Twisted cohomology of the Hilbert schemes of points on surfaces

TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF A CES MARC A. NIEPER- WISSKIR CHEN Abstract. W e calculate the co homology spaces of the Hilb ert sc hemes of points on surfaces with v alues in locally constan t systems. F or that purp ose, we generalise I. Gro jnoswki’ s and H. Nak a jim a’ s description of the ordinary cohomology in terms of a F ock space represen tation to the t wisted case. W e further generalise M. Lehn’s work on th e action of the Vir asoro algebra to the t wisted case. Building on work by M. Lehn and Ch. Sorger, we then give an explicit description of the cup-pro duct in the t wisted case whenev er the surface has a n umerically trivial canonical divisor. W e formulate our results in a wa y that they apply to the pro jectiv e and non-pro jective case in equal measure. As an application of our metho ds, we give explicit mo dels for the cohomol- ogy rings of the generalised Kummer v arieties and of a s er ies of certain ev en dimensional Calabi–Y au manifolds. Contents 1. Int ro duction and re s ults 1 2. The F o ck space descr iption 4 3. The Virasor o algebra in the twisted case 7 4. The b oundary op erator 9 5. The ring structure 10 6. The generalis ed Kummer v arie ties 14 References 15 1. Introduction and resul ts Let X be a quasi-pro jective smo oth surface ov er the complex num b ers . W e denote by X [ n ] the Hilb ert scheme of n p oints on X , parametris ing zero- dimensional subschemes of X of length n . It is a qua si-pro jective v a r iety ([Gro 61]) and s mo oth of dimensio n 2 n ([F o g 68]). Recall that the Hilb ert scheme X [ n ] can viewed as a resolution of the n -th s y mmetric power X ( n ) := X n / S n of the surface X by virtue of the Hilb ert–Chow morphism ρ : X [ n ] → X ( n ) , whic h ma ps eac h zer o-dimensional subscheme ξ o f X to its supp ort supp ξ co un ted with m ultiplicities. Let L b e a lo cally c onstant system (alwa ys ov er the complex n umbers and of rank 1) ov er X . W e can view it as a functor from the fundamen tal group oid Π of X to the categ ory o f o ne-dimensional complex vector s pa ces. The fundamental group oid Π ( n ) of X ( n ) is the quotient g roup oid of Π n by the natural S n -action by [Bro8 8 ] o r (in terms o f the fundamental gr o up) [B ea83]. Thus Date : Nov ember 21, 2018. The author would like to thank the Max Planck Institute for M athematics in Bonn f or its hospitalit y and supp ort during the preparation of this paper. 1 2 MARC A. NIEP ER-WISSKIRCHEN we can co nstruct fr o m L a lo c a lly co nstant system L ( n ) on X ( n ) by setting L ( n ) ( x 1 , . . . , x n ) := O i L ( x i ) , for ea ch ( x 1 , . . . , x n ) ∈ X ( n ) (for the no tio n o f the tensor pro duct ov er an un- ordered index set see , e .g ., [LS0 3]). This induces the lo cally free s ystem L [ n ] := ρ ∗ L ( n ) on X [ n ] . W e are interested in the c a lculation of the cohomolo gy space L n ≥ 0 H ∗ ( X [ n ] , L [ n ] [2 n ]). Besided the natural gr ading given by the coho mological degree it carries a weightin g given by the num b er of p oints n . Likewise, the sym- metric algebra S ∗ ( L ν ≥ 1 H ∗ ( X, L ν [2])) ca rries a g r ading by cohomo logical degr ee and a weigh ting, which is defined such that H ∗ ( X, L [2 ] ν ) is o f pur e weight ν . The firs t result of this pap er is the fo llowing: Theorem 1. 1. Ther e is a natu r al ve ctor sp ac e isomorphism M n ≥ 0 H ∗ ( X [ n ] , L [ n ] [2 n ]) → S ∗   M ν ≥ 1 H ∗ ( X, L ν [2]   that r esp e cts the gr ading and weighting. F or L = C , the triv ia l sy stem, this re s ult has already app eared in [Gro96] and [Nak 97]). Theorem 1 .1 is prov en by defining a Heisenberg Lie alg ebra h X,L , whose under- lying vector space is given by M n ≥ 0 H ∗ ( X, L n [2]) ⊕ M n ≥ 0 H ∗ c ( X, L − n [2]) ⊕ C c ⊕ C d and by showing that L n ≥ 0 H ∗ ( X [ n ] , L [ n ] [2 n ]) is an irr educible low est weigh t rep- resentation of this Lie algebra, as is done in [Nak97] for the un twisted cas e. Let G b e a finite subgroup of the group of lo cally cons tant sys tems on X . Via the ma pping L → L [ n ] , which is in fact an isomor phism b etw een the gr oups o f lo cally constant systems on X and X [ n ] , resp ectively , G becomes a subgroup of the group of lo cally co nstant systems on X [ n ] . Such a group naturally defines a Galois cov ering η : GX [ n ] → X [ n ] of degr ee | G | with η ∗ C = L L ∈ G L . Let us call this cov ering the G -covering of X [ n ] . (In cas e that G is the group of a ll loca lly constant systems on X [ n ] , the G -covering is the universal one for n ≥ 2.) Using the Ler ay sp ectral sequence for η , which already deg enerates at the E 2 -term, the cohomology of GX [ n ] can b e co mputed b y Theorem 1.1: Corollary 1.2. Ther e is a natura l ve ctor sp ac e isomorphism M n ≥ 0 H ∗ ( GX [ n ] , C [2 n ]) → M L ∈ G S ∗   M ν ≥ 1 H ∗ ( X, L ν [2])   that r esp e cts the gr ading and weighting. W e then pro ceed in the pap er by defining a twisted version v X,L of the Virasor o Lie a lgebra, who se under lying vector space will be given by M n ≥ 0 H ∗ ( X, L n ) ⊕ M n ≥ 0 H ∗ c ( X, L − n ) ⊕ C c ⊕ C d . (Note the different gra