The Poincaré series of some special quasihomogeneous surface singularities

In [E6] a relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. We study this relation for Fuchsian singularities and show that it is connected with the mirror symmetry of K3 surfaces and with automorphisms of the Leech lattice. We also indicate relations between other singularities and Conway’s group. Introduction K. Saito [Sa1, Sa2] has introduced a duality between polynomials which are products of cyclotomic polynomials. He has shown that V. I. Arnold’s strange duality between the 14 exceptional unimodal hypersurface singularities is related to such a duality between the characteristic polynomials of the monodromy operators of the singularities. Moreover, he has observed that the dual polynomials pair together to the characteristic polynomial of an automorphism of the Leech lattice. It is now well-known that Arnold’s strange duality is related to the mirror symmetry of K3 surfaces (see e.g. [D5]). The author [E4, E5] has shown that these features still hold in a certain way for the extension of Arnold’s strange duality discovered by C. T. C. Wall and the author [EW]. The 14 exceptional singularities and the singularities involved in the extension of Arnold’s strange duality are examples of Fuchsian singularities. By this we mean the following. Let Γ ⊂ PSL(2,R) be a finitely generated Fuchsian group of the first kind. Let Ak denote the C-vector space of Γ-automorphic forms of weight 2k, k ≥ 0, and let A = ⊕∞ k=0Ak be the algebra of Γ-automorphic forms. Then (X,x) := (SpecA,m) where m := ⊕∞ k=1 Ak is a normal surface singularity which is called a Fuchsian singularity. Partially supported by the DFG-programme ”Global methods in complex geometry” (Eb 102/4–1); 2000 Mathematics Subject Classification: Primary 14J17, 32S25, 32S40, 13D40; Secondary 14J28, 11H56