New derived autoequivalences of Hilbert schemes and generalised Kummer varieties

0.21 MB·English

by  Andreas Krug

NEW DERIVED AUTOEQUIVALENCES OF HILBERT SCHEMES AND
GENERALIZED KUMMER VARIETIES
ANDREASKRUG
Abstract. We show that for every smooth projective surface X and every n ≥ 2 the
push-forward along the diagonal embedding gives a n−1-functor into the Sn-equivariant
derived category of Xn. Using the Bridgeland–King–PReid–Haiman equivalence this yields
3 a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In
1 the case that the canonical bundle of X is trivial and n = 2 this autoequivalence coincides
0 with theknownEZ-sphericaltwist inducedbytheboundaryoftheHilbertscheme. Wealso
2 generalise the 16 spherical objects on the Kummer surface given by the exceptional curves
n to n4 orthogonal n−1-Objects on thegeneralised Kummervariety.
P
a
J

1
2 1. Introduction

] For every smooth projective surface X over and every n there is the Bridgeland–
G
C ∈ N
King–Reid–Haiman equivalence (see [BKR01] and [Hai01])
A
. Φ: Db(X[n]) ≃ Db (Xn)
h −→ Sn
at between the bounded derived category of the Hilbert scheme of n points on X and the Sn-
m equivariant derived category of the cartesian product of X. In [Plo07] Ploog used this to
[ give a general construction which associates to every autoequvalence Ψ Aut(Db(X)) an
∈
autoequivalence α(Ψ) Aut(Db(X[n])) on the Hilbert scheme. Recently, Ploog and Sosna
1
∈
v [PS12] gave a construction that produces out of spherical objects (see [ST01]) on the surface
0 n-objects(see[HT06])onX[n]whichinturninducefurtherderivedautoequivalences. Onthe
7 P
9 otherhand,thereareonlyveryfewautoequivalencesofDb(X[n])knowntoexistindependently
4 of Db(X):
.
1 There is always an involution given by tensoring with the alternating representation
0 • in Db (Xn), i.e with the one-dimensional representation on which σ S acts via
3 Sn ∈ n
1 multiplication by sgn(σ).
: Addington introduced in [Add11] the notion of a n-functor generalising the n-
v • P P
i objects of Huybrechts and Thomas. He showed that for X a K3-surface and n 2
X ≥
the Fourier–Mukai transform F : Db(X) Db(X[n]) induced by the universal sheaf
a
ar is a n−1-functor. This yields an autoeq→uivalence of Db(X[n]) for every K3-surface
P
X and every n 2.
≥
For X = A an abelian surface the pull-back along the summation map Σ: A[n] A
• →
is a n−1-functor and thus induces a derived autoequivalence (see [Mea12]).
P
The boundary of the Hilbert scheme ∂X[n] is the codimension 1 subvariety of points
•
representingnon-reducedsubschemesofX. Forn = 2itequalsX[2] := µ−1(∆)where
∆
µ: X[2] S2X denotestheHilbert-Chowmorphism. Forn = 2andX asurfacewith
→
trivial canonical bundle it is known (see [Huy06, examples 8.49 (iv)]) that every line
bundleon the boundaryof theHilbert scheme is an EZ-sphericalobject (see [Hor05])
1
2 ANDREASKRUG
and thus also induces an autoequivalence. We will see in remark 4.6 that the induced
[2]
automorphisms given by different choices of line bundles on X only differ by twists
∆
with line bundles on X[2]. Thus, we will just speak of the autoequivalence induced
by the boundary referring to the automorphism induced by the EZ-spherical object
( 1).
Oµ|X[2] −
∆
In this article we generalise this last example to surfaces with arbitrary canonical bundle and
to arbitrary n 2. More precisely, we consider the functor F: Db(X) Db (Xn) which is
S
≥ → n
defined as the composition of the functor triv: Db(X) Db (X) given by equipping every
S
→ n
object with the trivial S -linearisation and the push-forward δ : Db (X) Db (Xn) along
n ∗ S S
n → n
the diagonal embedding. Then we show in section 3 the following.
Theorem 1.1. For every n with n 2 and every smooth projective surface X the
functor F: Db(X) Db (Xn∈) isNa n−1-fu≥nctor.