ding compa red to h X,L .) W e define an action o f v X,L on L n ≥ 0 H ∗ ( X [ n ] , L [ n ] [2 n ]) by generalis ing r esults o f [Leh99] to the twisted, not nec- essarily pro jective ca se. As in [Leh99], we calculate the commutators of the op er a- tors in h X,L with the b oundary op era tor ∂ that is given b y multiplying with − 1 2 of TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 3 the exceptional divisor class of the Hilb ert– Chow mo rphism. It turns out that the same relations a s in the un twisted, pro jective case hold. The next main result of the pap er is a decriptio n of the ring structure when- ever X has a numerically trivial divisor. F ollowing ideas in [LS0 3], we intro duce a family of explicitely describ ed g raded unital a lgebras H [ n ] asso ciated to a G - weigh ted (non- c ounital) gra ded F r ob enius alg ebra H o f degree d . F or example, H = L L ∈ G H ∗ ( X, L [2 ]) is s uch a F rob enius a lgebra of deg ree 2 . The following holds for e a ch n ≥ 0: Theorem 1.3. Ass ume t hat X has a nu meric al ly t rivial c anonic al divisor. Then ther e is a natur al isomorphism M L ∈ G H ∗ ( X [ n ] , L [ n ] [2 n ]) → M L ∈ G H ∗ ( X, L [2 ]) ! [ n ] of ( G -weighte d) gr ade d algebr as of de gr e e 2 n . F or L = C , and X pro jectiv e, this theor em is the main r e sult in [LS03 ]. The idea of the pro of of Theo rem 1 .3 is not to r einv ent the wheel but to study how everything ca n a lready b e deduced from the more sp ecial case consider ed in [LS03]. Again by the Leray sp ectra l sequence, also T he o rem 1.3 has a na tural application to the co homology ring of the G - c overings of X [ n ] : Corollary 1.4. Ther e is a natura l isomorphism H ∗ ( GX [ n ] , C [2 n ]) → M L ∈ G H ∗ ( X, L [2 ]) ! [ n ] of gr ade d unital algebr as of de gr e e 2 n . W e want to point out a t least t wo applicatio ns of o ur results. The first one is the computation of the cohomolog y ring of certain families of Ca labi–Y a u manifolds of even dimension: Let X b e a n Enriques surface. Let G b e the group of all lo cally constant s ystems on X , i.e. G ≃ Z / (2 ). W e denote the non-trivial element in G by L . The Ho dge diamo nds of H ∗ ( X, C [2]) and H ∗ ( X, L [2 ]) ar e given by 1 0 0 0 10 0 0 0 1 and 0 0 0 1 10 1 , 0 0 0 resp ectively . Denote by X { n } the (tw o - fold) universal cov er o f X [ n ] . By Remar k 2.6, the isomorphism of Co r ollary 1.2 is in fact a n iso morphism of Ho dge structures. It follows that H k, 0 ( X { n } , C ) = ( C for k = 0 or k = 2 n , a nd 0 for 0 < k < 2 n . In conjunction w ith Coro lla ry 1.4, we hav e thus proven: Prop ositio n 1. 5 . F or n > 1 , t he ma nifold X { n } is a Calabi–Y au m anifold in the strict sense. Its c ohomolo gy ring H ∗ ( X { n } , C [2 n ]) is natur al ly isomorphic to ( H ∗ ( X, C [2]) ⊕ H ∗ ( X, L [2 ])) [ n ] . Our s e cond applica tio n is the calcula tio n of the cohomo logy ring of the g ener- alised Kummer v arieties X [[ n ]] for an abe lia n sur face X . (A slightly less explicit description of this ring ha s been obtained by more sp ecial methods in [Bri0 2].) Re- call from [Bea8 3] that the gener alised Kummer v arie t y X [[ n ]] is defined as the fibre 4 MARC A. NIEP ER-WISSKIRCHEN ov er 0 of the mor phism σ : X [ n ] → X , which is the Hilb ert–Chow morphism follow ed by the s umma tio n mor phism X ( n ) → X of the ab elian surface. The gener alised Kummer surface is smo oth and of dimension 2 n − 2 ([Bea 83]). As ab ov e, let H b e a G -weigh ted graded F r ob enius algebr a of degree d . Assume further that H is equipp ed with a co mpatible structure of a Hopf a lgebra of degree d . F or each n > 0, we ass o ciate to such an a lgebra an explicitely descr ibed g raded unital algebra H [[ n ]] of degree n . In the following Theo rem, we view H ∗ ( X, C [2]) a s such an a lgebra (the Hopf algebra s tructure given by the gro up structure o f X ), where we give H ∗ ( X, C [2]) the trivial G -weigh ting for the group G := X [ n ] ∨ , the ch ara cter g roup of the group of n - torsion p oints on X . W e prove the following: Theorem 1. 6. Ther e is a natu r al isomorphism H ∗ ( X [[ n ]] , C [2 n ]) → ( H ∗ ( X, C [2])) [[ n ]] of gr ade d unital algebr as of de gr e e 2 n . W e s hould rema rk that most the “ha rd work” tha t is hidden b ehind the scenes has alrea dy b een done by others ([Gro96], [Nak97], [Leh99], [LQ W02], [LS03], etc.), and our own contribution is to see how the ideas and results in the cited paper s can b e a pplied an g eneralised to the twisted and to the non-pro jective case. R emark 1.7 . Let us finally mention that the restrictio n to a lg ebraic, i.e. quasi- pro jected surfaces, is unnecessary . O ur metho ds work equally w ell when we replace X by any complex surface . In this case, the Hilb ert schemes b ecome the Douady spaces ([Dou66]). 