OF POINTS ON AN ENRIQUES SURFACE
ANDREAS KRUG AND PAWEL SOSNA
Abstract. We use semi-orthogonal decompositions to construct autoequiv-alences of Hilbert schemes of points on Enriques surfaces and of Calabi–Yau varieties which cover them. While doing this, we show that the derived cate-gory of a surface whose irregularity and geometric genus vanish embeds into the derived category of its Hilbert scheme of points.
1. Introduction
The bounded derived category of coherent sheaves on a smooth projective complex variety Z, denoted by Db(Z), is now widely recognized as an important invariant which can be used to study the geometry of Z. It is therefore quite natural to consider the group of autoequivalences Aut(Db(Z)). This group al-ways contains the subgroup Autst(Db(Z)) of so-called standard autoequivalences, namely those generated by automorphisms ofZ, the shift functor and tensor prod-ucts with line bundles. Note that all these equivalences send coherent sheaves to (shifts of) coherent sheaves. A classical result by Bondal and Orlov, see [8], states that Autst(Db(Z))'Aut(Db(Z)) ifωZ is either ample or anti-ample. Therefore, we expect the most interesting behaviour if the canonical bundle is trivial. For instance, ifωZ ' OZ and Hi(Z, ωZ) = 0 for all 0< i <dim(Z), then Aut(Db(Z))
contains so-called spherical twists, which are more interesting than the ones men-tioned above since, in general, they do not preserve the abelian category of coherent sheaves; see [29].
It is usually quite difficult to construct new autoequivalences of a given variety Z. However, if A is some triangulated category and an exact functor Φ :A //Db(Z) is a so-called spherical or
Pn-functor, see Subsection 2.5, we do
get an autoequivalence of Db(Z). For example, let Xe be a K3 surface and Xe[n] its Hilbert scheme of n points. It was shown in [1] that the Fourier–Mukai func-tor Db(Xe) //Db(Xe[n]) with the ideal sheaf of the universal family as kernel is a
Pn−1-functor whose associated autoequivalence is new. Interestingly, for an abelian
surfaceAthe corresponding functor is not aPn−1-functor, but pulling everything to the generalised Kummer variety does give one; see [22]. Another example was
2010Mathematics Subject Classification. 18E30, 14F05, 14J28.
Key words and phrases. derived categories, semi-orthogonal decompositions, Fourier–Mukai
given in [18] where it was shown that, given any surfaceS, there is aPn−1-functor
Db(S) //Db(S[n]) which is defined using equivariant methods.
In this paper we will construct new examples of spherical functors using En-riques surfaces and Hilbert schemes of points on them. Let X be an Enriques surface and X[n] the Hilbert scheme of n points on X. The canonical cover of
X[n] is a Calabi–Yau variety (see [23] or [24]) and will be denoted by CY
n.
Write π: CYn //X[n] for the quotient map. Consider the Fourier–Mukai
func-torF: Db(X) //Db(X[n]) induced by the ideal sheaf of the universal family.
Theorem 1.1. The functor e
F =π∗F: Db(X) //Db(CYn)
is split spherical for alln≥2and the associated twistTeis equivariant, so descends to an autoequivalence ofX[n]. The autoequivalence
e
T of Db(CY
n)is not standard
and not a twist around a spherical object.
Under some conditions we can also compare our twist Te to the autoequiva-lences constructed in [26]; see Proposition 3.18.
Once we establish Theorem 1.2 below, the first part of the theorem is an incarnation of the following general principle, see Theorem 3.4 and Remark 3.11: IfY is a smooth projective variety whose canonical bundle is of order 2,Ye its canonical cover with quotient map π: Ye //Y, and Aan admissible subcategory of Db(Y) with embedding functor i: A //Db(Y), then π∗i is a split spherical functor and the associated twist is equivariant.
The following result might be of independent interest.
Theorem 1.2. If S is any surface with pg = q = 0, then the FM-transform
F: Db(S) //Db(S[n]) whose kernel is the ideal sheaf of the universal family, is fully faithful, hence Db(S)is an admissible subcategory of Db(S[n]).
Since there are many semi-orthogonal decompositions of Db(X[n]), we have many, potentially non-standard, twists associated with them. Note that in general it seems difficult to describe the complement of the image ofF explicitly but see Remark 3.14 and Subsection 5.3 for statements related to this question.
The paper is organised as follows. In Section 2 we present some background information, before proving our main results in Section 3. In Section 4 we give a general construction of exceptional sequences on the Hilbert scheme S[n] out of exceptional sequences on a surface S. This construction has been independently considered by Evgeny Shinder. In particular, we have the following result which is probably well-known to experts.
Proposition 1.3. If a surfaceShas a full exceptional collection, then so too does
S[n].
for X an Enriques surface, and, in some cases,Pn-functors on smooth Deligne–
Mumford stacks. The last section also gives some background on the proof of Proposition 3.1, the main ingredient in the proof of Theorem 1.2.
Conventions. We will work over the complex numbers and all functors are as-sumed to be derived. Furthermore, all varieties are asas-sumed to be smooth and projective unless stated otherwise. We will write Hi(E) for the i-th cohomology
object of a complexE∈Db(Z) and H∗(E) for the complex ⊕
iHi(Z, E)[−i]. If F
is a functor, its right adjoint will be denoted byFRand its left adjoint by FL.
Acknowledgements.We thank Ciaran Meachan and David Ploog for comments. We are very grateful to the referee for many helpful comments which greatly improved the exposition. A.K. was supported by the SFB/TR 45 of the DFG (German Research Foundation). P.S. was partially financially supported by the RTG 1670 of the DFG.
2. Preliminaries
2.1. Hilbert schemes of surfaces with pg =q = 0. Let S be a surface with
pg=q= 0 and considerS[n], the Hilbert scheme ofnpoints onS. Then we have Hk(S[n],O
S[n]) = 0 for allk > 0, compare [24]. Indeed, by the K¨unneth formula
H∗(Sn,O
Sn) = H∗(S,OS)⊗n is concentrated in degree zero. As a consequence, the structure sheaf of then-th symmetric product has no higher cohomology, and the same then also holds for S[n], because the symmetric product has rational singularities.
For example, we can consider an Enriques surface, which is a smooth pro-jective surfaceX with pg =q= 0 such that the canonical bundleωX is of order 2.
2.2. Canonical covers. LetY be a variety with torsion canonical bundle of (min-imal) orderk. Thecanonical coverYe ofY is the unique (up to isomorphism) variety with trivial canonical bundle and an ´etale morphismπ: Ye //Y of degreeksuch thatπ∗OYe =
Lk−1
i=0 ω
i
Y. In this case, there is a free action of the cyclic groupZ/kZ
onYe such thatπis the quotient morphism.