S
→ n P
In section 4 we show that for n = 2 the induced autoequivalence coincides under Φ with
the autoequivalence induced by the boundary. In section 5 we compare the autoequivalence
inducedbyF tosomeotherderivedautoequivalences ofX[n] showingthatitdiffersessentially
from the standard autoequivalences and the autoequivalence induced by F . In particular,
a
the Hilbert scheme always has non-standard autoequivalences even if X is a Fano surface. In
the last section we consider the case that X = A is an abelian surface. We show that after
restricting our n−1-functor from A[n] to the generalised Kummer variety K A it splits
n−1
P
into n4 pairwise orthogonal n−1-objects. They generalise the 16 spherical objects on the
P
Kummer surface given by the line bundles ( 1) on the exceptional curves.
C
O −
Acknowledgements: TheauthorwantstothankDanielHuybrechtsandCiaranMeachan
for helpful discussions. This work was supported by the SFB/TR 45 of the DFG (German
Research Foundation). As communicated to the author shortly before he posted this article
on the ArXiv, Will Donovan independently discovered the n-functor F.
P
2. n-functors
P
A n-functor is defined in [Add11] as a functor F: of triangulated categories
P A → B
admitting left and right adjoints L and R such that
(i) There is an autoequivalence H of such that
A
RF id H H2 Hn.
≃ ⊕ ⊕ ⊕···⊕
(ii) The map
RεF
HRF ֒ RFRF RF
→ −−−→
with ε being the counit of the adjunction is, when written in the components
H H2 Hn Hn+1 id H H2 Hn,
⊕ ⊕···⊕ ⊕ → ⊕ ⊕ ⊕···⊕
of the form
∗ ∗ ··· ∗ ∗
1
∗ ··· ∗ ∗
0 1  .
 ··· ∗ ∗
... ... ... ... ...
 
0 0 1 
··· ∗
NEW DERIVED AUTOEQUIVALENCES OF HILBERT SCHEMES AND KUMMER VARIETIES 3
(iii) R HnL. If and have Serre functors, this is equivalent to S FHn FS .
B A
≃ A B ≃
Inthe following we always consider the case that and are(equivariant) derived categories
A B
of smooth projective varieties and F is a Fourier–Mukai transform. The n-twist associated
P
to a n-functor F is defined as the double cone
P
P := cone(cone(FHR FR) id) .
F
→ →
The map defining the inner cone is given by the composition
FjR εFR−FRε
FHR FRFR FR
−−−→ −−−−−−→
where j is the inclusion given by the decomposition in (i). The map defining the outer cone
is induced by the counit ε: FR id (for details see [Add11]). Taking the cones of the
→
Fourier–Mukai transforms indeed makes sense, since all the occurring maps are induced by
maps between the integral kernels (see [AL12]). We set kerR := B RB = 0 . By the
{ ∈ B | }
adjoint property it equals the right-orthogonal complement (imF)⊥.
Proposition 2.1 ([Add11, section 3]). Let F: be a n-functor.
A → B P
(i) We have P (B)= B for B kerR.
F
∈
(ii) P F Hn+1[2].
F
◦ ≃
(iii) The objects in imF kerR form a spanning class of .
∪ B
(iv) P is an autoequivalence.
F
Example 2.2. (i) Let = Db(X) for a smooth projective variety X. A n-object (see
[HT06]) is an objectBE such that E ω E and Ext∗(E,E) =PH∗( n, ) as
∈ B ⊗ X ≃ ∼ P C
-algebras (theringstructureontheleft-handsideistheYonedaproductandon the
C
right-hand side the cup product). A n-object can be identified with the n-functor
P P
F: Db(pt) , E
→ B C 7→
with H = [ 2]. Note that the right adjoint is indeed given by R = Ext∗(E, ). The
−
n-twist associated to the functor F is the same as the n-twist associated to the
P P
object E as defined in [HT06].
(ii) A 1-functor is the same as a spherical functor (see [Ann07]) where the unit
P
η
id RF H
−→ →
splits. In this case there is also the spherical twist given by
ε
T := cone FR id .
F (cid:16) (cid:17)
−→
It is again an autoequivalence with T2 = P (see [Add11, p. 33]).
F F
Lemma 2.3. (i) Let Ψ Aut( ) such that Ψ H H Ψ. Then F Ψ is again a
∈ A ◦ ≃ ◦ ◦
n-functor with the property
P
P P .
F◦Ψ F
≃
(ii) Let Φ: be an equivalence of triangulated categories. Then Φ F is again a
B → C ◦
n-functor with the property that
P
P Φ Φ P .