2. The Fock sp ac e description In this sec tion, we pr ov e Theore m 1.1 for a lo cally co nstant system L on X by the metho d tha t is us e d in [Nak 97] for the unt wisted case, i.e. by realising the cohomolog y spa ce of the Hilb ert schemes (with co efficients in a lo cally constant system) as a n irreducible represe n tation o f a Heisenberg Lie a lgebra. Let l ≥ 0 and n ≥ 1 b e tw o na tural num be rs. Set X ( l,n ) := n ( x ′ , x, x ) ∈ X ( n + l ) × X × X ( l ) | x ′ = x + nx o (w e write the union of uno rdered tuples a dditively). The preima ge in X [ n + l ] × X × X [ l ] of this closed subset under the Hilber t–Chow morphisms ρ is denoted b y X [ n,l ] (this incidence v ar ie ty has alr eady b een cons idered in [Nak97]). W e denote the pro jections of X ( l + n ) × X × X ( l ) onto its three factors by ˜ p , ˜ q and ˜ r , r esp ectively . L ikewise, we denote the three pr o jections of X [ l + n ] × X × X [ l ] by p , q and r . Lemma 2.1 . It is q ∗ L n ⊗ r ∗ L [ l ] | X [ n,l ] = p ∗ L [ l + n ] | X [ n,l ] . Pr o of. It is ˜ q ∗ L n ⊗ ˜ r ∗ L ( l ) | X ( n,l ) = ˜ p ∗ L ( l + n ) | X ( n,l ) . This follows fr o m ( ˜ q ∗ L n ⊗ ˜ r ∗ L ( l ) )( x + nx, x, x ) = L ( x ) ⊗ n ⊗ O x ′ ∈ x L ( x ′ ) = O x ′ ∈ x + nx L ( x ′ ) = ˜ p ∗ L ( l + n ) ( x + nx, x, x ) for every ( x + nx, x, x ) ∈ X ( l,n ) . By pulling bac k everything to the Hilbert sc hemes, the Lemma fo llows.  TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 5 Due to Lemma 2.1 and the fact that p | X [ l,n ] is prop er ([Nak97]), the op erator (a corres p o ndence , see [Nak97]) N : H ∗ ( X, L n [2]) × H ∗ ( X [ l ] , L [ l ] [2 l ]) → H ∗ ( X [ l + n ] , L [ l + n ] [2( l + n )]) , ( α, β ) 7→ PD − 1 p ∗ (( q ∗ α ∪ r ∗ β ) ∩ [ X [ l,n ] ]) is well defined. Her e, PD : H ∗ ( X [ l + n ] , L [ n + l ] [2( l + n )]) → H BM ∗ ( X [ l + n ] , L [ n + l ] [ − 2( l + n )]) is the Poincar´ e-duality iso mo rphism b etw een the cohomology and the B orel–Mo o r e homology . (The degree shifts a r e chosen in a wa y that N a n op erator of degree 0, see [L S03].) F urthermor e, q × r | X × X [ l ] is prop er ([Nak9 7]). Thus w e can a ls o define an op er - ator the o ther way round: N † : H ∗ c ( X, L − n [2]) × H ∗ ( X [ n + l ] , L [ l + n ] [2]) → H ∗ ( X [ l ] , L [ l ] [2 l ]) , ( α, β ) 7→ ( − 1) n PD − 1 r ∗ ( q ∗ α ∪ p ∗ β ∩ [ X [ l,n ] ]) As in [Nak97], we will use these op erators to define an a ction of a Heisenber g Lie a lgebra o n V X,L := M n ≥ 0 H ∗ ( X [ n ] , L [ n ] ) . F or this, let A b e a weighted, graded F ro b enius algebra o f degr e e d (over the complex nu mbers), that is a weighted and gra ded vector spa ce ov er C with a (g raded) commutativ e and ass o ciative multiplication of degree d and weight 0 and a unit element 1 (neces sarily o f degree − d and weight 0) to gether with a linear form R : A → C of degr ee − d and weight 0 s uch that for ea ch weight ν ∈ Z the induced bilinear form h· , ·i : A ( ν ) × A ( − ν ) → C , ( a, a ′ ) 7→ R A aa ′ is non- degenerate (of degree 0). Here A ( ν ) denotes the weigh t spa ce of w eight ν . In particular, all weigh t s paces are finite-dimensional. In the cas e of a tr ivial weigh ting, this notio n of a grade d F rob enius algebr a has alrea dy app eare d in [LS03]. Example 2.2 . The vector spac e A X,L := M ν ≥ 0 H ∗ ( X, L ν [2]) ⊕ M ν ≥ 0 H ∗ c ( X, L − ν [2]) is naturally a weight ed, graded F rob enius algebra of degree 2 (= dim X ) as follows: The gra ding is g iven by the co ho mology gr ading. T he weigh ting is defined by defining H ∗ ( X, L ν [2]) of pure w eight ν for ν ≥ 0 and H ∗ c ( X, L − ν [2]) of pure w eight − ν . The multiplication is given by the cup-pr o duct, whe r e we v iew the pr o duct of an ordinar y cohomolo gy class and of a coho mology class w ith co mpact supp ort as an ordinar y cohomo logy cla ss whenever the res ulting weight is s trictly p ositive, and as a cohomolog y class with compact s uppo rt otherwise. The linear form R is given by ev aluating a class with compact supp ort on the fundamental class of X . F or such a weigh ted, gr aded F r ob enius algebr a A w e set h A := A ⊕ C c ⊕ C d . W e define the structure of a weighted, graded Lie a lgebra o n h A by defining c to b e a central element o f weigh t 0 and degree 0, d an element of weigh t 0 a nd degree 0 and by setting the following commutator relations: [ d , a ] := n · a for ea ch element a ∈ A of weigh t n , and [ a , a ′ ] = h [ d , a ] , a ′ i c for elements a, a ′ ∈ A . Definition 2.3. T he Lie a lg ebra h A the Heisenb er g algebr a asso ciate d to A . 6 MARC A. NIEP ER-WISSKIRCHEN F or A = A X,L , we set h X,L := h A . W e define a linear map q : h X,L → End( V X,L ) as follows: Let l ≥ 0 and β ∈ V X,L ( l ) = H ∗ ( X [ l ] , L [ l ] [2 l ]). W e set q ( c )( β ) := β , and q ( d )( β ) := l β . F or n ≥ 0 , and α ∈ A X,L ( ν ) = H ∗ ( X, L ν [2]), we set q ( α )( β ) := N ( α, β ). F or α ∈ A X,L ( − ν ) = H ∗ c ( X, L − ν [2]), we set q ( α )( β ) := N † ( α, β ). Finally , we set q ( α )( β ) = 0 for α ∈ A X,L (0) = H ∗ ( X, C ) ⊕ H ∗ c ( X, C ). Prop ositio n 2.4. The map q is a weighte d, gr ade d action of h X,L on V X,L . Pr o of. This Propo sition is pr oven in [Nak97] for the un twisted c ase, i.e. for L = C . The pro of there is ba sed on calculating commutators on the level of cycles of the corres p o ndence s defined by the incidence schemes X [ l,n ] . These commutators are independent of the lo cally constant s ystem used. Thus the pro of in [Nak9 7] also applies to this mo re g eneral ca se.  