As an example, the canonical cover of an Enriques surfaceX is a K3 surface e
X andX is the quotient ofXe by the action of a fixed point free involution. Furthermore, the canonical bundle of X[n] has order 2 and the associated canonical cover, denoted here by CYn, is a Calabi–Yau variety; see [23, Prop. 1.6]
or [24, Thm. 3.1].
2.3. Fourier–Mukai transforms and kernels. Recall that given an objectEin Db(Z×Z0), where Z andZ0 are smooth and projective, we get an exact functor Db(Z) //Db(Z0),α //p
Z0∗(E ⊗p∗
Zα). Such a functor, denoted by FME, is called
∆ :Z //Z×Z for the diagonal map andL ∈Pic(Z). In particular, FMO∆ is the identity functor.
Convention.We will write ML for the functor FM∆∗L.
LetSbe any smooth projective surface,Zn⊂S×S[n]be the universal family
and consider its structure sequence
0 //IZn //OS×S[n] //OZn //0.
We can use the objects from the above sequence as kernels to get a triangle
F //F0 //F00 of functors Db(S) //Db(S[n]). Since all these functors are FM-transforms, they have left and right adjoints; see [16, Prop. 5.9].
2.4. Equivalences of canonical covers. The relation between autoequivalences of a variety Y with torsion canonical bundle and those of the canonical coverYe was studied in [10]. We recall some facts in the special case where the order ofωY
is 2. We will writeτ for the fixed-point free involution ofYe such thatY /e hτi 'Y andπfor the quotient map.
An autoequivalenceϕeof Db(Ye) isequivariant ifτ∗ϕe'ϕτe ∗.
By [10, Sect. 4], an equivariant functorϕedescends to a functorϕ∈Aut(Db(Y)) with functor isomorphismsπ∗ϕe'ϕπ∗andπ
∗ϕ'
e
ϕπ∗; moreover, the two descents
ϕ,ϕ0 ofϕediffer by tensoring withωY.
In the other direction, it is also shown in [10, Sect. 4] that every autoequiv-alence of Aut(Db(Y)) has an equivariant lift. Two lifts differ by the action ofτ in Aut(Db(
e
Y)).
2.5. Spherical functors. Now consider two triangulated categoriesAandBand any exact functorF:A //Bwith left and right adjointsFL, FR:B //A. Define
thetwist T =TF to be the cone on the counit:F FR //id
B of the adjunction and thecotwist C to be the cone on the unit η: idA //FRF.
Remark 2.1. Of course, one needs to make sure that the above cones actually exist. If one works with Fourier–Mukai-transforms, this is not a problem, because the maps between the functors come from the underlying kernels and everything works out, even for (reasonable) schemes which are not necessarily smooth and projective; see [3]. More generally, everything works out if one uses an appropriate notion of a spherical DG-functor; see [4].
So, as we will see, in the cases of interest to us, we have the triangles
F FR //idB //T and idA //FRF //C. Following [4], we call F spherical if
C is an equivalence and FR ' CFL. If A and B admit Serre functors SA and SB, the last condition is equivalent to SBF C 'FSA. If F is a spherical functor, thenT is an equivalence. If the triangle idA //FRF //C splits, we callF split spherical.
For an example of a (split) spherical functor consider ad-dimensional variety
Z and a spherical object E ∈ Db(Z), that is, E⊗ω
C⊕C[−d]. The functor
F =− ⊗E: Db(Spec(C)) //Db(Z)
is then spherical and the associated autoequivalence of Db(Z) is the spherical twist from the introduction, denoted by STE and called Seidel–Thomas twist in
the following.
2.6. Pn-functors. Following [1, Def. 3.1], a
Pn-functor is a functorF:A //Bof
triangulated categories such that
(1) There is an autoequivalenceHF =H ofAsuch that
FRF 'id⊕H⊕H2⊕ · · · ⊕Hn.
(2) The map
HFRF //FRF FRF −−FRF// FRF,
with being the counit of the adjunction is, when written in the compo-nents
H⊕H2⊕ · · · ⊕Hn⊕Hn+1 //id⊕H⊕H2⊕ · · · ⊕Hn,
of the form
∗ ∗ · · · ∗ ∗ 1 ∗ · · · ∗ ∗ 0 1 · · · ∗ ∗ ..
. ... . .. ... ... 0 0 · · · 1 ∗
.
(2.1)
(3) FR'HnFL. IfAandBhave Serre functors, this condition is equivalent toSBF Hn'F SA.
If F is a Pn-functor, then there is also an associated autoequivalence of B,
denoted by PF = P. A P1-functor is precisely a split spherical functor and for
the associated equivalences we have T2 ' P. If Xe is a K3 surface, the functor
F = FMIZn defined in Subsection 2.3 is aP
n−1-functor; see [1].
2.7. Semi-orthogonal decompositions. References for the following facts are, for example, [6] and [7].
Let T be a triangulated category. A semi-orthogonal decomposition of T is a sequence of strictly full triangulated subcategories A1, . . . ,Am such that (a) if Ai ∈ Ai and Aj ∈ Aj, then Hom(Ai, Aj[l]) = 0 for i > j and alll, and (b) the
Ai generateT, that is, the smallest triangulated subcategory of T containing all
theAi is alreadyT. We writeT =hA1, . . . ,Ami. Ifm= 2, these conditions boil
down to the existence of a functorial exact triangleA2 //T //A1for any object
T ∈ T.
and right admissible. Note that ifT admits a Serre functor, then the existence of one of the adjoints implies the existence of the other.
Given any subcategory A, the category A⊥ consists of objects b such that Hom(a, b[k]) = 0 for all a ∈ A and all k ∈ Z. If A is right admissible, then
T = hA⊥,Ai is a semi-orthogonal decomposition. Similarly, T = hA,⊥Ai is a semi-orthogonal decomposition ifAis left admissible, where ⊥Ais defined in the obvious way.
Examples typically arise from so-called exceptional objects. Recall that an object E ∈ Db(Z) (or any
C-linear triangulated category) is called exceptional
if Hom(E, E) = C and Hom(E, E[k]) = 0 for all k 6= 0. The smallest
triangu-lated subcategory containing E is then equivalent to Db(Spec(
C)) and this
cate-gory, by abuse of notation again denoted byE, is admissible, leading to a semi-orthogonal decomposition Db(Z) =hE⊥, Ei. We call a sequence of exceptional ob-jectsE1, . . . , En anexceptional collection if Db(Z) =h(E1, . . . , En)⊥, E1, . . . , Eni, where (E1, . . . , En)⊥ is the category of objectsF which satisfy Hom(Ei, F[k]) = 0 for alli, k. The collection is calledfull if (E1. . . , En)⊥= 0.
