p∗ I • is an injective resolution of p∗ F so the morphism f˜ is
represented by an element v ∈ HomXe (p∗ A, p∗ I 1 ). Decomposing v we can
write v = v0 + v1 where v0 is fixed by the action of G and v1 is in the span
of the other eigenspaces. The fact that f˜ is equivariant means that v1 lies in
the image of α. Thus f˜ is also represented by v0 . But then f˜ is represented
by an equivariant element of HomXe (p∗ A, p∗ I 1 ) and hence is equal to p∗ (u)
for some u ∈ HomX (M, I 1 ).
¤
The group actions we consider in this paper arise from the following standard cyclic cover construction.
Proposition 3.2. Let X be a non-singular projective variety with a line
bundle L of finite order n. Then there is a non-singular projective variety
e and an étale cover p : X
e → X of degree n, such that
X
(3)
p∗ (OXe ) =
n−1
M
L⊗i .
i=0
e is uniquely defined up to isomorphism, and there is a free
Furthermore, X
e such that X = X/G
e
e →X
action of the cyclic group G = Zn on X
and p : X
is the quotient morphism.
Proof. By the results of [1, §I.17] there exists a non-singular projective vae and a degree n unbranched cover satisfying (3). By [7, Exercise
riety X
6
e is isomorphic to Spec(A), where A is the OX -algebra
II.5.17], X
n−1
M
L⊗i ,
i=0
which proves uniqueness. The action of G is generated by the automorphism
e
⊗L of A, and clearly X = X/G.
¤
Proposition 3.3. Take assumptions as in Proposition 3.2. Thus X is the
e by the cyclic group G = Zn . Let g be a generator of G. Then
quotient of X
e of D(X)
e is isomorphic to p∗ (E) for some object
(a) A simple object E
e ∼
e
E of D(X) if and only if there is an isomorphism g ∗ (E)
= E.
e
(b) A simple object E of D(X) is isomorphic to p∗ (E) for some object
e of D(X)
e if and only if there is an isomorphism E ⊗ L ∼
E
= E.
e → g ∗ (E)
e is an isomorphism. Since
Proof. First, part (a). Suppose λg : E
e
G is cyclic and E is simple, replacing λg by a suitable scalar multiple we
e The result then follows from
can define the structure of a G-object on E.
Proposition 3.1.
Part (b) is entirely analogous. The functor p∗ effects an equivalence between the category of OXe -modules and the category of modules over the
sheaf of algebras A of Proposition 3.2. A module over this algebra is nothing but an OX -module E together with compatible isomorphisms
E ⊗ L⊗i → E ⊗ L⊗j
for all pairs i, j ∈ Zn . Again, if E is simple, it is enough to have an isomorphism E ⊗ L → E. The corresponding statement for objects of D(X)
follows as in Proposition 3.1.
¤
We make the following definition.
Definition 3.4. Let X be a non-singular projective variety whose canonical
bundle has finite order n. By the canonical cover of X we shall mean the
e of Proposition 3.2 together with
unique non-singular projective variety X
e → X.
the quotient morphism pX : X
e is
The reader should note that the action of Zn on the canonical cover X
only defined up to automorphisms of Zn .
4. Lifts of FM transforms
In this section we prove our main result, relating FM transforms on varieties with canonical bundles of finite order to equivariant FM transforms
on the canonical covers. Throughout we shall suppose that the cyclic group
e and Ye , and
G = Zn acts freely on two non-singular projective varieties X
e
e
denote the quotient morphisms by pX : X → X and pY : Y → Y respectively.
e : D(Ye ) → D(X)
e is called equivariant if there
Definition 4.1. A functor Φ
are isomorphisms of functors
e∼
e ◦ g∗
g∗ ◦ Φ
=Φ
for each g ∈ G.
7
Definition 4.2. Given a functor Φ : D(Y ) → D(X), a lift of Φ is a functor
e : D(Ye ) → D(X)
e such that the following two squares of functors commute
Φ
up to isomorphism
e
Φ
e
D(
xX)


p∗X 
ypX,∗
Φ
D(X),
e
D(
−→
xY )


p∗Y 
ypY,∗
D(Y )
−→
that is such that there are isomorphisms of functors
e∼
pX,∗ ◦ Φ
= Φ ◦ pY,∗ ,
(4)
e ◦ p∗Y .
p∗X ◦ Φ ∼
=Φ
e descends to give the functor Φ.
