Monodromy and betti numbers of weighted complete intersections

Abstract

LET (X, 0) be an isolated singularity of complete intersection in Cm defined by the weighted homogeneous polynomials h of degree d, with respect to the positive integer weights wt (Xj) =wjfori=l ,..., pandj=l,..., m. Letf: (X, 0) + (C, 0) be a function germ induced by a weighted homogeneous polynomial of degree d with respect to the weights w = (w, , . . . , w,) such that (X,, 0) = (f-‘(O), 0) is again an isolated singularity of complete intersection with n = dim X, = dim X - 1 2 1. If X, denotes the Milnor fiber of the singularity (X,, 0), then there is a natural (complex) monodromy operator h: H”(x,, C) -+ H”(8,, C) associated to the function f [lS]. In the first part of this note we show that this monodromy operator is diagonalizable and compute its characteristic polynomial A@) = det (A. Id -h) in terms of the weights w and the degrees d = (d,, . . . , dp) and d. In the special case of Brieskorn-Pham singularities this result is due to Hamm [93, not to mention the case when X is smooth, treated already by Milnor and Orlik [ 1 l] and Brieskorn VI. Our proof depends on the relation between the monodromy operator h and the Gauss-Manin connection of the function f (as suggested by an example in Looijenga [lo], p. 166) and on the knowledge of the Poincare series of S&/dRn~~ ’ computed by Greuel and Hamm [7]. In the second part we derive some topological consequences. Namely, there are two spaces naturally associated to the singularity (X, 0): its link K = X n S, where S is the unit sphere in Cm and the quasi-smooth weighted complete intersection Y defined by the polynomials A in the weighted projective space P(w) [4]. We show that the results in the first section allow one to compute the (middle) Betti numbers of K and Y in terms of w, d. Equivalently, we determine the rank of the intersection form of the Milnor lattice of (X, 0). We also prove that all the quasi-smooth weighted complete intersections of the same type (w, d) are homeomorphic. In the final section we determine the mixed Hodge structure on the cohomology of the Milnor fiber of the singularity (X, 0), when dim X I 3, using an idea due to Steenbrink [13].