On the Monodromy at Isolated Singularities of Weighted Homogeneous Polynomials

Abstract

Assume/: C" -» C is a weighted homogeneous polynomial with isolated singularity, and define <(>: S2™'1 -f~\0) -> Sl by <K?) = K£)/\Kz)\.If the monomials of/ are algebraically independent, then the closure F0 of <fr'( 1) in S2"1-1 admits a deformation into the subset G where each monomial of/has nonnegative real values.For the polynomial/(z,, ..., zm) = zx'z2 + • • ■ +z¡¡?-{zm + zj£zx, G is a cell complex of dimension m -1, invariant under a characteristic map h of the fibration <)>, and the inclusion G ->F0 induces isomorphisms in homology.To compute the homology of the link K = /"'(0) n S2"1-1 it thus suffices to calculate the action of A, oni/m_,(G).Letrf = axa2 ■ • ■ am + (-l)m_l.Letw,, w2,..., wm be the weights associated with / satisfying Oj/wj + l/wJ+l -1 tor j -1, 2, . . ., m -1 and am/wm + 1/w, = 1.Let n = d/wx, q = gcd(n, d), r = q + (-1)-.Then Hm_2(K) = Z' © Z^ and #m_,(/0 = Z'.The purpose of this paper is to calculate for the complex polynomial/defined by f(zx, z2, . . ., zm) = z?>z2 + z?z3 + ■ ■■ +z^\zm + z>, the integral homology of the (2m -3)-manifold K defined by K = {(z" . .., zm) G C": |z,|2 + • • • +|zj2 = 1 and/(z" . .., 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 = axa2 • • ■ am + (-l)m_1, and let q be the greatest common divisor of d and a2a3 ■ ■ ■ am -a3a4-■ ■ am + a4-■ ■ am-+(-l)m-2am + (-1)"-1.Then AT is (am -3)-connected, Hm_2(K) is the direct sum of a free abelian group of rank q + (-l)m and a cyclic group of order d/q, and Hm_x(K) is free abelian of rank q + (-l)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 /: C" -* C be a polynomial, and let V = f~x(0).If V has a singularity at z0, that is, all the partial derivatives of/vanish there, let St denote the (2m -l)-sphere in C" of radius e, centered at z0, 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.

Locations

• Transactions of the American Mathematical Society - View - PDF