We also say that Φ
We start with an easy lemma.
e : D(X)
e → D(X)
e be integral
Lemma 4.3. Let Φ : D(X) → D(X) and Φ
e lifts Φ.
functors such that Φ
e∼
e
(a) Suppose Φ ∼
= idD(X) . Then Φ
= g∗ for some g ∈ G. In particular Φ
is an equivalence.
e∼
(b) Suppose Φ
= idD(X)
e . Then Φ is an equivalence. If pX is a canonical
⊗i
cover, then Φ(−) ∼
⊗ −) for some integer i.
= (ωX
e and put
Proof. Write p = pX . We start with (a). Take a point x̃ ∈ X,
e
x = p(x̃). The object E = Φ(Ox̃ ) satisfies p∗ (E) = Ox , so E = Of (x̃) for
some point f (x̃) in the fibre p−1 (x). As we observed in Section 2, this implies
e →X
e is a morphism of varieties, and for some line bundle L on
that f : X
e
X,
∼
e
Φ(−)
= f∗ (L ⊗ −).
Since f (x̃) always lies in the fibre p−1 (x), one has f = g for some g ∈ G.
e also lifts the identity, and takes p∗ (OX ) = O e to L, so
The functor g∗−1 ◦ Φ
X
in fact L must be trivial.
e such that p(x̃) = x.
To prove (b), take a point x ∈ X, and a point x̃ ∈ X
Then
Φ(Ox ) = p∗ (Ox̃ ) = Ox
so that Φ ∼
= (L ⊗ −) for some line bundle L on X. We must have p∗ L = OXe ,
so if p is the canonical cover, the projection formula gives
L⊗(
n−1
M
⊗i
)
ωX
∗
∗
= L ⊗ p∗ (p OX ) = p∗ (p L) =
n−1
M
⊗i
,
ωX
i=0
i=0
and L is a power of ωX .
¤
e given by
Consider the skew-diagonal action of G on Ye × X
g(ỹ, x̃) = (g(ỹ), g −1 (x̃)).
e → Z and the group G acts freely
There is a quotient morphism f : Ye × X
on Z with quotient Y × X. There is a commutative diagram
8
e
Ye × X
πYe ¡¡
¡
¡
ª
f
@
@πXe
@
@
R
?
Ye
e
X
Z
j ¡¡
pY
¡
?¡
ª
Y ¾
@
k
h @
?
pX
@
@
R ?
- X
Y ×X
πY
πX
e = f ∗ (Q) and P =
Lemma 4.4. Let Q be an object of D(Z) and put P
e
P
P
e=Φ
h∗ (Q). Then Φ
e e is a lift of Φ = ΦY →X .
Y →X
Proof. Let E be an object of D(Y ). By [6, II.5.6, II.5.12], there are natural
isomorphisms
L
L
∗ ∗
∗
∗
∼
∼
e ∗Y (E))
e
e
Φ(p
= RπX,∗
e (P ⊗ πYe pY (E)) = RπX,∗
e (f (Q ⊗ j (E)))
L
L
∼
= p∗X Rk∗ (Q ⊗ j ∗ (E))) ∼
= p∗X RπX,∗ (h∗ (Q ⊗ h∗ πY∗ (E)))
L
∼
= p∗X RπX,∗ (P ⊗ πY∗ (E)) ∼
= p∗X Φ(E).
The other isomorphism of (4) can be proved in the same way, or by taking
adjoints.
¤
The following theorem is the main result of this paper.
Theorem 4.5. Let X and Y be non-singular projective varieties with canonical bundles of order n, and take canonical covers
e → X,
pX : X
pY : Ye → Y.
e and Ye by the cyclic group G = Zn . Then
Thus X and Y are quotients of X
any equivariant FM transform
e : D(Ye ) −→ D(X)
e
Φ
is the lift of some FM transform
Φ : D(Y ) −→ D(X),
e so that any
and conversely, we may choose the actions of G on Ye and X
e
FM transform Φ lifts to give an equivariant FM transform Φ.
e : D(Ye ) → D(X)
e be an equivariant transform with kernel P.
e
Proof. Let Φ
The equivariance is equivalent to the existence of isomorphisms
e ∼
e
(id e ×g)∗ (P)
= (g × id e )∗ (P)
Y
X
e = f ∗ Q for
for all g ∈ G. The argument of Proposition 3.3 implies that P
e descends to a functor Φ.
some object Q of D(Z). Thus Φ
e be a inverse of Φ.
e Then Ψ
e is
We must show that Φ is an equivalence. Let Ψ
equivariant and hence is the lift of some integral functor Ψ : D(X) → D(X).