Φ◦F F
◦ ≃ ◦
Proof. The proof is analogous to the proof of the corresponding statement for spherical func-
tors [Ann07, proposition 2]. (cid:3)
4 ANDREASKRUG
Corollary 2.4. Let E ,...,E be a collection of pairwise orthogonal (that means
1 n
∈ B
Hom∗(E ,E ) = 0 = Hom∗(E ,E ) for i = j) n-objects with associated twists p := P .
i j j i 6 P i Ei
Then
n Aut( ) , (λ ,...,λ ) pλ1 pλn
Z → A 1 n 7→ 1 ◦···◦ n
defines a group isomorphism n = p ,...,p Aut( ).
∼ 1 n
Z h i⊂ B
Proof. Bypart(ii)ofthepreviouslemmathep commutewhichmeansthatthemapisindeed
i
a group homomorphism onto the subgroup generated by the p . Let g = pλ1 pλn. Then
g(E ) = E [2nλ ] by proposition 2.1. Thus, g = id implies λ =i = λ =10.◦···◦ n (cid:3)
i i i 1 n
···
Lemma 2.5. Let X be a smooth variety, T Aut(Db(X)), and A,B Db(X) objects such
∈ ∈
that TA= A[i] and TB = B[j] for some i = j . Then A B and B A.
6 ∈ Z ⊥ ⊥
Proof. See [Add11, p. 11]. (cid:3)
Remark2.6. Thisshowstogetherwithproposition2.1thatfora n-functorF withH = [ ℓ]
P −
for some ℓ there does not exist a non zero-object A with T (A) = A[i] for any values of
F
∈ Z
i besides 0 and nℓ+2 because such an object would be orthogonal to the spanning class
−
imF kerR.
∪
3. The diagonal embedding
LetX beasmoothprojectivesurfaceover and2 n . Wedenotebyδ: X Xn the
diagonal embedding. We want to show that FC: Db(X≤) ∈DNb (Xn) given as the co→mposition
S
→ n
Db(X) triv Db (X) δ∗ Db (Xn)
S S
−−→ n −→ n
of the functor which maps each object to itself equipped with the trivial action and the
equivariant push-forward is a n−1-functor. Its right adjoint R is given as the composition
P
Db (Xn) δ! Db (X) [ ]Sn Db(Xn)
S S
n −→ n −−−→
of the usual right adjoint (see [LH09] for equivariant Grothendieck duality) and the functor
of taking invariants. We consider the standard representation ̺ of S as the quotient of the
n
regular representation n by the one dimensional invariant subspace. The normal bundle
C
sequence
0 T T N 0
X Xn|X
→ → → →
where N := N = N is of the form
δ X/Xn
0 T T⊕n N 0
→ X → X → →
where the map T T⊕n is the diagonal embedding. When considering T as a S -
X → X Xn|X n
sheaf equipped with the natural linearisation it is given by T n where n is the regular
X
representation. Thus, as a S -sheaf, the normal bundle equals⊗TC ̺. WeCalso see that the
n X
⊗
normal bundle sequence splits using e.g. the splitting
1
T n T , (v ,...,v ) (v + +v ).
X X 1 n 1 n
⊗C → 7→ n ···
Theorem 3.1 ([AC12]). Let ι: Z ֒ M be a regular embedding of codimension c such that
→
the normal bundle sequence splits. Then there is an isomorphism
c
(1) ι∗ι ( ) ( ) ( iN∨ [i])
∗ ≃ ⊗ M∧ Z/M
i=0
NEW DERIVED AUTOEQUIVALENCES OF HILBERT SCHEMES AND KUMMER VARIETIES 5
of endofunctors of Db(Z).
Corollary 3.2. Under the same assumptions, there is an isomorphism
c
(2) ι!ι ( ) ( ) ( iN [ i])
∗ ≃ ⊗ M∧ Z/M −
i=0
Proof. Tensorise both sides of (1) by ι! cN [ c]. (cid:3)
M Z/M
O ≃ ∧ −
Lemma 3.3. The monad multiplication ι!ει : ι!ι ι!ι ι!ι is given under the above isomor-
∗ ∗ ∗ ∗
→
phism (if chosen correctly) by the wedge pairing
c c c
( iN [ i]) ( jN [ j]) kN [ k].