Note that any fully faithful FM-transform i: A= Db(Z0) //Db(Z) gives a semi-orthogonal decomposition Db(Z) =hi(A)⊥, i(A)i.
We will need the following well-known and easy fact.
Lemma 2.2. If T has a Serre functor ST and A is an admissible subcategory, thenA has a Serre functorSA'iRSTi.
Proof. Givena, a0∈ A, we compute
HomA(a, a0)'HomT(i(a), i(a0))'HomT(i(a0),STi(a))∨
'HomA(a0, iRSTi(a))∨.
Remark 2.3. Assume Ais an admissible subcategory of Db(Z). The embedding functorilifts to DG-enhancements (see, for example, [21] for this notion). It can be checked that the adjoints iR and iL also lift to so-called DG quasi-functors,
which follows, for example, from [21, Lem. 4.4 and Prop. 4.10]. So the composition ofiwith any FM-transform will lift to the DG-level, hence we are in a position to use the results of [4] and all required cones of natural transformations will exist.
2.8. Group actions and derived categories. Let G be a finite group acting on a smooth projective variety Z. The equivariant derived category, denoted by Db
G(Z), is defined as Db(Coh G
(Z)), see, for example, [25] for details. Recall that for every subgroupH ⊂Gthe restriction functor Res : Db
G(Z) //D
b
H(Z) has the
inflation functor Inf : DbH(Z) //DbG(Z) as a left and right adjoint (see e.g. [25, Sect. 1.4]). It is given forA∈Db(Z) by
Inf(A) = M [g]∈H\G
g∗A
(2.2)
IfGacts trivially onZ, there is also the functor triv : Db(Z) //DbG(Z) which equips an object with the trivial G-linearisation. Its left and right adjoint is the functor (−)G: Db
G(Z) //D
b(Z) of invariants.
Convention. When working with Fourier–Mukai transforms, we will frequently identify the functor with its kernel.
3. Proofs of the main results
3.1. Surfaces with pg =q= 0. Recall the FM-transforms from Subsection 2.3. To computeFRF in the examples known so far, one usually works out the various
compositions such as, for example,F00RF0. In our case,FRF has a rather simple
shape.
Proposition 3.1. If S is a surface with pg =q= 0, then the composition FRF
is isomorphic to the identity.
Proof. Firstly, we note that, using results in Section 6 of [22] and the isomorphism H∗(S[n],O
S[n])'C, the following holds:
F00RF0 ' OS×S,
F0RF0 '(OSωS)[2], F00RF00' O∆⊕ OS×S,
F0RF00'(OSωS)[2].
Next,F0RF'0 andF00RF ' O∆[−1]. Indeed, the mapF0 //F00induces an isomorphismF0RF0 //F0RF00and the componentOS×S //OS×S of the induced
mapF00RF0 //F0RF00is an isomorphism too; see [22, Sect. 6] or Section 5. Hence, the first assertion follows from the triangleF0RF //F0RF0 //F0RF00.
We then consider the triangle F00RF //F00RF0 //F00RF00 and check that the cokernel of the mapF00RF0 //F00RF00is isomorphic toO∆. Indeed, if
ϕ= (ϕ1, ϕ2) :A //A⊕B
is a map in an abelian category such that the first component is an isomorphism (so ϕ is an injection), the cokernel has to be isomorphic to B. To see this, just note that the map A⊕B //B defined by (−ϕ2ϕ−1,idB) satisfies the universal
property of the cokernel.
To finally prove the claim, use the triangle F00RF //F0RF //FRF and
what was proved above.
Proof of Theorem 1.2. SinceFRF 'id,F is fully faithful. On the other hand,F
has adjoints, soF(Db(S)) is an admissible subcategory of Db(S[n]).
Remark 3.3. There exist surfacesS of general type withpg =q= 0 such that Db(S) contains an admissible subcategory whose Hochschild homology is trivial and whose Grothendieck group is finite or torsion, see, for example, [5]. Therefore, by Theorem 1.2, Db(S[n]) also contains such a (quasi-)phantom category.
3.2. Application to Enriques surfaces.
Theorem 3.4. Let X be an Enriques surface,CYn the canonical cover ofX[n], π: CYn //X[n] the covering map andτ: CYn //CYn the deck transformation.
Ifi:A //Db(X[n])is the embedding of an admissible subcategoryAofDb(X[n]), thenπ∗iis a split spherical functor whose induced twist TeA:=Tπ∗i is equivariant
and thus descends to an autoequivalence ofDb(X[n])for all n≥2. Proof. First, the functorπ∗: Db(X[n]) //Db(CY
n) is split spherical with cotwist C = (−)⊗ωX[n] and twist Tπ∗ =τ∗[1]. Indeed, this follows from the identities
π∗π∗'(−)⊗(OX[n]⊕ωX[n]),π∗ωX[n]'ωCYn, andπ∗π∗'id⊕τ∗.
Next, π∗i is also split spherical as follows from the first step together with Lemma 2.2; compare [1, Prop. 1.1].
Finally, to see that TeAis equivariant, we note that π∗iiRπ∗τ∗ 'π∗iiRπ∗ '
τ∗π∗iiRπ∗, soτ∗TeA'TeAτ∗, since both are a cone ofπ∗iiRπ∗ //τ∗.
Example 3.5. If A 'Db(Spec(
C)) is the category generated by an exceptional
object E, then the twist TeA associated to π∗i is the Seidel–Thomas twist STA,
whereA'π∗i(E) is a spherical object by [29, Prop. 3.13].
Example 3.6. Setting i = F: Db(X) //Db(X[n]) as the FM transform along the universal ideal sheaf gives the first part of Theorem 1.1.
Remark 3.7. The functor Fb: Db(Xe) //Db(X[n]) defined as F πX∗ fails to be spherical in an interesting way. Note that π!X = πX∗ in this case, so RbFb '
πX∗πX∗ ' id⊕τX∗ and C ' τX∗ is an autoequivalence of D
b( e
X). But the con-dition FbS
e
X ' SX[n]F Cb is not satisfied. Indeed, FbS
e
X(α) ' Fb(α)[2] whereas SX[n]F Cb (α)'Fb(α)⊗ωX[n][2n], so these objects are not isomorphic forα∈Db(Xe) since their non-vanishing cohomologies lie in different degrees.
One could callFb aspherelike functor in analogy with the spherelike objects of [14].
Furthermore, it is easy to check that the functor F: Db(Xe) //Db(CYn)
defined asπ∗F πX∗is not spherical as well.
