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HIRZEBRUCH χy GENERA OF THE HILBERT SCHEMES OF SURFACES BY LOCALIZATION FORMULA KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU Abstract. We use the Atiyah-Bott-Berline-Vergne localization formula to calculate the Hirzebruch χy genus χy (S [n] ), where S [n] is the Hilbert schemes of points of length n of a surface S . Combinatorial interpretation of the weights of the fixed points of the natural torus action on (C2 )[n] is used. 1. Introduction 1.1. Genera of complex manifolds. Let Ω = ΩU ⊗ Q be the complex bordism ring with coefficients in Q. Given a ring R, a complex genera with values in R is a ring homomorphism φ : Ω → R. Recall that according to Milnor [11] and Novikov, two complex manifolds are com- plex cobordant if and only if they have the same Chern numbers. For a closed complex manifold M , Hirzebruch [10] defined its χy genus by p q q p ∗ p p ∗ χy (M ) = y (−1) dim H (M, Λ T M ) = y χ(M, Λ T M ). p,q ≥0 p ≥0 Recall that for any holomorphic vector bundle V → M , χ(M, V ) = (−1)q dim H q (M, V ). q≥0 This can be extended to Ω to obtain a complex genus with values in C[y]. Dijkgraaf, Moore, Verlinde and Verlinde [4] introduced a complex genus as follows. For any vector bundle V over M , define the formal sums Λ V = qkΛk V, S V = qk Sk V, q q k≥0 k≥0 where Λk and Sk denote the k-th exterior and symmetric product respectively. Let dimC M = d, set