Automorphisms of P 8 singularities and the complex crystallographic groups

The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in [14, 12]. We classify smoothable automorphisms of P8 singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in [20]. Singularity theory has been maintaining close relations with reflection groups since its early days, starting with the famous works of Arnold and Brieskorn [1, 8]. The first to emerge, as simple function singularities, were the Weyl groups Ak, Dk and Ek. They were followed by the Bk, Ck and F4 as simple boundary singularities or, equivalently, functions invariant under the involution [2, 22]. Since then appearance of Weyl groups in a singularity problem became a kind of criterion of naturalness of the problem [3, 4, 5]. The next to receive a singularity realisation was the list of all Coxeter groups, in the classification of stable Lagrangian maps [9]. Some of the Shephard-Todd groups appeared in [10, 11, 13] in the context of simple functions equivariant with respect to a cyclic group action. And finally, the first examples of complex crystallographic groups came out in [14, 12] in connection with the symmetries of parabolic functions J10 and X9. The affine reflection groups appeared there as monodromy groups, which is similar to the first realisations of other classes of reflection groups. This time it was the equivariant monodromy corresponding to the symmetry eigenspaces Hχ in the vanishing second homology on which the intersection form σ has corank 1. This paper completes the study of cyclically equivariant parabolic functions started in [14, 12] and considers the P8 singularities. The approach we are developing here is considerably shorter, with less calculations and more self-contained. This is allowed by a preliminary observation that the modulus parameter may take on only exceptional values if corank(σ|Hχ) = 1: the j-invariant of the underlying elliptic curves must be either 0 or 1728 (see Section 2.2). The structure of the paper is as follows. Section 1 describes the crystallographic groups which will be involved. Section 2 gives a classification of smoothable cyclic symmetries of P8 singularities possessing eigenspaces Hχ with the property as already mentioned. In Section 3, we obtain distinguished sets of generators in such eigenspaces and calculate the intersection numbers of the elements in the sets. This allows us to describe the PicardLefschetz operators generating, as complex reflections, the equivariant monodromy action on the Hχ.