Remark 3.8. LetB =⊥Aso that Db(Y) = hA,Biis a semi-orthogonal decom-position. Then, by [2, Thm. 11] we haveTeATeB'τ∗[1].
Remark 3.9. One of the two descents ofTeAis
TA:= cone(iiR⊕Mω
X[n]ii
RM ω
where is the counit of the adjunction. To see this, first note that the twistTeA along the spherical functorπ∗iis given by
e
TA:= cone(π∗iiRπ∗− // id). Sinceπ∗π∗'id⊕MωX[n] andπ
∗M
ωX[n] 'π
∗, we have
(π∗iiRπ∗)π∗'π∗iiR⊕π∗iiRMωX[n] 'π
∗(iiR⊕M
ωX[n]ii
RM ωX[n])
from which TeAπ∗ 'π∗TA follows. Using Mω
X[n]π∗ 'π∗ one can similarly show thatπ∗TeA'TAπ∗.
Remark 3.10. As in the case of spherical or Pn-twists (see [18, Lem. 2.3]), we
have for any Ψ∈Aut(Db(X[n])) the relation ΨT
A'TΨ(A)Ψ. For this one uses the cone description ofTA given by Equation (3.1) and the fact that the embedding functor of Ψ(A) is Ψi.
Remark 3.11. The reader will note that the proof of Theorem 3.4 shows more generally that the functor π∗i is split spherical with equivariant twist for any canonical coverπ:Ye //Y of degree 2 and any fully faithful admissible embedding
i:A //Db(Y). More generally,π∗ is aPn−1-functor ifπ: Ye //Y is a canonical cover of ordern≥2. But forn≥3 the compositionπ∗i is in general not aPn−1
-functor for a fully faithful admissible embedding i: A //Db(Y). The reason is that Lemma 2.2 does not generalise to powers of the Serre functor, that is, in general,Sk
A6'iRSTkifork≥2.
3.3. Comparison to known autoequivalences. LetX be an Enriques surface and F: Db(X) //Db(X[n]) the FM-transform induced by the ideal sheaf of the universal family. We denote the twist along the spherical functorFe=π∗F byTe∈ Aut(Db(CY
n)) and its descent as described in Remark 3.9 byT ∈Aut(Db(X[n])).
The second descent is given by Mω
X[n]T. Now we want to compare these new autoe-quivalences to the known ones. Of course, on any variety there are the standard autoequivalences. On CYn there are also Seidel-Thomas twists and the
equiva-lences constructed in [26], while onX[n] there is the
Pn−1-functor constructed in
[18]. We will need the following statement.
Lemma 3.12. LetSbe any surface and letk(ξ)∈Db(S[n])be the skyscraper sheaf of a point ξ∈S[n]. ThenF FR(k(ξ))has rankχ−2nwhereχ:=χ(ω
S) =χ(OS).
Proof. The kernel ofFRis given by I∨
Zn⊗p∗SωS[2]. SinceIZn is flat overX[n]we getFR(k(ξ)) =I∨
ξ ⊗ωS[2] whereξ on the left-hand side denotes a point of S[n]
andξon the right-hand side denotes the subscheme ofSrepresented by this point. Note that for any objectE ∈Db(M) on a smooth varietyM andx∈M we have rkE =χ(E, k(x)). Thus,
rkF FR(α) =χ(F FR(α), α) =χ(FR(α), FR(α)) =χ(Iξ,Iξ) =χ−n−n+ 0
where the last equality uses the equationIξ=OS− Oξ in the Grothendieck group
Proposition 3.13. The autoequivalence T is not contained in the subgroup of Aut(Db(X[n])) generated by the standard autoequivalences Autst(Db(X[n])) and the equivalence P arising from the Pn−1-functor constructed in [18]. Similarly,
e
T /∈ hAutst(Db(CY
n)),Pei, wherePe is a lift ofP.
Proof. The first assertion follows from the second, since standard autoequivalences lift to standard autoequivalences.
The twistP is rank-preserving since all the objects in the image of the cor-respondingPn−1-functor are supported on a proper subset ofX[n]; see [18, Rem.
4.7]. Thus, the lift Pe is rank preserving too. The same holds for every standard autoequivalence (up to the sign -1 occurring for odd shifts). But by Lemma 3.12 we have forξe∈CYn withξ:=π(eξ)∈S[n] the equalities
2·rk(Te(k(eξ)) = rk(T(k(ξ))) = 4n−26= 0. Before proceeding further with the comparisons, we need to compute the values ofTe andT on certain spanning classes (for details on spanning classes see [16, Sect. 1.3]). Following the proof of Theorem 3.4 we see that the cotwist of Fe isC=SX[−2n] = MωX[2−2n]. By [1, Sect. 1.4]Teis given on the spanning class im(Fe)∪im(Fe)⊥ by
e
TFe'F Ce [1]'FeMωX[3−2n], Te(β) =β forβ∈im(Fe)
⊥.
(3.2)
Note that im(Fe)⊥ = ker(FeR) ={β∈Db(CYn)|FeR(β) = 0}. Using the cone description (3.1) of T we also get
T F 'Mω
X[n]FMωX[3−2n], TMωX[n]F'FMωX[3−2n] (3.3)
by Lemma 2.2. Also,T acts as the identity onhA,A⊗ωX[n]i⊥ = ker(FR)∩ker(FL)
whereA= im(F). So we have a description of the restriction ofT to the spanning class C := A ∪ A ⊗ωX[n] ∪ hA,A ⊗ωX[n]i⊥. To see that C is indeed a spanning
class, note that if Hom(β, γ) = 0 for all γ ∈ C, then, in particular, Hom(β, α) = 0 = Hom(β, α⊗ωX[n]) for all α∈ A. By Serre duality, β∈ hA,A ⊗ωX[n]i⊥⊂ C,
henceβ'0. Similarly, one proves that Hom(γ, β) = 0 implies thatβ '0. Remark 3.14. We will see in Section 5.3 that there exist non-trivial objects in ker(FR)∩ker(FL). By applying π∗ we also get non-trivial objects in ker(
e
FR)∩
ker(FeL). So there are 06=β∈Db(X[n]) and 06=βe∈Db(CYn) such thatT(β) =β
andTe(eβ) =βe.
Lemma 3.15. Let E ∈ Db(CY
n) be a spherical object and STE the induced
Seidel–Thomas twist. Let 0 6=α∈Db(CYn) with STE(α) =α[`]. Then `= 0 or `= 1−2n.
