Numerically Trivial Involutions of Kummer Type of an Enriques Surface

There are two types of numerically trivial involutions of an Enriques surface according as their period lattice. One is U(2) ⊥ U(2)-type and the other is U ⊥ U(2)-type. An Enriques surface with an involution of U(2) ⊥ U(2)-type is doubly covered by a Kummer surface of product type, and such involutions are classified again into two types according as the parity of the corresponding Göpel subgroups. Involutions of odd U(2) ⊥ U(2)type are constructed from the standard Cremona involutions of the quadric surface and closely related with quartic del Pezzo surfaces. It is known that a nontrivial automorphism of a K3 surface acts nontrivially on its cohomology group. But this is not true for an Enriques surface. An automorphism of an Enriques surface S is said to be numerically trivial (resp. cohomologically trivial) if it acts on the cohomology group H(S,Q) (resp. H(S,Z)) trivially. In this paper we classify the numerically trivial involutions, correcting [3]. Let S be a (minimal) Enriques surface, that is, a compact complex surface with H(OS) = H(OS) = 0 and 2KS ∼ 0, and σ a numerically trivial (holomorphic) involution of S. We denote the covering K3 surface of S by S̃ and the covering involution by ε. Then the period lattice NR of (S, σ) is isomorphic to either U(2) ⊥ U(2) or U ⊥ U(2) as a lattice ([3, