On a conjecture by A. Durfee

Abstract

This note provides a negative answer to the following question of A. H. Durfee [Invent. Math. 28 (1975), 231–241; ]: Is it true for arbitrary polynomials F(x,y,z) having an isolated singularity at the origin that the local monodromy is of finite order if and only if a resolution of F(x,y,z)=0 contains no cycles? Here "the monodromy'' means the action on the cohomology of the Milnor fiber of F corresponding to the degeneration F(x,y,z)=t. The authors consider the following example: F(x,y,z)=(xz−y2)3−((y−x)x2)2+z6. They calculate the graph of the resolution (which is a tree) and invariant polynomials of the monodromy (showing the presence of Jordan blocks of a size greater than 1). The key point in these calculations is that this singularity belongs to the class of superisolated (SIS) surface singularities which was studied in detail by the first named author [Mem. Amer. Math. Soc. 109 (1994), no. 525, x+84 pp.;]. SISs are the singularities of the form F(x,y,z)=f(x,y,z)+lN, where l is a generic linear form, N is a sufficiently large integer and f(x,y,z)=0 is a projective plane algebraic curve, the cone over which is the tangent cone of the singularity F(x,y,z). The main step in detecting that the order of the monodromy of a SIS is infinite is the calculation of the Alexander polynomial [A. S. Libgober, Duke Math. J. 49 (1982), no. 4, 833–851;] of the plane curve f(x,y,z)=0. In the authors' example, the plane sextic (xz−y2)3−((y−x)x2)2 has two singularities with local types u3=v10 and u2=v3 respectively and has as its Alexander polynomial t2−t+1. The latter yields that the monodromy of F has an infinite order. The paper is concluded with a series of other interesting observations on the relation between the topology of resolution and monodromy of SIS singularities.