Example 2 .5 . Let α = P α (1) ⊗ · · · ⊗ α ( n ) ∈ H ∗ ( X ( n ) , L ( n ) [2 n ]) = S n H ∗ ( X, L [2 ]) (w e use the Sweedler notation to denote elements in tensor pro ducts). The pull-back of α by the Hilber t–Chow morphism ρ : X [ n ] → X ( n ) is then given b y ρ ∗ α = 1 n ! X q ( α (1) ) · · · q ( α ( n ) ) | 0 i , where | 0 i is the unit 1 ∈ H ∗ ( X [0] , C ) = C . W e will use Pro po sition 2.4 to prov e our fir s t Theor em. Pr o of of The or em 1.1. The vector space ˜ V X,L := S ∗ ( L ν ≥ 1 H ∗ ( X, L ν [2])) carr ies a unique structure as an h X,L -mo dule such that c acts as the identit y , d acts by m ultiplying with the weigh t, α ∈ H ∗ ( X, L n ) for n ≥ 1 acts by multiplying with α , and α ∈ H ∗ ( X, C ) ⊕ H ∗ c ( X, C ) acts by zer o. By the represe ntation theor y of the Lie a lgebras of Heisenber g-type, this is an ir reducible lowest weigh t re pr esentation of h V ,L , which is generated b y the low est weight vector 1, which is of weight 0. The h V ,L -mo dule V X,L also has a vector of weight 0, namely | 0 i . Thus, there is a unique mo rphism Φ : ˜ V L → V L of h L -mo dules that ma ps 1 to | 0 i . This will b e the inv er se o f the isomorphism men tioned in Theorem 1.1. It remains to show tha t Φ is bijectiv e. The injectivit y follows fro m the fact that ˜ V X,L is irr educible as an h X,L -mo dule. In or de r to prov e the surjectivit y , we will deriv e upp er bounds on the dimensions of the weigh t spaces of the right hand side V X,L (see also [Leh04] ab out this pr o of metho d). By the Leray sp ectral sequence a sso ciated to the Hilber t–Chow morphism ρ : X [ n ] → X ( n ) , such an upper b ound is provided b y the dimension of the sp ectral sequence’s E 2 -term H ∗ ( X ( n ) , R ∗ ρ ∗ L [2 n ]). As shown in [GS93], it follows fro m the Beilinson–Ber nstein–Deligne–Gabb er deco mp os ition theor em that R ∗ ρ ∗ Q [2 n ] = M λ ∈ P ( n ) ( i λ ) ∗ Q [2 ℓ ( λ )] . Here, P ( n ) is the set o f all partitions of n , ℓ ( λ ) = r is the length of a par tition λ = ( λ 1 , λ 2 , . . . , λ r ), X ( λ ) := { P r i =1 λ i x i | x i ∈ X } ⊂ X ( n ) , and i λ : X ( λ ) → X ( n ) is the inclusio n map. Set L ( λ ) := i ∗ λ L ( n ) . By the pro jection for mula, it follows that R ∗ ρ ∗ L [2 n ] = L λ ∈ P ( n ) ( i λ ) ∗ L ( λ ) [2 ℓ ( λ )]. Thu s, an upper b ound o n the dimensio n of H ∗ ( X [ n ] , L [ n ] [2 n ]) is pro vided b y the dimension of L λ ∈ P ( n ) H ∗ ( X ( λ ) , L ( λ ) [2 ℓ ( λ )]). By [GS93], this ca n b e see n to b e TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 7 isomorphic to M P i ≥ 1 iν i = n O i ≥ 1 S ν i H ∗ ( X, L i [2]) , where each ν i ≥ 0. It follows that the upp er bo und given by the E 2 -term is exa ctly the dimension of the n -th weigh t spa ce o f ˜ V X,L . Thus the dimensio n of the weight spaces of V X,L cannot b e greater than the dimensions of the weight spaces of ˜ V X,L . Thu s the Theorem is prov en.  R emark 2.6 . Assume that X is pro jective. In this cas e, the (t wisted) cohomolo gy spaces of X and its Hilb ert s chemes X [ n ] carry pur e Ho dge structure s. A s the isomorphism of Theorem 1.1 is defined by algebraic corresp ondences (i.e. by corre- sp ondences of Ho dge type ( p, p )), it follows that the isomorphism in Theorem 1.1 is co mpatible with the natural Ho dge structures on b oth sides. In terms of Ho dge num ber s, the following equation enco des our result: X n ≥ 0 Y i,j h i,j ( X [ n ] , L [ n ] [2 n ]) p i q j z n = Y m ≥ 1 Y i,j (1 − ( − 1 ) i + j p i q j z m ) − ( − 1) i + j h i,j ( X,L m [2]) 3. The Virasoro algebra in the twisted case T o e a ch w eighted, graded F r o b e nius algebra A of degree d , we asso ciate a skew- symmetric form e : A × A → C of degree d as follows: Let n ∈ Z . W e note that A ( n ) and A ( − n ) a re dual to ea ch other via the linear form R . Th us we can co nsider the linear map ∆( n ) : C → A ( n ) ⊗ A ( − n ) dual to the bilinear form h· , ·i : A ( n ) ⊗ A ( − n ) → C . W rite ∆( n )1 = P e (1) ( n ) ⊗ e (2) ( n ) in Sweedler notation. Then we define e b y setting e ( α, β ) := n X ν =0 ν ( n − ν ) 2 Z X e (1) ( ν ) e (2) ( ν ) αβ for all α ∈ A ( n ) whenever n ≥ 0. W e shall call this form the Euler form of A . Example 3.1 . Assume that A ( n ) ≡ A (0) for all n ∈ Z . In this case, we have e ( α, β ) = n 3 − n 12 Z eαβ for α ∈ A ( n ) with e := R P e (1) (0) e (2) (0) ([Leh99]). W e use the Euler form to define another Lie algebra asso cia ted to A . W e set v A := A [ − 2 ] ⊕ C c ⊕ C d . W e define the str ucture of a weigh ted, gr aded Lie algebra on v A be defining c to be a central elemen t or w eight 0 and degree 0, d an alement o f w eight 0 and degree 0 and by introducing the following commut ator r elations: [ d , a ] := n · a for each element a ∈ A [ − 2] of weigh t n , and [ a, a ′ ] := ( d a ) a ′ − a ( d a ′ ) − e ( a, a ′ ) for elements a, a ′ ∈ A . Definition 3.2. T he Lie a lg ebra v A is the V ir asor o algebr a asso ciate d to A . F or A = A X,L , we set v X,L := v A . The whole cons truction is a g e neralisation to the t wisted cas e of the Viraso ro algebr a found in [Leh99]. 