On the monodromy at isolated singularities of weighted homogeneous polynomials

Abstract

Assumef: C' -M C is a weighted homogeneous polynomial with isolated singularity, and define 4: S2m -1f(0) -5 SI by z4) = f(z)/lIf(z)I. If the monomials of f are algebraically independent, then the closure Fo of 4-1(l) in S2'-I admits a deformation into the subset G where each monomial of f has nonnegative real values. For the polynomialf(z1, . . ., Zm) = Z 1'Z2 + * * * + Z,:-Jz_ + z,:z1, G is a cell complex of dimension m 1, invariant under a characteristic map h of the fibration S, and the inclusion G -*Fo induces isomorphisms in homology. To compute the homology of the link K = f'l(O) n S21 it thus suffices to calculate the action of h* on Hni (G). Let d = a1a2 * * * am + (-j)'-1. Let wl, w2,... W Wm be the weights associated with f, satisfying aj/wj + l/wj+l = 1 for j = 1, 2, .r.-., m-1 and am/wm + l/wl = 1. Let n = d/wl, q = gcd(n, d), r = q + (-)m. Then Hm-2(K) = Zr ( Zd/q and Hm-i(K) = Zr. The purpose of this paper is to calculate for the complex polynomialf defined by f(z, Z2 ... * Zm) = Z'Z2 + Z22Z3 + ***+ Zm-m + Z'mZ the integral homology of the (2m 3)-manifold K defined by K = (z .. , Zm) E Cm: Iz112 + ... v +IZm12 = 1 and Af(zi, ... ., Zm) = 0). This contributes to a project begun by Milnor [3] and continued by Milnor and Orlik [4], Orlik and Wagreich [9], Orlik [6], Orlik and Randell [8], and others, to compute invariants which will help to describe the topology of a hypersurface defined by a complex polynomial near an isolated singularity. The results of this paper can be described briefly. Let d = a,a2 . . . am + (-I)m-f, and let q be the greatest common divisor of d and a2a3 . am a3a4 . . . am + a4 . * am * +(1)m 2am + (l 1) Then K is (m 3)-connected, Hm-2(K) is the direct sum of a free abelian group of rank q + (-I)m and a cyclic group of order d/q, and Hm 1(K) is free abelian of rank q + (I1)m. I am grateful to Peter Orlik for the survey article [7] which brought the problem to my attention and which has a good bibliography on this and related subjects. 1. Preliminaries. Let f: Cm -C be a polynomial, and let V = fr'(O). If V has a singularity at Z'o, that is, all the partial derivatives of f vanish there, let Se denote the (2m 1)-sphere in Cm of radius e, centered at z4, and write K = V n Se, Then for all e sufficiently small, the portion of V inside Se is homeomorphic to the cone over K [3, p. 18], and so we are interested in the topology of K. Received by the editors September 3, 1980 and, in revised form, January 28, 1981. 1980 Mathematics Subject Classification. Primary 14B05; Secondary 32C40.