M∧ Z/M − ⊗ M∧ Z/M − → M∧ Z/M −
i=0 j=0 k=0
Proof. For E Db(M) the object ι!E can be identified with om(ι ,E) considered as an
∗ Z
∈ H O
object in Db(Z). Under this identification the counit map om(ι ,E) E is given by
∗ Z
H O →
ϕ ϕ(1) (see [Har66, section III.6]). Now we get for F Db(Z) the identifications
7→ ∈
ι!ι ι!ι F om(ι ,ι ) om(ι ,ι ) F
∗ ∗ ≃ H ∗OZ ∗OZ ⊗OZ H ∗OZ ∗OZ ⊗OZ
and ι!ι F om(ι ,ι ) F under which the monad multiplication equals the
∗ ≃ H ∗OZ ∗OZ ⊗OZ
Yoneda product. It is known (see [LH09, p. 442]) that the Yoneda product corresponds to
the wedge product when choosing the right isomorphism. (cid:3)
In the case that ι = δ from above this yields the isomorphism of monads
2(n−1)
(3) δ!δ ( ) ( ) ( i(T ̺)[ i]).
∗ ≃ ⊗ M ∧ X ⊗ −
i=0
Lemma 3.4 ([Sca09a, Appendix B]). Let V be a two-dimensional vector space with a basis
consisting of vectors u and v. Then the space of invariants [ i(V ̺)]Sn is one-dimensional
∧ ⊗
if 0 i 2(n 1) is even and zero if it is odd. In the even case i= 2ℓ the space of invariants
≤ ≤ −
is spanned by the image of the vector ωℓ, where
k
ω = ue ve 2(V n),
X i ∧ i ∈ ∧ ⊗C
i=1
under the projection induced by the projection n ̺.
C →
Corollary 3.5. For a vector bundle E on X of rank two and 0 ℓ n 1 there is an
≤ ≤ −
isomorphism
[ 2ℓ(E ̺)]Sn = ( 2E)⊗ℓ.
∼
∧ ⊗ ∧
Proof. The isomorphism is given by composing the morphism
( 2E)⊗ℓ ℓ(E n) , x x x e x e
∧ → ∧ ⊗C 1⊗···⊗ ℓ 7→ X 1 i1 ∧···∧ ℓ iℓ
1≤i1<···<iℓ≤n
with the projection induced by the projection n ̺. (cid:3)
C →
We set H := 2T [ 2] = ω∨[ 2] = S−1.
∧ X − X − X
Corollary 3.6. There is the isomorphism of functors
RF id H H2 Hn−1.
≃ ⊕ ⊕ ⊕···⊕
6 ANDREASKRUG
Proof. This follows by formula (3) and corollary 3.5. (cid:3)
Lemma 3.7. The map HRF RF defined in condition (ii) for n-functors is for this pair
F ⇋ R given by the matrix → P
0 0 0 0
 ··· 
1 0 0 0
···
0 1 0 0 .
 ··· 
... ... ... ... ...
 
0 0 1 0
···
Proof. The generators ωℓ from lemma 3.4 are mapped to each other by wedge product. By
lemma 3.3 the monad multipliction is given by wedge product. (cid:3)
Lemma 3.8. There is the isomorphism SXnFHn−1 FSX.
≃
Proof. For Db(X) there are natural isomorphisms
E ∈
SXnFHn−1(E) = ωXn[2n]⊗δ∗(E ⊗ωX−(n−1)[−2(n−1)]) ≃ ωX⊠n⊗δ∗(E ⊗ωX−(n−1))[2]
PF
δ ( ω [2]) =FS ( ).
∗ X X
≃ E ⊗ E
(cid:3)
All this together shows theorem 1.1, i.e. that F = δ triv is indeed a n−1-functor.
∗
◦ P
4. Composition with the Bridgeland–King–Reid–Haiman equivalence
The isospectral Hilbert scheme InX X[n] Xn is defined as the reduced fibre product
⊂ ×
InX := (cid:0)X[n]×SnX Xn(cid:1)red with the defining morphisms being the Hilbert–Chow morphism
µ: X[n] SnX and the quotient morphism π: Xn SnX. Thus, there is the commutative
→ →
diagram
InX q Xn
−−−−→
p π
 
Xy[n] SnyX.
−−−µ−→
The Bridgeland–King–Reid–Haiman equivalence is the functor
Φ := FM triv = p q∗ triv: Db(X[n]) Db (Xn).