8 MARC A. NIEP ER-WISSKIRCHEN W e now define a linear map L : v X,L → End( V X,L ) as follo ws: W e define L ( c ) to be the identit y , L ( d ) to b e multiplication w ith the weigh t, and for α ∈ A [ − 2] we set L ( α ) := 1 2 X X ν ∈ Z : q ( e (1) ( ν )) q ( e (2) ( ν ) α ): , where the n ormal or der e d pr o duct : aa ′ : of tw o op erators is defined to be aa ′ if the weigh t of a is gre a ter or equal to the weight of a ′ and is defined to b e a ′ a if the weigh t of a ′ is grea ter than the weigh t of a . The following Lemma is prov en for the unt wis ted case in [Leh99]. Lemma 3.3 . F or α ∈ A V ,L [ − 2] and β ∈ A V ,L , we have [ L ( α ) , q ( β )] = − q ( α [ d , β ]) . Pr o of. Let α ∈ A V ,L [2]( n ) and β ∈ A V ,L ( m ) with n, m ∈ Z . In the following calculations we omit a ll Ko s zul signs ar ising from commuting the gra ded elements α and β . By definition, we have [ L ( α ) , q ( β )] = 1 2 P P ν [: q ( e (1) ( ν )) q ( e (2) ( ν ) α ): , q ( β )], where ν r uns thro ugh all integers. As the commutator o f tw o op er ators in h V ,L is central, w e do not hav e to pay a tten tion to the o r der of the factors of the normally ordered pro duct when calculating the commutator: [: q ( e (1) ( ν )) q ( e (2) ( ν ) α ): , q ( β )] = ν h e (1) ( ν ) , β i q ( e (2) ( ν ) α ) + ( n − ν ) h e (2) ( ν ) α, β i q ( e (1) ( ν )) . As h· , ·i is of w eight zer o, the first summand is only non- zero for ν = − m , while the second summand is only non-zero for ν = n + m . Thus we have [ L ( α ) , q ( β )] = − m 2 X  h e (1) ( − m ) , β i q ( e (2) ( − m ) α ) + h e (2) ( n + m ) α, β i q ( e (1) ( n + m ))  . As e (1) ( · ) is the dual basis to e (2) ( · ), the right ha nd side simplifies to − mq ( αβ ), which prov es the Lemma.  W e use Lemma 3.3 to prov e the following Pro po sition, which has alrea dy ap- pea red in [Leh99] for the unt wisted, pro jective cas e: Prop ositio n 3.4 . The map L is a weighte d, gr ade d action of t he Vir asor o algebr a v X,L on V X,L . Pr o of. Let α ∈ A [ − 2]( m ) and β ∈ A [ − 2 ]( n ) with m, n ∈ Z . W e hav e to prove that [ L ( α ) , L ( β )] = ( m − n ) L ( αβ ) − e ( α, β ). W e follow ideas in [FLM88]. In all summations be low, ν runs through all integers if not sp ecified otherwis e. W e b egin with the case n 6 = 0 a nd m + n 6 = 0. In this ca se, by Lemma 3.3, it is [ L ( α ) , L ( β )] = 1 2 " L ( m ) , X X ν q ( e (1) ( ν )) q ( e (2) ( ν ) β ) # = 1 2 X X ν ( − ν ) q ( e (1) ( ν ) α ) q ( e (2) ( ν ) β ) + X X ν ( ν − n ) q ( e (1) ( ν )) q ( e (2) ( ν ) αβ ) ! . As P q ( e (1) ( ν )( α ) q ( e (2) ( ν ) β ) = q ( e (1) ( ν + m )) q ( e (2) ( ν − m ) αβ ), the r ight ha nd side is eq ual to 1 2 X X ν  ( − ν ) q ( e (1) ( ν + m )) q ( e (2) ( ν + m ) αβ ) + ( ν − n ) q ( e (1) ( ν )) q ( e (2) ( ν ) αβ )  , which is nothing e lse than ( m − n ) L ( αβ ). Note that e ( α, β ) = 0 in this case. TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 9 The next case we s tudy is m > 0 and n = − m . In o rder to ensur e conv ergence in the following calculations we hav e to split up L ( β ) as follows: L ( β ) = X X ν ≥ m q ( e (1) ( ν ) β ) q ( e (2) ( ν )) + X X ν 0: Let β ∈ A X, C ( − m ) = H ∗ c ( X, C [2]). Consider an op en embedding j : X → ˆ X of X int o a s mo oth, pro jective surface ˆ X . W e denote the corre s po nding embeddings X [ n ] → ˆ X [ n ] also by the letter j . Denote the 1 in A ˆ X , C ( m ) = H ∗ ( X, C [2]) b y 1( m ). As all constructio ns co ns idered so far a r e functor ia l (in the appr o priate sense s ) with resp ect to op en embedding s , we hav e j ∗ [ q ′ (1( m )) , q ( j ∗ β )] | 0 i = [ q ′ ( j ∗ 1( m )) , q ( β )] | 0 i . The right hand side is given by m 2 R K m β , where K m is the class corr esp onding to X . By the c a lculations in [Leh9 9], the left hand side is giv en b y m 2 | m |− 1 2 R K ˆ X j ∗ β , where K ˆ X is the ca no nical divisor class of ˆ X . As j ∗ K ˆ X = K X , we s e e that K m = | m |− 1 2 K also holds in the non- pr o jective case, which proves the P rop osition.  Corollary 4.3. F or al l α ∈ A X,L , t he fol lowing holds: q ′ ( α ) = L ([ , dα ]) + q ( K ([ , dα ])) . Pr o of. This can b e deduced from 4 .2 as the resp e c tiv e statement for the un twisted, pro jective case in [Leh99] is prov en.  5. The ring structure F rom now on, we assume that the canonical divisor of X is num erica lly triv ial. Let H b e a non-c oun ital gr ade d F r ob enius algebr a of de gr e e d (over t he c omplex numb ers) , that is a g r aded vector spac e ov er C with a (gra de d) commutativ e a nd asso ciative m ultiplicatio n of deg ree d a nd a unit element 1 (of degre e − d ) together with a co asso ciative and co commutativ e H -mo dule homomor phism ∆ : H → H ⊗ H of degree d . (W e rega rd H ⊗ H as an H -alge br a b y multiplying on the left factor.) The map ∆ is ca lle d the diagonal . Example 5.1 . Let H b e a gr aded F r ob enius a lgebra of degr ee d . The dual ∆ to the m ultiplication map H ⊗ H → H with resp ect to h· , ·i makes H a non-counital, graded F r ob enius a lgebra of deg r ee d . In this context, the integral R of the F r o be nius algebra is the c ounit of H . Let G b e a finite ab elian group, which will b e written a dditively in the sequel. A G -weigh ting o n H is a n a ction of the character group G ∨ of G o n H . In o ther words, H comes together with a