Proof. We have STE(E) =E[1−2n] and STE(β) =βfor allβ∈E⊥. The assertion
follows by [1, Prop. 1.2] together with the fact that E∪E⊥ is a spanning class of Db(CY
n). See also Lemma 3.17 below for a similar statement with a similar
Proposition 3.16. The twistTedoes not equal a shift of a Seidel–Thomas twist. Proof. By (3.2) we haveTe(Fe(α)) =Fe(α)[3−2n] forα∈Db(X) withωX⊗α'α, e.g.α=k(x) a skyscraper sheaf. By Remark 3.14 there is also a 06=β∈Db(CYn)
such that Te(β) = β. Thus, the assumption that Te[m] = STE for some m ∈ Z
contradicts the previous lemma.
Recall that ifZ, Z0 are smooth projective varieties anda < btwo integers, an exact functor G: Db(Z) //Db(Z0) is said to have cohomological amplitude [a, b] if for every complex E ∈ Db(Z) whose cohomology is concentrated in degrees betweenpandq, the cohomology ofG(E) is concentrated in degrees betweenp−a
andq+b. We will need the following result.
Lemma 3.17. Let 0 6=α∈Db(X[n]) with T(α) = α[`] or T(α) =α⊗ωX[n][`].
Then`= 0 or`= 3−2n. Similarly, ifγ∈Db(CYn)with Te(γ) =γ[`]then `= 0 or`= 3−2n.
Proof. Letα∈Db(X[n]) withT(α) =α[`], ` /∈ {0,3−2n}. To simplify notation write Φ = MωX[n], Ψ = MωX and j = 3−2n. By Equation (3.3) we have T F ' ΦFΨ[j]. Hence, for anyβ ∈Db(X) we have
Hom α, F(β)[k]'Hom Tm(α), TmF(β)[k]
'Hom α[m`],ΦmFΨm(β)[mj+k] 'Hom Ψ−mFLΦ−m(α), β[(j−`)m+k] for every k∈Z. This vanishes form 0 since Φ = Mω
X[n] and Ψ = MωX have cohomological amplitude [0,0] andFL has finite cohomological amplitude by [20,
Prop. 2.5]. Therefore,α∈⊥AwhereA= imF. Similarly, we getα∈⊥(A⊗ω
X[n]).
The objectαis also orthogonal toB:=hA,A ⊗ωX[n]i⊥ =A⊥∩⊥Aon which
T acts trivially. Indeed, forβ∈ Bandk∈Zwe have
Hom(α, β[k]) = Hom(Tm(α), Tm(β)[k]) = Hom(α[m`], β[k]) which vanishes form0.
Therefore,αis orthogonal to the spanning classA∪A⊗ωX[n]∪hA,A⊗ωX[n]i⊥,
hence is zero. The proof in the case that T(α) = α⊗ωX[n][`] is similar and the
statement aboutTeandγ follows by applyingπ∗.
Now we want to compare our autoequivalence to those described in [26]. While recalling the construction of the latter, we also introduce some more general facts which will be useful in the next section.
LetZ be a smooth projective variety andn≥2. We consider the cartesian powerZn equipped with the naturalS
n-action given by permuting the factors.
ForE ∈Db(Z) an exceptional object, the box productEn is again
excep-tional, since by the K¨unneth formula Ext∗Db
Sn(Zn)(E
n
, En)'Ext∗Xn(E n
, En)Sn'SnExt∗
More generally, for ρ an irreducible representation of Sn, the object En⊗ρ
is exceptional and the objects obtained this way are pairwise orthogonal, i.e. Ext∗(En⊗ρ, En⊗ρ0) = 0 forρ6=ρ0.
The case of biggest interest is ifZis a surface. Then there is the Bridgeland– King–Reid–Haiman equivalence (see [9] and [13])
Φ : Db(Z[n])−' // DbS n(Z
n).
In particular, when Z =X is an Enriques surface, we get for every excep-tional object E ∈ Db(X) further induced autoequivalences given by the Seidel– Thomas twists Tρe := STπ∗Φ−1(En⊗ρ) ∈ Aut(Db(CYn)) and their descentsTρ ∈
Aut(Db(X[n])). We assume from now on that there exists an object 0 6= F ∈ hE, E⊗ωXi⊥. This is equivalent to
e
E⊥ being non-trivial, whereEe:=π∗E is the corresponding spherical object on the K3 coverXe ofX (for example, ifE is a line bundle, we can takeEe⊗Ix∨⊗Iy ∈ Ee⊥ for two distinct pointsx, y). In this case
Fn is orthogonal to every En⊗ρ and ωX
n⊗En⊗ρ. Thus, Tρe(α) = α for
α=π∗Φ−1(Fn). Since also
e
Tρ(β) =β[1−2n] for β =π∗Φ−1(En⊗ρ) we see
thatTρe and thus also its descentTρ are non-standard.
By construction, the object Ee is invariant under τX, hence the associated Seidel-Thomas twist descends to an equivalence ΦE of Db(X). In turn, this gives
an equivalence ΦEn = FMPn ∈Aut(Db(X[n])), whereP ∈Db(X×X) denotes the Fourier–Mukai kernel of ΦE; see [25]. It is possible to lift this equivalence to
g Φn
E ∈Aut(CYn); see [26] for details.
Recall that the isomorphism classes of irreducible representations ofSn are
in bijection to the setP(n) of partitions ofn. For an exceptional objectE∈Db(X) we set
GE:=h[1],ΦEn, Tρ: ρ∈P(n)i ⊂Aut(Db(X[n])),
e
GE:=h[1],Φgn
E ,Tρe : ρ∈P(n)i ⊂Aut(Db(CYn)).
Proposition 3.18. Let E ∈Db(X) be exceptional and 0 6=F ∈ hE, E⊗ωXi⊥. We have GE ' Zp(n)+2 'GeE, wherep(n) =|P(n)|. Furthermore, T /∈ GE and
e
T /∈GEe .
Proof. For 0≤k≤nwe consider the objects
Ek·Fn−k := InfSn
Sk×Sn−k(E
k
Fn−k)∈DbSn(X
n).
We have ΦE(E) =E[−1] and ΦE(F) =F. It follows that
ΦEn:E k
·Fn−k //Ek·Fn−k[−k], En⊗ρ //En⊗ρ[−n].
(3.4)
By Remark 3.10, the latter shows that Φn
E commutes with all theTρ. Note that
fork < nandρ6'ρ0 we have
Similarly to (3.3) we get by the cone description (3.1) ofTρ
Tρ:Ek·Fn−k //Ek·Fn−k fork < n ,
(3.5)
Tρ:En⊗ρ0 //
(
En⊗ρ0 ifρ6=ρ0
ωXn⊗En⊗ρ0[−(2n−1)] ifρ=ρ0 which, in particular, shows that theTρ pairwise commute.