OIXn ◦ ∗◦ ◦ −→ Sn
By the results in [BKR01] and [Hai01] it is indeed an equivalence. The isospectral Hilbert
scheme can be identified with the blow-up of Xn along the union of all the pairwise diagonals
∆ = (x ,...,x ) Xn x = x (see [Hai01]). By lemma 2.3 the functor composition
ij 1 n i j
{ ∈ | }
Φ−1 F: Db(X) Db(X[n]) is again a n-functor and thus yields an autoequivalence of the
◦ → P
derived category of the Hilbert scheme.
Lemma 4.1. Let G: be a right-exact functor between abelian categories such that
A → B A
has a G-adapted class and let X• D−( ) be a complex such that q(X•) is G-acyclic for
∈ A H
every n . Then n(LG(X•)) = G n(X•) holds for every n .
∈ Z H H ∈ Z
Proof. This follows from the spectral sequence
Ep,q = LpG q(X•)= En = n(LG(X•)).
2 H ⇒ H
(cid:3)
NEW DERIVED AUTOEQUIVALENCES OF HILBERT SCHEMES AND KUMMER VARIETIES 7
If the surface X has trivial canonical bundle, it is known that any line bundle L on the
boundary ∂X[2] = X[2] of the Hilbert scheme of two points on X is an EZ-spherical object
∆
(see [Huy06, examples 8.49 (iv)]). That means that the functor
F˜ : Db(X) Db(X[2]) , j (L µ∗ E)
L → E 7→ ∗ ⊗ ∆
is a spherical functor where the maps j and µ come from the fibre diagram
∆
X[2] j X[2]
∆ −−−−→
µ∆ µ
 
Xy S2yX
−−−d−→
with d being the diagonal embedding. The map µ is a 1-bundle.
∆
P
Proposition 4.2. Let X be a smooth projective surface (with arbitrary canonical bundle).
Then there is an isomorphism of functors Φ−1 F F˜ , where Φ−1 is the inverse of
◦ ≃ Oµ∆(−1)
the BKRH-equivalence and F = δ triv: Db(X) Db (X2) from the previous section.
∗ S
◦ → 2
Proof. The functor Φ−1 is given by the composition [ ]S2 FMX2→X[2] with Fourier–Mukai
◦ Q
kernel = ∨ q∗ω [4]. The isospectral Hilbert scheme I2X is the blow-up of X2 along
Q OI2X⊗ X[2]
the diagonal. In particular, it is smooth. Let E = p−1(∆) be the exceptional divisor of the
blow up. The 1-bundles p : E X and µ : X[2] X are isomorphic via q: I2X X[2].
P ∆ → ∆ ∆ → →
The canonical bundle of the blow-up is given by ω = p∗ω (E). Let N be the normal
I2X ∼ X2⊗O
bundle of the codimension 4 regular embedding I2X X[2] X2. By adjunction formula
→ ×
4N = ω∨ ω = q∗ω∨ (E).
∧ ∼ X[2]×X2|I2X ⊗ I2X ∼ X[2] ⊗O
It follows by Grothendieck-Verdier duality for regular embeddings that
= ∨ q∗ω [4] 4N[ 4] q∗ω [4] (E).
Q OI2X ⊗ X[2] ≃ ∧ − ⊗ X[2] ≃ O
Here, the line bundle (E) is equipped with the natural S -linearisation which is trivial
2
over E, i.e. (E) =O ( 1) carries the trivial S -action. We need the following slight
OE Op∆ − 2
generalisation of [Huy06, Proposition 11.12] for a blow-up
E i X˜
−−−−→
π p
 
y y
Y X
−−−j−→
of a smooth projective variety X along a smooth subvariety Y of codimension c.
Lemma 4.3. For every Coh(Y) and every k there is an isomorphism
F ∈ ∈ Z
k(p∗j ) = i π∗ −kΩ ( k) .
H ∗F ∼ ∗(cid:16) F ⊗∧ π ⊗Oπ − (cid:17)
Proof. This can be proven locally. Hence, we may assume that Y = Z(s) is the zero locus of
a global section of a vector bundle of rank c. Thus, the blow-up diagram can be enlarged
E
8 ANDREASKRUG
to
E i X˜ ι ( )
−−−−→ −−−−→ P E
π p g
  
y y y
Y X X
−−−j−→ −−−id−→
where ι is a closed embedding of codimension c 1 such that the normal bundle M has the
−
property kM∨ = −kΩ ( k) (see [Huy06, p. 252]). The outer square is a flat base
∧ |E ∧ π ⊗Oπ −
change. It follows that
(4) ι p∗j ι ι∗g∗j ι ι∗ι i π∗ ι i (π∗ i∗ι∗ι )
∗ ∗F ≃ ∗ ∗F ≃ ∗ ∗ ∗ F ≃ ∗ ∗ F ⊗ ∗OX˜
where the last isomorphism is given by applying the projection formula two times. Now
k(ι∗ι )= −kM∨ is locally free for every k . By lemma 4.1 it follows that
H ∗OX˜ ∼ ∧ ∈ Z
k(π∗ i∗ι∗ι ) = π∗ i∗ k(ι∗ι ) = π∗ kM∨ = π∗ −kΩ ( k).
