Invariant Symmetries of Unimodal Function Singularities

Abstract

We classify finite order symmetries g of the 14 exceptional unimodal function singularities f in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of f should not be contained in one eigenspace of g?. We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising. One of the most famous classical results in singularity theory is the Arnold and Brieskorn discovery of the close relationship between simple function singularities and Weyl groups Aμ, Dμ, Eμ [1, 6]. A few years after it, Arnold extended the relationship to simple singularities with the Z2 reflection symmetry and Weyl groups Bμ, Cμ, F4 [2] (see also Slodowy’s book [24]). Consideration of Zm symmetries of simple functions led in [10, 11, 12, 25] to the appearance of Shephard-Todd groups within function singularities. The emphasis there was on realisations of the complex reflection groups as equivariant monodromy groups acting on the appropriate character subspaces in the homology of invariant Milnor fibres, and on the diffeomorphisms between the discriminants of the reflection groups and of the Zm-equivariant functions. A further series of papers [13, 14, 15], on cyclic symmetries of the parabolic functions, brought in similar singularity realisations of certain complex crystallographic groups [22]. In this paper, we are naturally expanding the programme to cyclic symmetries of the 14 exceptional unimodal function singularities on one hand, and complex hyperbolic reflection groups on the other. The basic idea is as follows. In the 3-variable case, the intersection form on the vanishing homology of an exceptional unimodal function f is non-degenerate and has positive signature 2. Assume g is an automorphism of C of finite order m, and our function is g-invariant. Then g acts on the second homology of the Milnor fibre f−1(e), and decomposes it into a direct sum of the character subspaces Hχ, χ = 1, on which g acts as multiplication by χ. Assume the rank 2 positive subspace of the intersection form splits between two character summands. Then the monodromy within a g-invariant versal deformation of f acts as a complex hyperbolic reflection group on each of them. Developing further the technique introduced in papers on cyclically symmetric functions [10, 11, 12], we construct vanishing bases in the hyperbolic summands and obtain the generating reflections as the corresponding Picard-Lefschetz operators. The main result of the paper is a complete classification of the invariant symmetries of the 14 singularities, which split the positive subspace in the vanishing homology, and the description – via constructing the corresponding Dynkin diagrams – of the complex hyperbolic groups arising. All the rank 2 reflection groups obtained projectivise to the triangle groups of the Poincare disk. The task of identification of higher dimensional groups is left for a future paper, along with the consideration of the equivariant symmetry setting. It should be noted that it is the first time when complex hyperbolic reflection groups are appearing in a singularity theory context. The approach introduced may be useful for constructing new complex hyperbolic lattices (cf. [8, 20]). The paper is organised as follows. Section 1 introduces the notion of singularities with symmetry, recalls the definitions and constructions given in [10, 11, 12]. Section 2 contains classification of splitting invariant symmetries of the 14 singularities. In Section 3.3 we construct Dynkin diagrams of the hyperbolic monodromy groups associated with the symmetric functions. Projectivisations of the rank 2 monodromy groups are considered in Section 4. More details of the constructions may be found in [16].