Documenta Math. 749Twisted Cohomology of theHilbert Schemes of Points on SurfacesMarc A. Nieper-Wißkirchen1Received: February 1, 2008Revised: April 23, 2009Communicated by Thomas PeternellAbstract. We calculate the cohomology spaces of the Hilbertschemes of points on surfaces with values in local systems. For thatpurp ose, we generalise I. Grojnoswki’s and H. Nakajima’s descriptionof the ordinary cohomology in terms of a Fock spac e representationto the twisted cas e. We make further non-trivial generalisations ofM. Lehn’s work on the action of the Virasoro algebra to the twistedand the non-projective case.Building on work by M. Lehn and Ch. Sorger, we then give an ex-plicit description of the cup-product in the twisted case whenever thesurface has a numerically trivial canonical divisor.We formulate our results in a way that they apply to the projectiveand non-projective case in equal measure.As an application of o ur methods, we give explicit models for thecohomology rings of the generalised Kummer varieties and of a seriesof certain even dimensional Calabi–Yau manifolds.2000 Mathematics Subject C lassification: 14C05; 14C15Keywords and Phrases: Hilbert schemes of points on surfaces, ratio-nal cohomology ring, loc ally constant systems, generalised Kummervarieties1The author would like to thank the Max Planck Institute for Mathematics in Bonn forits hospitality and support during the preparation of this paper. He would also like to thankthe referee for some very helpful suggestions.Documenta Mathematica 14 (2009) 749–770750 Marc A. Nieper-WißkirchenContents1. Introduction and results 7502. The Fock space description 7543. The Viraso ro algebra in the twisted case 7584. The boundary operator 7605. The ring structure 7626. The generalised Kummer varieties 767References 7691. Introduction and resultsLet X be a quasi-projective smooth surface over the complex numbers. Wedenote by X[n]the Hilbert scheme of n points on X, parametrising zero-dimensional subschemes of X of length n. It is a quasi-projective variety([Gro61]) and smooth of dimension 2n ([Fog68]). Recall that the Hilbert schemeX[n]can be viewed as a resolution of the n-th symmetric power X(n):= Xn/Snof the surface X by virtue of the Hilbert–Chow morphism ρ: X[n]→ X(n),which maps each zero-dimensional subscheme ξ of X to its support supp ξcounted with multiplicities.Let L be a local system (always over the complex numbers and o f rank 1) overX. We can view it as a functor from the fundamental groupoid Π of X to thecategory of one-dimensiona l complex vector spaces.The fundamental groupoid Π(n)of X(n)is the quotient groupoid of Πnby thenatural Sn-action by [Bro88]. (Recall from [Bro88] that the quotient groupoidof a groupoid P on which a group G is acting (by functors) is a groupoid P/Gtogether with a functor p: P → P/G that is invaria nt under the G-action andso that p: P → P/G is universal with respect to this pro perty.)Readers who prefer to think in terms of the fundamental g roup (as opp osed tothe fundamental groupoid) can find a desc ription of the fundamental group ofL(n)in [Bea83].By the universal property of Π(n), we can thus construct from L a local systemL(n)on X(n)by settingL(n)(x1, . . . , xn) :=OiL(xi),for ea ch (x1, . . . , xn) ∈ X(n)(for the notion of the tensor product over anunordered index set see, e.g., [LS03]). This induces the locally free systemL[n]:= ρ∗L(n)on X[n].We are interested in the calculation of the direc t sum of cohomology spacesLn≥0H∗(X[n], L[n][2n]). Besided the natural grading given by the cohomo-logical degree it carries weighting (see remark 1.1 below) given by the numb e rof points n . Likewise, the symmetric algebra S∗(Lν≥1H∗(X, Lν[2])) carriesDocumenta Mathematica 14 (2009) 749–770Twisted Cohomology of the . . . 751a gr ading by c ohomological degree and a weighting, which is defined so thatH∗(X, L[2]ν) is of pure weight ν.Remark 1.1. Here, a weighting is just another name for a second grading. Aweight space is a homogeneous subspace to a given deg ree with res pect to thissecond grading. B e ing of pure weight means b e ing homogeneous with respectto the second grading.In the context of super vector spaces, however, me make a difference between agrading and a weighting: Write V = V0⊕ V1for the decomposition of a supervector space into its even and odd part. Recall that for a grading V =Ln∈ZVnon V we have Vi=LnVi+n (mod 2)n.For a weighting, on the contrary, we want to adopt the following convention:If V =Ln∈ZV (n) is the decomposition of a weighted super vector space intoits weight spaces , one has Vi=LnV (n)i, i.e. the weighting does not interferewith the Z/(2)-grading.This difference is important, for example, for the notion of (super-)commutativity.The first result of this paper is the following:Theorem 1.2. There is a natural vector space isomorphismMn≥0H∗(X[n], L[n][2n]) → S∗Mν≥1H∗(X, Lν[2])that respects the grading and weighting.For L = C, the trivial system, this res ult has already appeared in [Gro96]and [Nak97]).Theorem 1.2 is proven by defining a Heisenberg Lie algebra hX,L, whose un-derlying vector space is given byMn≥0H∗(X, Ln[2]) ⊕Mn≥0H∗c(X, L−n[2]) ⊕ Cc ⊕ Cdand by showing thatLn≥0H∗(X[n], L[n][2n]) is an irreducible lowest weightrepresentation of this Lie algebra, as is done in [Nak97] for the untwisted case.Let p :ˆX → X be a finite abelian Galois covering over the surface X withGalois group G. The direct image M := p∗C of the trivial local system onˆXis a local system on X of rank |G|, the order of G. Note that G acts naturallyon M . As G is abelian, there is a decomposition M∼=Lχ∈G∨Lχ, whereG∨= Hom(G, C×) is the character group of G and Lχis the subsystem of Mon which G acts via χ. In fact, e ach Lχis a local system of rank one.Consider M[n]:=Lχ∈G∨L[n]χ. This is a local system of rank |G| on X[n]. Letq :dX[n]→ X[n]be a finite abelian Galois covering of X[n]such that q∗C = M[n].Using the Ler ay s pectral sequence for q, which already degenerates at the E2-term, the cohomology ofdX[n]can be computed by Theorem 1.2:Documenta Mathematica 14 (2009) 749–770752 Marc A. Nieper-WißkirchenCorollary 1.3. There is a natural vector space isomorphismMn≥0H∗(dX[n], C[2n]) →Mχ∈G∨S∗Mν≥1H∗(X, Lνχ[2])that respects the grading and weighting.We then proceed in the paper by defining a twisted version vX,Lof the