weigh t decomp osition of the form L L ∈ G H ( L ), where each χ ∈ G ∨ acts on H ( L ) by multiplication with χ ( L ). TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 11 Example 5.2 . Let G b e a finite subgr o up of the group of lo cally constant systems on X , written additively . The G -weigh ted vector spa c e H X,G := M L ∈ G H ∗ ( X, L [2 ]) is natura lly a non- counital G -weigh ted, graded F rob enius a lg ebra of degr ee 2 as follows: the grading is given by the cohomolog ical g r ading. The multiplication is given b y the cup pr o duct. T he diago nal is given b y the pr o p e r push-forward δ ∗ : H X,G → H X,G ⊗ H X,G that is induced by the diagona l map δ : X × X → X . By iterated applicatio n, ∆ induce maps ∆ : H → H ⊗ n with n ≥ 1. W e deno te the restr iction o f ∆ : H → H ⊗ n to H ( nL ), L ∈ G , follow e d by the pro jection o nt o H ( L ) ⊗ n by ∆( L ) : H ( nL ) → H ( L ) ⊗ n . The element e := ( ∇ ◦ ∆(0))(1) ∈ H is called the Euler class of H , where ∇ : H ⊗ H → H is the multiplication map. There is a construction given in [LS03] that ass o ciates to eac h graded F rob enius algebra H o f degr ee of d a sequence of graded F ro benius algebr a s H [ n ] (whose degrees are given by nd ). W e ex tend this cons tr uction to G -weigh ted not necessar ily counital F rob enius a lgebras as follows: F o r each L ∈ G , set H n ( L ) := M σ ∈ S n   O B ∈ σ \ [ n ] H ( | B | L )   σ and H n := M L ∈ G H n ( L ) , where [ n ] := { 1 , . . . , n } a nd σ \ [ n ] is the set o f orbits of the action of the cyclic gro up generated by σ on the set [ n ]. (Note that H n (0) = H (0) { S n } in the ter minology of [LS03].) The symmetric group S n acts on H n . The g raded vector space o f inv ar ia nt s, H S n n , is denoted by H [ n ] . Let f : I → J a surjection of finite sets a nd ( n i ) i ∈ I a tuple of integers. Fibre-wise m ultiplication yields ring homo morphisms ∇ I ,J := ∇ f : O i ∈ I H ( n i L ) → O j ∈ J H   X f ( i )= j n i L   of degree d ( | I | − | J | ). (These corres po nd to the ring homomorphism f I ,J in [LS03].) Dually , by using the dia gonal morphisms ∆( L ) and r elying on their coa s so ciativity and co commu tativity , we can define ∇ f -mo dule ho momorphisms ∆ J,I := ∆ f : O j ∈ J H   X f ( i )= j n i L   → O i ∈ I H ( n i L ) , which are also o f deg ree d ( | I | − | J | ). (These cor resp ond to the mo dule homomor - phisms f J,I in [LS03]). Let σ, τ ∈ S n be tw o p er m utations. By h σ , τ i we denote the subgro up of S n generated by the tw o p ermutations. Note that there are natur al sur jections σ \ [ n ] → h σ , τ i\ [ n ], τ \ [ n ] → h σ , τ i\ [ n ], and ( σ τ ) \ [ n ] → h σ, τ i\ [ n ]. The co rresp onding ring and mo dule ho momorphism are deno ted by ∇ σ, h σ ,τ i , etc., and ∆ h σ,τ i ,σ , etc. Let L, M ∈ G . W e define a linear map m σ,τ : O B ∈ σ \ [ n ] H ( | B | L ) ⊗ O B ∈ τ \ [ n ] H ( | B | M ) → O B ∈ ( σ τ ) \ [ n ] H ( | B | ( L + M )) by m σ,τ ( α ⊗ β ) = ∆ h σ,τ i , ( σ τ ) ( ∇ σ, h σ ,τ i ( α ) ∇ τ , h σ,τ i ( β ) e γ ( σ,τ ) ) , where the expression e γ ( σ,τ ) is defined a s in [LS0 3] (we have to use our Euler cla ss e , which is defined a bove). This defines a pr o duct H n ⊗ H n → H n which is given 12 MARC A. NIEP ER-WISSKIRCHEN by ( α σ ) · ( β σ ) := m σ,τ ( α, β ) σ τ for α σ ∈ H n ( L ) a nd β σ ∈ H n ( M ). This pro duct is asso ciative, S n -equiv ar iant, and of degree nd , which ca n be pr ov e n exactly a s the c o rresp onding statements ab out the pro duct of the rings H { S n } , which a re defined in [LS03]. The pr o duct bec omes (grade d) co mm utative when restr ic ted to H [ n ] . Thus we hav e made H [ n ] a g raded co mm utative, unital algebr a of degr ee nd . Definition 5.3. T he alg ebra H [ n ] is the n -th Hilb ert algebr a of H . In case G is trivial, the n - Hilber t a lgebra of H de fined here is ex actly the a lg ebra H [ n ] of [LS03]. F or non-tr ivial G , this is no long e r true. The underlying g raded vector space of P n ≥ 0 H [ n ] ( L ) is na turally isomor phic to S ( L ) := S ∗ ( L n ≥ 1 H ( nL )), namely as follows: Firs tly , w e int ro duce linear maps H n ( L ) → S ( L ), which are defined b y mapping a n element o f the form P σ ∈ S n N B ∈ σ \ [ n ] α σ,B σ to 1 n ! P σ ∈ S n Q B ∈ σ \ [ n ] α σ,B . The r estrictions o f these mor- phisms to the S ∗ -inv ar iant parts define a linear map L n ≥ 0 H [ n ] ( L ) → S ( L ). This map is an isomorphism, whic h ca n b e prov en exactly a s it is in [LS03] for trivial G . Recall that H ∗ ( X, C [2]) is a (trivially weigh ted) gra de d non-co unital F rob enius algebra of deg ree d . Lemma 5.4 . Ther e is a natu r al isomorphism H ∗ ( X, C [2]) [ n ] → H ∗ ( X [ n ] , C [2 n ]) of gr ade d unital algebr as of de gr e e nd . Pr o of. Recall the just defined iso morphism b et ween the spaces L n ≥ 0 H ∗ ( X, C [2]) [ n ] and S ∗ ( L ν > 0 H ∗ ( X, C [2])) (for L = C ). The comp osition of this is o morphism with isomorphism b etw een S ∗ ( L ν > 0 H ∗ ( X, C [2])) and L n ≥ 0 H ∗ ( X [ n ] , C [2 n ]) of Theo- rem 1 .1 induces by restriction the claimed isomorphism o f the Lemma on the level of g r aded vector spaces. That this isomor phism is in fact an iso morphism of unital algebr as, is proven in [LS03] for X being pro jective. The pr o of there does not use the fact that H ∗ ( X, C [2]) has a counit, in fact it only uses its diagona l map. It r elies o n the earlier work in [Leh9 9], which has b een ex tended to the no n-pro jective ca se ab ov e, and [LQ W02], which can similarly b e extended. Thus the pro o f in [L S0 3] also works in the non-pr o jective case, when we replac e the notion of a F rob enius algebra by the notion of a non-counital F ro benius alg ebra.  