lnventiones math, 39. 199-211 (1977) Ill verl tjorl e$ mathematicae (t~; by Springer-Verlag 1977 The Monodromy of Weighted Homogeneous Singularities P. Orlik and R. Randelt* University of Wisconsin, Madison, Wt.53706, USA The Institute for Advanced Study, School of Mathematics, Princeton, N.J. 08540, USA Varieties defined by weighted homogeneous (or quasihomogeneous) polynomials are of particular interest because they admit actions of the multiplicative group of non-zero complex numbers [24] and because the topological properties of many general singularities may be computed from quasihomogeneous or semi- quasihomogeneous normal forms [4]. In this paper we use hyperplane sections and relative monodromy to study such singularities, and we compute explicitly the local integral monodromy of a general class in any number of variables. 1. Introduction Let w = (wo ..... w,) be an (n + l)-tupte of positive rational numbers. A polynomial f(z0 ..... z,,) is said to be a weighted homogeneous polynomial (whp hereafter) , ,i, of f satisfies io/wo +... + i,/w, = 1. with weights w if each monomiat c~z'~' .... , It has an isolated critical point at 0c117 "+~ if gradf=(?~f/~zo,..., ~?f/(?z,) is zero at 0ct12 "+~, but gradflz4:0 for all z+0 in a neighborhood of 0. Unless other- wise stated, f will indicate a whp with isolated critical point at 0. Thus if we set I~= ~(f)= {zE~"+llf(z)-_t}, then V~ will be a hypersurface with an isolated singularity at 0, and 1~ will be nonsingutar for small t+0. In Section 2 we use results of L0, [t 7, ! 8] to compute the integral monodromy for weighted homogeneous polynomials of the form p(Zo ..... z,)-~o +~o~l +...+z, l~,, _ This is the first computation known of this invariant for indecomposabte poly- nomials (see (2.2)) in more than 3 variables. Recall how the integral monodromy is defined, Let S 2"+~ be the sphere of radius e centered at 0et[ ""~, and let 2,+t K=S~ ca V o. Then there is the well- known Milnor fibration f/[[']: S~:-K--~ S ~, see [19]. The fibre F is diffeomorphic * The authors were partially supported by NSF grants