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