W e will now deduce Theorem 1.3 from Lemma 5.4: Pr o of of The or em 1.3. Let L, M ∈ G . Let λ = ( λ 1 , . . . , λ l ) b e a partition of n . Let ν i the m ultiplicit y o f i in λ , i.e. λ = P i ν i · i . Set X ( λ ) := Q i X ( ν i ) , and L ( λ ) := Q i pr ∗ i L ( ν i ) , where the pr i denote the pro jections onto the fa ctors X ( ν i ) . Let α = P α (1) · · · α ( r ) ∈ H ∗ ( X ( λ ) , L ( λ ) [2 l ]) = N i S ν i H ∗ ( X, L [2 ]). W e set | α i := X q ( α (1) ) · · · q ( α ( r ) ) | 0 i . By Theor e m 1.1, the co homology spac e H ∗ ( X [ n ] , L [ n ] [2 n ]) is linearly spanned by classes of the form | α i . Let µ = ( µ 1 , . . . , µ m ) b e another par tition of n and β ∈ H ∗ ( X [ µ ] , M [ µ ] [2 m ]). In order to describ e the ring str ucture of H ∗ ( X [ n ] , L [ n ] [2 n ]), we have to ca lc ulate the classes | α ∪ β i := | α i ∪ | β i in terms of the vector spa ce description g iven b y Theorem 1.1. This mea ns that we have to ca lc ulate the num b ers h γ | α ∪ β i := q ( γ ) | α ∪ β i ∈ H ∗ ( X [0] , C ) = C TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 13 for all γ ∈ H ∗ c ( X ( κ ) , (( LM ) − 1 ) ( κ ) [2 k ]) for all par titions κ = ( κ 1 , . . . , κ k ) of n , and w e hav e to sho w that they ar e equal to the num b ers that would come out if w e calculated the pro duct of α and β b y the right hand s ide of the claimed iso morphism of the Theo rem. The class | α i is given b y applying a sequence of cor resp ondences to the v acuum vector: Recall from [Nak9 7] how to comp ose co rresp ondences. It follows that | α i is g iven b y PD − 1 (pr 1 ) ∗ (pr ∗ 2 α ∩ ζ λ ) , where the symbols have the following meaning: The maps pr 1 and pr 2 are the pro jections o f X [ n ] × X ( λ ) onto its factors X [ n ] and X ( λ ) . F urther , ζ λ is a certain class in H BM ∗ ( Z λ ), where Z λ is the incidence v ariety Z λ := ( ( ξ , ( x 1 , x 2 , . . . )) ∈ X [ n ] × X ( λ ) | supp ξ = X i ix i ) in X [ n ] × X ( λ ) . (Note tha t pr ∗ 1 L [ n ] | Z λ = pr ∗ 2 L ( λ ) | Z λ , and tha t p | Z λ is prop er.) F or | β i and | γ i we g et similar express io ns. B y definition of the cup-pr o duct (pull-back a long the diagonal), it follows that h γ | α ∪ β i = h r ∗ γ ∪ p ∗ α ∪ q ∗ β , ζ λ,µ,κ i , where p , q , and r are the pro jections fro m X ( λ ) × X ( µ ) × X ( κ ) onto its three factor s, and ζ λ,µ,κ is a cer tain cla ss in H BM ∗ ( Z λ,µ,κ ) with Z λ,µ,κ :=    (( x 1 , x 2 , . . . ) , ( y 1 , y 2 , . . . ) , ( z 1 , z 2 , . . . )) | X i ix i = X j j y j = X k k z k    . (The incidence v ariety is prop er ov er an y of the thre e factors, so everything is well- defined.) The main p oint is now that the incidence v a riety Z λ,µ,κ and the homolog y class ζ λ,µ,κ are indep endent o f the lo cally constan t systems L and M . In particula r , we ca n calculate ζ λ,µ,κ once w e know the cup-pro duct in the case L = M = C . But this is the c a se tha t is desc r ib ed in Lemma 5 .4, which w e will analyse now. First o f all, the incide nce v ariety is given by Z λ,µ,κ = X σ,τ Z σ,τ where σ and τ run thr ough all per mu tations with cycle t yp e λ and µ , resp ectively , such that ρ := στ has cycle type κ . The v a rieties Z σ,τ are defined as follows: As the orbits of the group actio n o f h σ i on [ n ] c orresp ond to the entries of the par tition λ , there exists a natur a l map X σ \ [ n ] → X ( λ ) , whic h is given b y symmetrising. F urthermore the natural s urjection σ \ [ n ] → h σ, τ i\ [ n ] induces a diagonal embedding X h σ,τ i\ [ n ] → X σ \ [ n ] . Comp osing b oth ma ps , we g et a natur al map X h σ,τ i\ [ n ] → X ( λ ) . Analoguously , we get maps from X h σ,τ i\ [ n ] to X ( µ ) and X ( κ ) . T ogether, these maps define a diago nal embedding i τ ,σ : X h σ,τ i\ [ n ] → X ( κ ) × X ( λ ) × X ( µ ) . W e define Z σ,τ to b e the image of this map. By Lemma 5.4, the class ζ λ,µ,κ is given by P σ,τ ( i σ,τ ) ∗ ζ σ,τ , wher e ea ch clas s ζ σ,τ ∈ H BM ∗ ( X h σ,τ i\ [ n ] ) is Poincar´ e dual to c σ,τ e γ ( σ,τ ) . Here, c σ,τ is a certain combinatorial factor (po s sibly dep ending on σ and τ ), whos e precise v alue is of no concern for us. Having de r ived the v alue of ζ λ,µ,κ from Lemma 5 .4, we hav e thus calculated the v alue h γ | α ∪ β i . Now we hav e to compare this v alue with the one that is predicted by the des c rip- tion of the cup-pro duct given by the right hand side of the claimed iso morphism of the The o rem. With the sa me ana lysis as above, we find this v alue is also given by a 14 MARC A. NIEP ER-WISSKIRCHEN corres p o ndence on Z λ,µ,κ with the cla ss P τ ,σ ( i σ,τ ) ∗ c σ,τ PD( e γ ( σ,τ ) ) with the sa me combinatorial factor s c σ,τ as ab ov e. W e th us find that the claimed ring s tr ucture yields the correct v alue of h γ | α ∪ β i .  