H F ⊗ ∗OX˜ ∼ F ⊗ H ∗OX˜ ∼ F ⊗∧ |E ∼ F ⊗∧ π ⊗Oπ −
By (4) also
ι k(p∗j ) = k(ι p∗j ) = k ι i (π∗ i∗ι∗ι ) = ι i k(π∗ i∗ι∗ι )
∗H ∗F ∼ H ∗ ∗F ∼ H (cid:0) ∗ ∗ F ⊗ ∗OX˜ (cid:1) ∼ ∗ ∗H F ⊗ ∗OX˜
which proves the assertion since ι : Coh(X˜) Coh( ( )) is fully faithful. (cid:3)
∗
→ P E
Remark 4.4. If X carries an action by a finite group G and Y is invariant under this action,
Galso acts ontheblow-upX˜. ThebundleM oftheproofisaquotientofthenormalbundle
|E
N = π∗N . In the case that there is a group action this quotient is G-equivariant.
E/ (E) ∼ Y/X
ThuPs, the formula of the lemma is in this case also true for Coh (X) with the action on
G
F ∈
the right hand side induced by the linearization of the wedge powers of M respectively N .
Y/X
In the case of the blow-up p: I2X X2 the center ∆ of the blow-up has codimension 2.
→
Thus,p∗δ is cohomologically concentrated indegree 0and 1. Sincep∗δ is concentrated
∗ ∗
on E wherFe S is acting trivially, one can take the invariants−even before apFplying the push-
2
forward along q: I2X X[2]. The group S acts on 0N trivially and on N
2 ∆/X2 ∆/X2
→ ∧
alternating. Hence, by the previous remark we have for Coh(X) equipped with the
trivial group action that 0(p∗δ )S2 = p∗ and 1(p∗δF ∈)S2 = 0. In particular, every
Coh(X)isacyclicundHerthef∗uFnctor[ ]S∆2Fp∗ δ Htrivwit∗hF[ ]S2 p∗ δ triv( ) p∗ ( ).
FTh∈is implies that [ ]S2 p∗ δ triv( •) ◦p∗◦( ∗•◦) for every •◦ D◦b(∗X◦). TFoge≃ther∆wFith
= (E) = ( 1◦) =◦ ∗ ◦( 1) tFhis p≃rov∆esFproposition 4.2F. ∈ (cid:3)
Q|E ∼ OE ∼ Op∆ − ∼ Oµ∆ −
Remark 4.5. The proposition says in particular that F˜ is also a spherical functor in
Oµ∆(−1)
the case that ω is not trivial. One can also prove this directly and for general L instead of
X
( 1).
Oµ∆ −
Remark 4.6. Since X[2] is a 1-bundle over X, every line bundle on it is of the form
∆ P
L = µ∗ K (i) for some K PicX and i . The canonical bundle of X[2] is given by
∼ ∆ ⊗Oµ∆ ∈ ∈ Z ∆
µ∗ ω ( 2). TheHilbert-Chowmorphismµisacrepantresolution,i.e. ω = µ∗ω .
∆ X⊗Oµ∆ − X[2] ∼ S2X
Thus,
ω =µ∗ (ω ) = µ∗ ω2 .
X[2]|X[2] ∼ ∆ S2X|∆ ∼ ∆ X
∆
Let N = (X[n]) be the normal bundle of X[2] in X[2]. By adjunction formula it is
OX[n] ∆ ∆
∆
given by µ∗ ω∨ ( 2). There is a line bundle D PicX[2] (namely the determinant
∆ X ⊗Oµ∆ − ∈
NEW DERIVED AUTOEQUIVALENCES OF HILBERT SCHEMES AND KUMMER VARIETIES 9
[2] [2] [2]
of the tautological sheaf ) such that 2c (D) = [X ] = c ( (X )) (see [Leh99, lemma
OX − 1 ∆ 1 O ∆
3.8]). Its restriction D is of the form µ∗ M (1) for some M PicX. Using this,
|X[2] ∆ ⊗Oµ∆ ∈
∆
we can rewrite for a general L = µ∗ K (i) PicX[2] the spherical functor F˜ as
∆ ⊗ Oµ∆ ∈ ∆ L
F˜ = Mi+1 F˜ M for some Q PicX where M is the autoequivalence given
L D ◦ Oµ∆(−1) ◦ Q ∈ Q
by tensor product with Q. The analogous of lemma 2.3 for spherical functors thus yields
t = Mi+1 t M−(i+1) where t is the spherical twist associated to F˜ .