e ◦Φ
e ∼
But then the composite functor Ψ
= idD(Ye ) lifts Ψ ◦ Φ, so by Lemma
9
4.3, twisting Ψ by some power of ωX , one has Ψ ◦ Φ ∼
= idD(Y ) . Similarly
∼
Φ ◦ Ψ = idD(X) , so Φ is an equivalence.
e to be such
For the converse, we may take the actions of G on Ye and X
that
n−1
M
∗
h∗ (OZ ) =
(πX
ωX ⊗ πY∗ ωY−1 )⊗i .
i=0
Let Φ : D(Y ) → D(X) be a FM transform with kernel P. It follows from
the isomorphism (1) and Proposition 3.3 that P = h∗ Q for some object Q
e and furthermore Φ
e is equivariant
of D(Z). By Lemma 4.4, Φ has a lift Φ,
with respect to our chosen G-actions.
Let Ψ be the inverse of Φ. Then Ψ is also a FM transform and hence lifts
e : D(Ye ) → D(X)
e by the same argument. The functor Ψ
e ◦Φ
e
to a functor Ψ
e with g ∗ for
is a lift of Ψ ◦ Φ ∼
= idD(Y ) , hence, by Lemma 4.3, composing Ψ
e ◦Φ
e∼
e ◦Ψ
e∼
some g ∈ G, we can assume that Ψ
= idD(Ye ) . Similarly, Φ
= idD(X)
e
and we are done.
¤
It is easy to see using Lemma 4.3 that, in the situation of the theorem, if
e 1, Φ
e 2 lift a given transform Φ, then Φ
e2 ∼
e 1 for
two FM transforms Φ
= g∗ ◦ Φ
some g ∈ G. Similarly, if FM transforms Φ1 , Φ2 both lift to give the same
⊗i
e then Φ2 ∼
transform Φ,
⊗ Φ1 for some integer i.
= ωX
A couple of points remain. Let X be a non-singular projective variety
e → X be the canonical
whose canonical bundle has order n, and let pX : X
e
cover. Thus X is the quotient of X by a free action of G = Zn .
Firstly, suppose there is a FM transform Φ relating X to another variety
Y . Then, by Lemma 2.1, ωY also has order n, and taking canonical covers
of X and Y we are in the situation of Theorem 4.5.
Secondly, suppose there is a non-singular projective variety Ye with a free
e relating X
e and
G-action, and that there is an equivariant FM transform Φ
Ye . Then we claim that the quotient morphism pY : Ye → Y is a canonical
cover of Y = Ye /G, so we are again in the situation of Theorem 4.5.
To prove the claim, note that by the argument used in the proof of Theoe descends to give a FM transform Φ : D(Y ) → D(X).
rem 4.5, the functor Φ
Lemma 2.1 shows that ωY has order n. Taking a canonical cover Y 0 of Y
e
e −1 ◦ Φ0 is
we can lift Φ to a FM transform Φ0 : D(Y 0 ) → D(X).
Now Φ
an equivariant FM transform relating Y 0 and Ye which lifts the identity on
D(Y ). It follows that Y 0 and Ye are isomorphic as G-spaces, hence the claim.
5. Examples
e be a non-singular projective surface with a fixed polarisation and
Let X
let Ye be a complete, fine, non-singular, two-dimensional moduli space of
e Then there is a universal sheaf P
e on Ye × X,
e and by
stable sheaves on X.
Theorem 2.2, the resulting functor
e = ΦPe
e
e
Φ
e : D(Y ) −→ D(X)
Ye →X
is a FM transform.
e has trivial canonical bundle (so is of either abelian or
Assume that X
K3 type). By Lemma 2.1 the variety Ye also has trivial canonical bundle.
10
e and let
Suppose further that the cyclic group G = Zn acts freely on X
e
pX : X → X denote the quotient morphism.
e induces an action of G on the moduli space Ye such
The action of G on X
that for each point ỹ ∈ Ye , and each g ∈ G,
eỹ ) ∼
eg(ỹ) .
g ∗ (P
=P
(5)
If this action of G on Ye is free then we can form the quotient Y = Ye /G, and
e descends to give a FM transform Φ : D(Y ) →
Theorem 4.5 shows that Φ
D(X). The following lemma gives a purely numerical criterion for when this
happens.