Now consider Ψ = (Φn E )a◦
Q
ρT bρ
ρ [c]∈GEand assume that Ψ'id. We have
Ψ(Fn) =Fn[c] and thus c= 0. It follows that Ψ(E1·Fn−1) =E1·Fn−1[−a] and thus a= 0. Finally, Ψ(En⊗ρ) = ωbρ
Xn⊗En⊗ρ[−bρ(2n−1)] shows that
bρ = 0 for allρ. The assertion thatT /∈GE follows in a similar way using Lemma 3.17.
The identities (3.4) and (3.5) lift to identities in Db(CYn). Therefore, one
can analogously show thatGeE'Zp(n)+2andT /e∈GE.
Remark 3.19. Letabe the sign representation, i.e. the one-dimensional represen-tation on whichSn acts by multiplication by sgn. By Remark 3.10 we haveTa'
Ma◦TC◦MawhereCdenotes the trivial representation and Ma∈Aut(DbSn(X
n)) is
the involution (−)⊗a. Note that for higher dimensional irreducible representations
ρthere is no such relation since Mρ is not an equivalence.
4. Exceptional sequences on X[n]
Let Z be a smooth projective variety and n ≥ 2. In this section we will construct exceptional sequences in the equivariant derived category Db
Sn(Z
n) out
of exceptional sequences in Db(Z).
Proposition 4.1([27, Cor. 1]). Let Z andZ0 be smooth projective varieties with full exceptional sequencesE1, . . . , Ek andF1, . . . , F` respectively. Then
E1F1, E1F2, . . . , E1F`, E2F1, . . . , EkF`
is a full exceptional sequence ofDb(Z×Z0).
Let nowE1, . . . , Ek be an exceptional sequence onZ. We consider for every multi-indexα= (α1, . . . , αn)∈[1, k]n:={1, . . . , k}n the object
E(α) :=Eα1· · ·Eαn ∈Db(Zn).
Remark 4.2. Letα, β∈[1, k]n. By the K¨unneth formula (4.1) Ext∗(E(α), E(β)) = Ext∗(Eα1, Eβ1)⊗ · · · ⊗Ext
∗(E
αn, Eβn).
Theorem 4.3([11]). LetGbe a finite group acting on a varietyM. Consider an exceptional sequence ofDb(M)of the form
E1(1), . . . , Ek(1)
1, E
(2) 1 , . . . , E
(2)
k2 , . . . , E
(`) 1 , . . . , E
(`)
k`
such thatGacts transitively on every blockE(1i), . . . , Ek(i)
i, i.e. for everyi∈[`]and every paira, b∈[1, ki]there is ag∈Gsuch thatg∗E
(i)
a 'Eb(i)(and conversely for
everyg∈Gand everya∈[1, ki] there is an elementb∈[1, ki]such thatg∗E(ai)' Eb(i)). Let Hi := StabG(E
(i)
1 ) and assume that E (i)
1 carries an Hi-linearisation, i.e. there exists an E(i)∈Db
Hi(M)such that Res(E
(i)) =E(i) 1 . Then
InfGH1(E(1)⊗V(1)
1 ), . . . ,Inf
G H1(E
(1)⊗V(1)
m1), . . . ,
InfGH `(E
(`)
⊗V1(`)), . . . ,InfGH `(E
(`) ⊗Vm(`)
`) is an exceptional sequence of Db
G(M) with V
(i) 1 , . . . , V
(i)
mi being all the irreducible representations of Hi. The induced exceptional sequence of DbG(M) is full if and only if the original exceptional sequence ofDb(M)is full.
Proof. In [11] the Theorem is only stated in the case of full exceptional sequences. But one can easily infer from the proof that non-full exceptional sequences also
induce non-full exceptional sequences.
In order to apply Theorem 4.3 we have to reorder the sequence consisting of the E(α) as follows. For a multi-index α∈ [1, k]n we denote the unique non-decreasing representative of itsSn-orbit by nd(α). Then we define a total order
Cof [1, k]n by
αCβ:⇐⇒ (
nd(α)<lexnd(β) or nd(α) = nd(β) andα <lexβ
Now the groupSn acts transitively on the blocks consisting of allE(α) with fixed
nd(α) because ofσ∗E(α)'E(σ−1·α). Furthermore, everyE(α) has a canonical Stab(α)-linearisation given by permutation of the factors in the box product. It remains to show that (E(α))αwith the ordering given byCis still an exceptional
sequence. This follows by Remark 4.2 and the last item of the following lemma.
Lemma 4.4. Let α, β∈[1, k]n.
(1) Let nd(α) = nd(β) but α 6= β. Then there exists an i ∈ [n] such that
αi< βi.
(2) Let σ∈Sn. Then there exists an i∈[1, n] such thatαi< βi if and only
if there exists a j∈[1, n] such that(σ·α)j <(σ·β)j.
(3) If nd(α)<lexnd(β), then there exists ani∈[1, n]with αi < βi. (4) Let αCβ. Then there exists ani∈[1, n]such that αi< βi.
Proof. If nd(α) = nd(β), we have α1+· · ·+αn = β1+· · ·+βn. This shows
using (2) that α= nd(α). Let σ∈ Sn be such that β =σ·nd(β) and letm :=
min{`∈[1, n]|nd(α)`6= nd(β)`}. Then nd(α)m<nd(β)m. Ifσ(m)≤m, we have ασ(m)= nd(α)σ(m)≤nd(α)m<nd(β)m=βσ(m). Ifσ(m)> m, there exists` > m
such thatσ(`)≤m. This yields
ασ(`)= nd(α)σ(`)≤nd(α)m<nd(β)m≤nd(β)`=βσ(`).
Finally, (4) follows from (1) and (3).
We summarize all of the above in the following
Proposition 4.5. If α ∈ [1, k]n is a non-decreasing multi-index and Vi(α) is an irreducible representation of Hα = Stab(α), then the collection of objects E(α, Vi(α)) := InfSn
Hα
E(α)⊗Vi(α) forms an exceptional sequence of Db
Sn(Z
n).
The induced sequence is full if and only if the original sequence onDb(Z)is full.
Remark 4.6. An exceptional sequence is called strong if all the higher exten-sion groups between its members vanish. Using Equation (4.1) one can show that (E(α, Vi(α)))α,i is strong if and only if (E`)` is strong. Thus, in the case
that the full exceptional sequence E1, . . . , Ek of Db(Z) is strong, there is an
equivalence of triangulated categories Db
Sn(Z
n)'Db(Mod−End
Sn(M)) where M:=L
α,iE(α, V
(α)
i ); see [6].
