Complex reflection groups and K3 surfaces I

Abstract

We construct here many families of K3 surfaces that one can obtain as quotients of algebraic surfaces by some subgroups of the rank four complex reflection groups. We find in total 15 families with at worst $ADE$--singularities. In particular we classify all the K3 surfaces that can be obtained as quotients by the derived subgroup of the previous complex reflection groups. We prove our results by using the geometry of the weighted projective spaces where these surfaces are embedded and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. This construction generalizes a previous construction by W. Barth and the second author. Comment: 26 pages

Complex reflection groups and K3 surfaces II. The groups $G_{29}$, $G_{30}$ and $G_{31}$

“…In the first paper of this series [5], the authors have explained how to build K3 surfaces from invariants of complex reflection groups of rank 4 generated by reflections of order 2. In this second part and the upcoming third part [7], we complete this qualitative result by investigating more precisely the examples given by the primitive groups (see [5, §2] for the definition), i.e.…”

“…In this second part and the upcoming third part [7], we complete this qualitative result by investigating more precisely the examples given by the primitive groups (see [5, §2] for the definition), i.e. the groups G 28 , G 29 , G 30 and G 31 (as in [5], we follow Shephard-Todd numbering for complex reflection groups [24]). In particular, we investigate the following questions:…”

“…authors who constructed highly singular surfaces from invariants of complex reflection groups [20], [3]. Most (but not all) of the singularities constructed this way can be obtained from [5,Corollary 2.4]. In §6.5, we also revisit Boissière-Sarti example of the smooth octic surface containing 352 lines [2], using Springer theory [5, Theorem 3.13].…”

“…Finally, note that the first part [5] was free of computer calcutations, as the arguments were pretty general: in this second part, we study very specific examples, for which the determination of geometric features (singularities, transcendental lattices, branch locus,etc.) requires computer calculations.…”

“…From now on, and until the end of this paper, we assume moreover that W is one of the three primitive groups G 29 , G 30 or G 31 . I some specific data for these three groups that were contained in [5,Table I].…”