Réseaux de Kummer et surfaces K3. (Kummer lattices and K3 surfaces)
The author characterizes lattice-theoretically what he calls hyperelliptic K3 surfaces which are K3 surfaces (i.e. complex analytic surfaces with \(c_ 1(X)=0\), \(H^ 0(X,\Omega^ 1_ X)=0)\) with rational correspondence with an abelian surface. All such K3 surfaces are obtained as minimal resolutions of quotients of abelian surfaces by finite groups and their complete classification is known [cf. \textit{I. Katsura}, J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 1-41 (1987; Zbl 0664.14023); \textit{A. Fujiki}, Publ. Res. Inst. Math. Sci. 24, No.1, 1-97 (1988; Zbl 0654.32015)]. The author treats the case when the finite group is cyclic (then of order 2, 3, 4, 6). The main result is that in each case r=3, 4, 5 or 6, there is a negative definite lattice \(M_ r\) of rank 18 such that a K3 surface is of this type r iff its Néron-Severi group (in \(H^ 2(X,{\mathbb{Z}}))\) contains \(M_ r\) as a primitive sublattice. The case \(r=2\) (Kummer surface) has been known before (due to Nikulin), and the method of proof is also very similar. One of the main points (known before) is that the action of such automorphisms of abelian surfaces on the cohomology group is essentially unique. We want to remark that the algebraicity condition on K3 surfaces seems to be not necessary. (Of course abelian surfaces should be replaced by complex tori of dimension 2.)
scientific article; zbMATH DE number 4127391
scientific article