Remark 4.7. Using [12] one can also construct semi-orthogonal decompositions of Db
Sn(Z
n) out of general semi-orthogonal decompositions of Db(Z) in a similar way. Let us describe the special case that Db(Z) = hA, Ei with E ∈ Db(Z) an exceptional object. Then there is an induced semi-orthogonal decomposition of DbSn(Zn) with components
B(k, ρ) :=hInfSn
Sk×Sn−k (E
k⊗ρ)
B
|Res(B)'Ak+1· · ·An, Ai∈ Ai
for k = 0, . . . , n and ρ an irreducible representation ofSk. Note that B(n, ρ) is
spanned by the exceptional objectEn⊗ρandB(n−1, ρ)' A.
Remark 4.8. IfB∈Db(Z) is a tilting object, so is⊕
ρ∈P(n)(Bn⊗ρ)∈DbSn(Z
n).
Remark 4.9. Any Enriques surface has a completely orthogonal exceptional se-quence (Ei)i of length 10 consisting of line bundles; see [30]. The induced
ex-ceptional sequence in Db
Sn(X
n) 'Db(X[n]) is again completely orthogonal and the same holds for the corresponding sequence of spherical objects on CYn. By
[18, Cor. 2.4] it follows that the associated spherical twists give an embedding
Z`(10,n) //Aut(Db(CYn)) where `(10, n) denotes the length of the induced
5. The truncated universal ideal functor
The arguments in this section follow those of [22, Sect. 6]. Recall that for any surfaceZ and anyn≥2 there is the Bridgeland–King–Reid–Haiman equivalence
Φ : Db(Z[n])−' // DbS n(Z
n).
Note that Z[n] is smooth under our assumptions, while this is not true anymore when dimZ ≥3 and n ≥ 3. The key new observation is that the functor ˆF := ΦF: Db(Z) //Db
Sn(Z
n) forZ =S a surface can be truncated to a functorGin
such a way thatGRG'FˆRFˆ and that this functor generalises in a nicer way to varieties of arbitrary dimension than ˆF does.
IfZ =Sis a surface, then ˆF00= Φ FMOZn = FMC•, whereC
•is the complex concentrated in degrees 0, . . . , n−1 given by
0 //
n
M
i=0
ODi // M
|I|=2
ODI⊗aI // M
|I|=3
ODI⊗aI. . . //OD[n]⊗a[n] //0 ;
see [28]. ForI⊂[1, n] :={1, . . . , n}, the reduced subvarietyDI ⊂S×Sn is given
byDI ={(y, x1, . . . , xn)|y=xi∀i∈I} andaI is the sign representation ofSI.
Furthermore, ˆF0 = Φ FMO
S×S[n] 'FMOS×Sn and the induced map ˆF
0 //Fˆ00 is
given by the morphism of kernels OS×Sn //C0 =⊕iODi whose components are given by restriction of sections. We setG00:= FMC0. As explained in [22, Sect. 6],
the main steps in the computation of the formulas of [28] and [17] can be translated into the following statement
ˆ
F0RFˆ00'Fˆ0RG00, Fˆ00RFˆ0'G00RFˆ0, Fˆ00RFˆ00'G00RG00.
(5.1)
Definition 5.1. Let Z be a smooth projective variety of arbitrary dimension d
andn≥2. Thetruncated universal ideal functor G= FMG: Db(Z) //DbS n(Z
n)
is the Fourier–Mukai transform whose kernel is the complex
G:=G•:= (0 //OZ×Zn // ⊕ni=1OD
i //0)∈D b
Sn(Z×Z
n).
Thus, there is the triangle of FM transformsG //G0 //G00 with
G0:= FMG0 = ˆF0= H∗(Z,−)⊗ OZn, G00= FMG00= InfSn
Sn−1p
∗
ntriv.
For E ∈ Db(Z) we have G00(E) = ⊕n
i=1p∗iE (see also Subsection 2.8 for details
about the inflation functor Inf and its right adjoint Res). The right adjoints are
G0R= FMG0R = H∗(Zn,−)Sn⊗ωZ[d], G0R=OZnωZ[d],
G00R= FMG00R= [−]Sn−1◦pn∗◦Res
Sn−1
Sn , G
00R=⊕n
i=0ODi. In the surface case Equation (5.1) gives the following
Lemma 5.2. If Z=S is a surface, FˆRFˆ'GRG.
See [19, Sec. 5.5] for a further investigation of the relation between the func-torsF andG.
We compute the compositions of the kernels: G0RG0 = (O
ZωZ)⊗SnH∗(OZ)[d],
G0RG00= (OZωZ)⊗Sn−1H∗(OZ)[d],
G00RG0 = (O
ZOZ)⊗Sn−1H∗(OZ),
G00RG00=O
∆⊗Sn−1H∗(OZ)⊕(OZOZ)⊗Sn−2H∗(OZ).
The induced mapG0RG0 //G0RG00under these isomorphisms is given by evaluation as follows. Let (ei)ibe a basis of H∗(OZ) = Hom(OZ,OZ[∗]). Then the component
(OZωZ)·ei1· · ·ein[d] //(OZωZ)·ei1· · ·eˆik· · ·ein[d] iseikid[d]. The component
(OZOZ)⊗Sn−1H∗(OZ) //(OZOZ)⊗Sn−2H∗(OZ)
ofG00RG0 //G00RG00 is given in the same way and the component (OZOZ)⊗Sn−1H∗(OZ) //O∆⊗Sn−1H∗(OZ)
is the restriction map. The mapG00RG0 //G0RG0 is given as follows. Let (θi)
i be
the basis of H∗(OZ)∨ dual to (ei)i. Furthermore, let θi correspond to the
mor-phism θie ∈ Hom(OZ, ωZ[d− ∗]) ' Hom(OZ,OZ[∗])∨ under Serre duality. Then
the component
(OZOZ)·ei1· · ·eˆik· · ·ein //(OZωZ)·ei1· · ·ein[d] isθiek.
In the following we will use the commutative diagram
G00RG //
G00RG0 //
G00RG00
G0RG //
G0RG0 //
G0RG00
GRG //GRG0 //GRG00. (5.2)
with exact triangles as columns and rows in order to deduce formulae forGRG=
FMGRG.
5.1. The case of an even dimensional Calabi–Yau variety. IfZis a Calabi– Yau variety of even dimensiond, then H∗(OZ) =C[0]⊕C[−d] andSkH∗(OZ) = C[0]⊕C[−d]⊕ · · · ⊕C[−dk]. Let u ∈ Hd(OZ) be the basis vector whose dual θ∈Hom(OZ,OZ[d])∨ corresponds to id∈Hom(OZ,OZ) under Serre duality. We
denote the induced degreed`basis vector ofSkH∗(O
Lemma 5.3. For a Calabi–Yau variety Z of even dimension dwe have
GRG'id⊕[−d]⊕ · · · ⊕[−d(n−1)].