Lemma 5.1. The action of G on Ye defined above is free if and only if
eỹ ), as F varies through all locally-free sheaves on X,
the integers χ(p∗X (F ), P
have no common factor.
e
Proof. Assume first that the action of G on Ye is free. As we noted above Φ
descends to a transform Φ : D(Y ) → D(X). Let Ψ be the inverse of Φ, take
e=P
eỹ . By the adjunction p∗ a pX,∗
a point ỹ ∈ Ye and put E
X
e = χ(F, pX,∗ (E)).
e
χ(p∗X F, E)
e so for any locally-free sheaf F on X
If y = pY (ỹ), Φ(Oy ) = pX,∗ (E),
e = χ(Ψ(F ), Oy ).
χ(F, pX,∗ (E))
Since Ψ is an equivalence these integers have no common factor.
For the converse suppose that the G-action is not free, so that for some
e=P
eỹ is some subgroup H ⊂ G of order m > 1.
ỹ ∈ Ye , the stabilizer of E
∗
e
Then E = f (A) for some A, where f is the intermediate quotient map
e → X/H.
e
e
f: X
For any bundle F on X/H
e = χ(f ∗ (F ), f ∗ (A)) = m χ(F, A)
χ(f ∗ (F ), E)
and the result follows.
¤
e with
Example 5.2. Let X be an Enriques surface. There is a K3 surface X
e
an involution σ such that X is the quotient of X by the group Z2 generated
e one has
by σ. For any point x̃ ∈ X
σ ∗ (Ix̃ ) = Iσ(x̃) ,
so the reflection functor of Example 2.3 descends to give a FM transform
Φ : D(X) → D(X). This has the property that for each x ∈ X there is an
exact sequence
0 −→ Φ(Ox ) −→ OX ⊕ ωX −→ Ox −→ 0.
It is this transform which was studied by S. Zube [14, §3.7].
Example 5.3. Let X be a bielliptic surface which is a quotient of a product
e = C1 × C2 by a cyclic group Zn . The quotient map
of elliptic curves X
e
p : X → X is then the canonical cover of X.
The original Fourier-Mukai functor of [10] never descends because the
sheaf OXe = F(O0 ) is G-invariant.
11
e of Chern
Consider instead the moduli space Ye of stable sheaves on X
character (4, 2`, 1), where ` = C1 +C2 is a principal polarisation. By Lemma
5.1, the FM transform of Example 2.4 descends to give a FM transform
Φ : D(Y ) → D(X), such that for all y ∈ Y the object Φ(Oy ) is a locally-free
sheaf on X of rank 4n.
Remark 5.4. In [4] we show that if X and Y are Enriques or bielliptic
surfaces, and Φ : D(Y ) → D(X) is a FM transform, then X and Y are
isomorphic.
References
[1] W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Ergebnisse Math.
Grenzgeb. (3), vol. 4, Springer-Verlag, 1984.
[2] T. Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. reine angew. math.
498 (1998) 115-133, also alg-geom 9705002.
[3] T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms,
Bull. Lond. Math. Soc. 31 (1999) 25-34, also alg-geom 9809114.
[4] T. Bridgeland, A. Maciocia, Complex surfaces with equivalent derived categories, to
appear in Math. Z.
[5] A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. 9
(1957) 119–221
[6] R. Hartshorne, Residues and duality, Lect. Notes Math. 20, Springer-Verlag, 1966.
[7] R. Hartshorne, Algebraic Geometry, Grad. Texts Math. 52, Springer-Verlag, 1977.
[8] M. Kontsevich, Homological algebra of mirror symmetry, Preprint alg-geom 9411018.
[9] A. Maciocia, Generalized Fourier-Mukai transforms, J. reine angew. Math. 480
(1996), 197-211, also alg-geom 9705001.
[10] S. Mukai, Duality between D(X) and D(X̂) with its application to Picard sheaves,
Nagoya Math. J. 81 (1981), 153-175.
[11] S. Mukai, On the moduli space of bundles on K3 surfaces I, in: Vector Bundles on
Algebraic Varieties, M.F. Atiyah et al., Oxford University Press (1987), 341-413.
[12] D. Naie, Special rank two vector bundles over Enriques surfaces, Math. Ann. 300
(1994) 297-316.
[13] D. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (NY), 84
(1997) 5 1361-1381, also alg-geom 9606006.
[14] S. Zube, Exceptional vector bundle on Enriques surfaces, Mat. Zametki 61 (1997) 6
825-834 also alg-geom 9410026.
Department of Mathematics and Statistics, The University of Edinburgh,
King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
email: [email protected]
[email protected]