L D ◦ Oµ∆(−1)◦ D L L
Remark 4.7. For general n 2 every object in the image of Φ−1 F still is supported on
≥ ◦
X[n] = µ−1(∆).
∆
5. Comparison with other autoequivalences
In the following we will denote the n-twist associated to F respectively Φ−1 F by
b ∈ Aut(DbSn(Xn)) ∼= Aut(Db(X[n])). InPthe case that n = 2 the functor F is spheric◦al (see
example 2.2 (ii)). We denote the associated spherical twist by √b.
Proposition 5.1. The automorphism b is not contained in the group of standard automor-
phisms
Aut(Db(X[n])) DAut (X[n]) = Aut(X[n])⋉Pic(X[n])
⊃ st ∼ Z×(cid:16) (cid:17)
generated by shifts, push-forwards along automorphisms and taking tensor products by line
bundles. The same holds in the case n = 2 for √b.
Proof. Let [ξ] X[n] X[n], i.e. suppξ 2. Then by remark 4.7 and proposition 2.1 (i),
∈ \ ∆ | | ≥
we have b( ([ξ])) = ([ξ]). Let g = [ℓ] ϕ M DAut (X[n]) where M is the functor
∗ L st L
C C ◦ ◦ ∈
E E L for an L PicX[n]. Then g( ([ξ])) = (ϕ([ξ]))[ℓ]. Thus, the assumption b = g
7→ ⊗ ∈ C C
implies ℓ = 0 and also ϕ= id, since X[n] X[n] is open in X[n]. Thus, the only possibility left
\ ∆
for b DAut (X[n]) is b = M for some line bundle L which can not hold by proposition 2.1
st L
∈
(ii). The proof that √b / DAut (X[n]) is the same. (cid:3)
st
∈
In [Plo07] Ploog gave a general construction which associates to derived autoequivalences
of the surface X derived autoequivalences of the Hilbert scheme X[n]. Let Ψ Aut(Db(X))
with Fourier–Mukai kernel Db(X X). Theobject ⊠n Db(Xn Xn) ca∈rries a natural
S -linearisation given by pPer∈mutation×of the box factorPs. T∈hus, it ind×uces a S -equivariant
n n
derived autoequivalence α(Ψ) := FMP⊠n of Xn. This gives the following.
Theorem 5.2 ([Plo07]). The above construction gives an injective group homomorphism
α: Aut(Db(X)) → Aut(DbSn(Xn)) ∼= Aut(Db(X[n])).
Remark 5.3. For every ϕ Aut(X) we have α(ϕ ) = (ϕn) where ϕn is the S -equivariant
∗ ∗ n
∈
automorphism of Xn given by ϕ(x ,...,x ) = (ϕ(x ),...,ϕ(x )). Furthermore, ϕ acts on
1 n 1 n
X[n] by the morphism ϕ[n], which is given by ϕ[n]([ξ]) = [ϕ(ξ)], and on Xn by the morphism
ϕn. Since the Bridgeland–King–Reid–Haiman equivalence is the Fourier–Mukai transform
with kernel the structural sheaf of InX, it is Aut(X)-equivariant, i.e. Φ (ϕ[n]) (ϕn) Φ.
∗ ∗
◦ ≃ ◦
Thus, α(ϕ ) Aut(Db (Xn)) corresponds to ϕ[n] Aut(Db(X[n])). For L PicX we have
∗ S ∗
α(ML) = ML∈⊠n wherenL⊠n is considered as a Sn-∈equivariant line bundle w∈ith the natural
lineariPzaictiXon[n.]UisntdheerlΦinethbeuanudtloemorp:h=isµm∗(M(LL⊠⊠nn)Sconr)r(essepeon[Kdrsut1o2M, leDmLm∈aA9u.2t(])D.b(X[n])) where
L L
D ∈ D
10 ANDREASKRUG
Lemma 5.4. (i) For every automorphism ϕ Aut(X) we have b α(ϕ ) = α(ϕ ) b
∗ ∗
∈ ◦ ◦
and for n = 2 also √b α(ϕ ) = α(ϕ ) √b.