Proof. By the description of the last subsection, the componentsOZ2·u` //OZ2·
u` of G00RG0 //G00RG00 andG0RG0 //G0RG00 equal the identity. By [15, Lem. 5], the top of the diagram (5.2) is isomorphic to
G00RG //
OZ2[−d(n−1)] //
O∆([0]⊕[−d]⊕ · · · ⊕[−d(n−1)])
G0RG //O
Z2[−d(n−1)] //0
where the middle vertical map is the component OZ2 ·un−1 //OZ2un[d] of the
morphism G00RG0 //G0RG0. By the above description it is the identity. Now the claim follows by the octahedral axiom. Alternatively, chase through long exact sequences of cohomology as in [1, Sect. 2.4].
Theorem 5.4. IfZis a Calabi–Yau variety of even dimension, then the truncated universal ideal functorG: Db(Z) //Db
Sn(Z
n)is a
Pn−1-functor.
Proof. By the previous lemma, condition (1) of the definition of aPn−1-functor is
satisfied.
The proof that condition (2) is satisfied is analogous to the proof in the case of the non-truncated functor when X is a K3 surface. Basically, one has to go through [1, Sect. 2.5] and replaceF byG,q∗ byp∗n◦triv : Db(Z) //DbSn−1(Z
n)
and its right adjointq∗by (−)Sn−1◦pn∗,g∗ by Inf : DbSn−1(Z
n) //Db
Sn(Z
n) and
its right adjoint g! by Res, H∗(O
S[n]) by H∗(OZn)Sn, 2 byd, and the symplectic formσ, which occurs in [1, Sect. 2.5], by a generator of Hd(OZ).
Very roughly, the idea is the following: GRG is identified with the direct summand (p∗n ◦ triv)R(p∗n ◦triv) ' id ⊗H∗(Zn−1,OZn−1)Sn−1 of G00RG00 and
the monad structure of (p∗n ◦triv)R(p∗n ◦triv) is given by the cup product on
H∗(Zn−1,OZn−1)Sn−1 which has the right form.
Condition (3) is easy to check since all occurring autoequivalences are simply
shifts.
Remark 5.5. If Z is an odd dimensional Calabi–Yau variety, then G fails to be spherical in an interesting way (compare Remark 3.7). Indeed, in this case H∗(OZ) =C[0]⊕C[−d] andSkH∗(OZ) =C[0]⊕C[−d] fork≥1. The reason for
the vanishing of the higher degrees is that the symmetric product is taken in the graded sense.
We can then check that for n ≥ 3 we have GRG ' O∆([0]⊕[−d]) by
us-ing that the componentOZ2⊗Sn−1H∗(OZ) //OZ2⊗Sn−2H∗(OZ) of the map
5.2. The caseH∗(OZ) =C[0]. In this case also SkH∗(OZ) =C[0] fork≥1.
Proposition 5.6. The functor Gis fully faithful, i.e.GRG=O∆.
Proof. The maps G00RG0 //G00RG00 and G0RG0 //G0RG00 are the identity on the componentsOZ2. Hence,G00RG ' O∆⊗Sn−1H∗(OZ)[−1] andG0RG= 0. It follows
thatGRG=O∆⊗Sn−1H∗(O
Z) =O∆.
Remark 5.7. In particular, whenZ =S is a surface, the reader will recognize, using Lemma 5.2, the proof of Theorem 1.2.
5.3. The orthogonal complement ofimG. Let, as in the previous subsection,
Z be a smooth projective variety such that the structure sheafOZ is exceptional.
We set A := O⊥
Z which means that we have the semi-orthogonal decomposition
Db(Z) =hA,OZi. For A∈ Awe have G0(A) = 0 and thusG(A) = Inf(OZn−1
A)[1]. Let ρn be the standard representation of Sn which is given by the short
exact sequence
0 //C−ι // Cn //ρn //0
whereι:C //Cndenotes the diagonal embedding of the trivial representation into
the permutation representation. We haveG0(OZ) =OZnandG00(OZ) =OZn⊗Cn from whichG(OZ) =OZn⊗ρn follows. We see that im(G) equals the component hB(n−1,C),OZn⊗ρniof the semi-orthogonal decomposition described in Remark 4.7. Thus, we get a description of im(G)⊥ and⊥im(G). For example, in the case
n= 2 we have im(G)⊥=hωZn,B(0,C)iwith
B(0,C) =hInf(A1A2)|A1, A2∈ Ai.
Let nowZ=X be an Enriques surface. By tensoring the exceptional sequence of Remark 4.9 byE10∨ we get the completely orthogonal exceptional sequence of line bundlesL1, . . . ,L9,OX in Db(X). Using the above description of the image of G
one can check thatLn
j ∈im(G)⊥∩⊥im(G) = ker(GR)∩ker(GL) forj= 1, . . . ,9.
The objects Ln
j are in fact also contained in ker(F
R)∩ker(FL) which proves
Remark 3.14. Indeed, fori= 2, . . . , nthe FM transform FMCi−1is the composition
Db(X)−−−Maitriv// Db
Si×Sn−i(X)
p∗X
− // DbS
i×Sn−i(X×X
n−i)
(δi×id)∗
−−−− // DbSi×Sn−i(Xi×Xn−i)−Inf// DbSi×Sn−i(Xi×Xn−i)
whereai is the sign representation ofSi andδi:X //Xi is the diagonal
embed-ding. The left adjoint FMLCi is
Db(X) (−)
Si×Sn−iMa i
←−−−−−−−−−−−DbSi×Sn−i(X)←−−pX! DbSi×Sn−i(X×Xn−i) (δi×id)∗
←−−−−−DbSi×S n−i(X
i
×Xn−i)←−−Res DbSi×S n−i(X
i
×Xn−i).
(5.3)
Now, letLbe one of theLj. Then (δi×id)∗L=L⊗iLi. TheSi-action onL⊗iis
given by permuting the tensor factors. SinceLis a line bundle, this action is trivial. Thus, after tensoring by Mai the Si-invariants vanish, hence FM
L
Ci−1(Ln) = 0.
FMRCi−1 is given by the composition (5.3) with (δi×id)∗ replaced by (δi×id)!and
pX! replaced bypX∗. Thus, also FMRCi−1(Ln) = 0 and henceLn ∈ker(FR).
References
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Mathematisches Institut, Universit¨at Bonn, Endenicher Allee 60, 53115 Bonn, Germany
E-mail address:[email protected]
Fachbereich Mathematik der Universit¨at Hamburg, Bundesstraße 55, 20146 Ham-burg, Germany
