International Congress of Mathematicians

486 L. GöttscheA partition a = (ni,...,n r ) £ P(n) can also be written as a = (l ai ,... ,n a "),where a, is the number of occurences of i in (m,..., n r ). We put \a\ = r = ^ a^.Then (2.1) can be reformulated asp(S^,z)= Y, p(S {ai) x ...xS {a - ) ,z)z 2{n -^).(2.2)a£P(n)This result has been refined to Hodge numbers in [30], [11] and this was generalizedin [13] to the Douady space of a complex surface. It has been further refined todetermine the motive and the Chow groups [14] and the element in the Grothendieckgroup of varieties of S M [28].Partially motivated by (2.1) and using arguments from physics in [15] a conjecturalrefinement to the Krichever-Höhn elliptic genus is given. We restrict ourattention to the case that Kx = 0 when the elliptic genus is a Jacobi form. For acomplex vector bundle F on a complex manifold X and a variable t we putA t (F) :=0A'(E)t', St(E) := (£)S k (E)t k .k>0 k>0For the holomorphic Euler characteristic we write x(A^, A t (F)) := ^x(X,and similarly for St(E). Then the elliptic genus is defined byA k E)t k!Writing (S) := ^2 m>0 i c(m,l)q m y l ,the conjecture is£ 4>(S^)p N = u Ti n-^^-^ - LJ M yfiQniyl\c(nniN>0 n>0,m>0 - (1 — p n q m: l)3. Infinite dimensional Lie algebras and the cohomologyringWe saw that one gets nice generating functions in n for the Betti numbers of theS^. Now we shall see that the direct sum of all the cohomologies of the S M carriesa new structure which governs the ring structures of the Hilbert schemes. We onlyconsider cohomology with rational coefficients and thus write H*(X) for H*(X, Q).We write H := H*(S); for n > 0 let H„ := H*(S^). and H := ® n > 0 H„.We shall see that H is an irreducible module under a Heisenberg algebra. Thiswas conjectured in [54] and proven in [45],[32]. H contains a distingished element1 £ H 0 = Q. We denote by J s and J s[n] the evaluation on the fundamental classof S and S^. Define for n > 0 the incidence varietyZ hn := {(Z,x,W) £ SW xSx SV +n^ \ Z c W,p(W) - p(Z) = n[x]},