R emark 5.5 . One c a n also define a natural diagonal map for the Hilb ert alg ebras H [ n ] making them int o grade d, non-co unital F ro b e nius alg ebras o f degr ee nd . The isomorphism of Theo rem 1.3 then b eco mes an iso mo rphism of gra ded non-counital F rob enius algebr as. 6. The generalised Kummer v arieties Finally , we wan t to use Theor em 1.3 to study the cohomolo gy r ing of the g ener- alised Kummer v arieties. Let H b e a non-counital gra ded F ro b enius a lgebra of degr ee d that is moreov er endow ed with a compa tible s tructure of a co commutativ e Hopf alge br a of degree d . The com ultiplication δ o f the Hopf algebra structure is of degree − d . The counit o f the Ho pf a lg ebra structure is denoted by ǫ and is of deg ree d . W e further a ssume that H is also equipp ed with a G -weigh ting for a finite gro up G . Example 6.1 . Let X be a n ab elian surface . The group structur e o n X induces naturally a gr aded Hopf a lgebra struc tur e of degree 2 on the graded F rob enius algebra H ∗ ( X, C [2]). This alg ebra is also trivia lly X [ n ] ∨ -weigh ted, where X [ n ] ∨ ≃ ( Z / ( n )) 4 is the character g roup o f the group of n -tor sion p oints o n X . (T rivia lly weigh ted means that the only non-trivial X [ n ] ∨ -weigh t space of H ∗ ( X, C [2]) is the one c orresp onding to the iden tity element 0 .) Let n be a p ositive integer. Recall the definition o f the ( G -w eighted) Hilb ert algebra H [ n ] . Repea ted applica tion of the comultiplication δ induces a ma p δ : H → H ⊗ n = H id \ [ n ] , which is of degr ee − ( n − 1) d . Its image lies in the subspace of symmetric tens o rs. Thus we can define a map φ : H → H [ n ] with φ ( α ) := δ ( α ) id . One c a n easily check that this map is an alg ebra homomor phism of deg ree − ( n − 1) d , making H [ n ] int o an H -alg ebra. Define H [[ n ]] := H [ n ] ⊗ H C , where we view C a s an H -algebr a of degr ee d via the Hopf counit ǫ . It is H [[ n ]] a ( G -weigh ted) gra ded F rob enius a lgebra of degr ee n d . Definition 6.2. T he alg ebra H [[ n ]] is the n -th Kummer algebr a of H . The r e ason of this na ming is of co urse Theor em 1.6. Pr o of of The or em 1.6. Let n : X → X denote the mo r phism that maps x to n · x . There is a natural ca rtesian square (1) X × X [[ n ]] ν − − − − → X [ n ] p   y   y σ X − − − − → n X , where p is the pro jection o n the fir st factor and ν maps a pair ( x, ξ ) to x + ξ , the subscheme that is g iven by translating ξ by x ([Bea8 3]). Let G b e the character group of the Galo is gro up o f n , i.e. G = X [ n ] ∨ . E a ch element L of G co r resp onds to a lo ca lly constant system L o n X , a nd we hav e n ∗ C = L L ∈ G L . It follows that ν is the G -cov ering o f X [ n ] with ν ∗ C = L L ∈ G L [ n ] . TWISTED COHOMOLOGY OF THE HILBER T SCHEMES OF POINTS ON SURF ACES 15 T oge ther with Theorem 1.3, this lea ds to the claimed descr iption of the c oho- mology r ing of X [[ n ]] : Fir stly , there is a natura l iso morphism H ∗ ( X [[ n ]] , C [2 n ]) → H ∗ ( X × X [[ n ]] , C [2 n ]) ⊗ H ∗ ( X, C [2]) C of unital a lgebras (the tensor pro duct is taken with re s pec t to the map p ∗ and the Hopf c o unit H ∗ ( X, C [2]) → C ). By the Leray sp ectral sequence for ν a nd by (1), the right hand s ide is na turally isomorphic to H ∗ ( X [ n ] , ν ∗ C [2 n ]) ⊗ H ∗ ( X, C [2]) C = M L ∈ G H ∗ ( X [ n ] , L [ n ] [2 n ]) ⊗ H ∗ ( X, C [2]) C (where the tens o r pro duct is taken with resp ect to the map σ ∗ and the Hopf counit). By Theor e m 1.3, the algebr a L L ∈ G H ∗ ( X [ n ] , L [ n ] [2 n ]) is natura lly isomor phic to L L ∈ G H ∗ ( X, L [2 ]) [ n ] . Now H ∗ ( X, L [2 ]) = 0 unless L is the tr ivial bundle, whic h follows fr o m the fact that all classes in H ∗ ( X, C ) ar e inv a riant under the action of the Galo is gr o up of n , i.e. cor resp ond to the trivial character. Thus there is a natural isomorphis m M L ∈ G H ∗ ( X [ n ] , L [ n ] [2 n ]) → H ∗ ( X, C [2]) [ n ] , of G -w eighted algebras, where we endow H ∗ ( X, C [2]) with the tr ivial G -weigh ting. Under this isomor phism, the map σ ∗ corres p o nds to the homomor phism φ defined ab ov e by E xample 2.5. Th us we hav e proven the e xistence of a na tural is omorphism H ∗ ( X [[ n ]] , C [2 n ]) → H ∗ ( X, C [2]) [ n ] ⊗ H ∗ ( X, C [2]) C of unital, graded algebra s. B ut the rig ht hand side is nothing but H [[ n ]] , thus the Theorem is pr ov en.  References [Bea83] Arnaud Beauville, V ari´ et´ es k¨ ahleriennes dont la pr e mi` er e classe de Chern e st nul le. , J. Differ. Geom. 18 (1983), 755–782 (F renc h). [Bri02] Mic hael Britze, On the Cohomolo gy of Genera lize d Kummer V arietie s , Ph.D. thesis, Mathematisc h-Naturwissenscha ftliche F akult¨ at der U niv ersit¨ at zu K¨ oln, 2002. [Bro88] Ronald Brown, T op olo gy: A geo metric ac co unt of g ener al top olo gy, homotopy typ e s and the fundamental g r oup oid. R e v., up date d and exp ande d ed . , Ellis Horwoo d Series in Mathematics and i ts Applications. Chic hester (UK): El lis Horw oo d Ltd.; New Y ork etc.: Halsted Press., 19 88. [Dou66] A. 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