International Congress of Mathematicians

Hilbert Schemes of Points on Surface 485that p : S^ —t S^ is crêpant. In the special case that S is an abelian surface or aK3 surface one can get a better result: A complex manifold X is called holomorphicsymplectic if there exists an everywhere non-degenerate holomorphic 2-form onX. If furthermore is unique up to scalar, X is called irreducible holomorphicsymplectic. A Kahler manifold X of real dimension in is called hyperkähler if itsholonomy group is Sp(n). Compact complex manifolds are holomorphic symplecticif and only of they admit a hyperkähler metric. In [7] it is shown that for a K3 surfaceS the Hilbert scheme S^ is irreducible holomorphic symplectic. There also, foran abelian surface A, the generalized Kummer varieties are constructed from AM.They form another series of irreducible holomorphic symplectic manifolds. The onlyotherexamples of compact hyperkähler manifolds, known not to be diffeomorphicto one in the above two series are the two isolated examples of resolutions of singularmoduli spaces of sheaves on K3 and abelian surfaces in [47],[48].2. Betti number, Euler numbers, elliptic genusFor many questions about the Hilbert schemes S^ one should look at alln at the same time. The first instance of this are the Betti numbers and Eulernumbers, for which we can find generating functions in terms of modular forms.Let % := {T £ C | S(r) > 0}. A modular form of weight k on Sl(2, Z) is a function/ : H -• C s.th.'(£ï!)='Furthermore, writing q = e 2 " T , we require that, in the Fourier development /(r) =12nez a nQ n ,a ll the the negative Fourier coefficients vanish. If also cto = 0, /is called a cusp form. The most well-known modular form is the discriminantA(T) := g[] n>0 (l — q n ) 24 , the unique cusp form of weight 12. The Dirichlet etafunction is n = A 1 / 24 .For a manifold X we denote by p(X, z) := '^2i(^l) t bi(X)z t the Poincaré polynomialand by e(X) = p(X, 1) the Euler number. The Betti numbers and Eulernumbers of the S^ have very nice generating functions [24]:j2p(s [n] ,z)t n =nn(i-^- 2+ v) ( - i),+iSi(s) . (2.1)n>0 k>\ i=0In particular Y, n >o e ( sln] )Q n ~