∗ ∗
◦ ◦
(ii) For every line bundle L Pic(X) we have b α(M ) = α(M ) b and for n = 2 also
L L
∈ ◦ ◦
√b α(M ) = α(M ) √b.
L L
◦ ◦
Proof. We have α(ϕ ) F F ϕ and α(M ) F F Mn. The assertions now follow by
∗ ◦ ≃ ◦ ∗ L ◦ ≃ ◦ L
lemma 2.3 (for √b one has to use the analogous result [Ann07, proposition 2] for spherical
twists). (cid:3)
Let G Aut(Db(X[n])) be the subgroup generated by b, shifts, and α(DAut (X)).
st
⊂
Proposition 5.5. The map
S: (Aut(X)⋉Pic(X)) Aut(Db (Xn)) , (k,ℓ,Ψ) bk [ℓ] α(Ψ)
S
Z×Z× → n 7→ ◦ ◦
defines a group isomorphism onto G.
Proof. By the previous lemma, b indeed commutes with α(Ψ) for Ψ DAut (X). Together
st
∈
with theorem 5.2 and the fact that shifts commute with every derived automorphism, this
shows that S is indeed a well-defined group homomorphism with image G. Now consider
g = bk [ℓ] α(ϕ ) α(M ) and assume g = id. For every point [ξ] X[n] X[n] we have
◦ ◦ ∗ ◦ L ∈ \ ∆
g(C([ξ])) = C([ϕ(ξ)])[ℓ] which shows ℓ = 0 and ϕ = id, i.e. g = bk ◦ ML⊠n. Hence, for
A Db(X) its image under F gets mapped to g(FA) = F(A N)[k(2n 2)] for some line
bu∈ndle N on X, which shows that k = 0. Finally, g = ML⊠n is⊗trivial only−if L = OX. (cid:3)
Remark 5.6. Again, the analogous statement with b replaced by √b holds.
Letnow X bea K3-surface. Inthis case Addingtonhas shown in[Add11]that theFourier–
Mukai transform F : Db(X) Db(X[n]) with kernel the universal sheaf is a n−1 functor
a Ξ
→ I P
with H = [ 2]. Here, Ξ X X[n] is the universal family of length n subschemes. We
− ⊂ ×
denote the associated n−1-twist by a and in case n = 2 the spherical twist by √a.
P
Lemma 5.7. For every point [ξ] X[n] ∂X[n], i.e. ξ = x1,...,xn red with pairwise distinct
∈ \ { }
x , the object a( ([ξ])) is supported on the whole X[n]. In case n = 2 the same holds for the
i
C
object √a( ([ξ])).
C
Proof. WesetforshortA = ([ξ]). WewillusetheexacttriangleofFourier–Mukaitransforms
C
F F′ F′′ with kernels ′ = and ′′ = . The right adjoints form the exact
→ → P OX×X[n] P OΞ
triangle R′′ R′ R with kernels ′′ = ∨[2] and ′ = [2]. Over the open subset
→ → Q OΞ Q OX×X[n]
X[n] ∂X[n], the universal family Ξ is smooth and thus on Ξ the object ∨ is a line
\ |X[n]\∂X[n] OΞ
bundle concentrated in degree 2. This yields
R′′(A) = [0] , R′(A) = H∗(X[n],A) [2] = [2].
ξ X X
O ⊗O O
SettingHi = i(R(A))thelongexactcohomologysequencegivesH−2 = ,H−1 = ,and
X ξ
H O O
Hi = 0forallothervaluesofi. TheonlyfunctorinthecompositionF = pr (pr∗ ( ) )
X[n]∗ X ⊗IΞ
that needs to be derived is the push-forward along pr . The reason is that the non-derived
X[n]
functors pr∗ as well as pr∗ ( ) are exact (see [Sca09b, proposition 2.3] for the latter).
X X ⊗OΞ
Thus, there is the spectral sequence
Ep,q = p(F(Hq)) = En = n(FR(A))
2 H ⇒ H
associated to the derived functor pr . It is zero outside of the 1 and 2 row. Now
X[n]∗ − −
F′( )= H∗(X, ) = ⊕n [0] and F′′( ) is also concentrated in degree zero since
Oξ Oξ ⊗OX[n] OX[n] Oξ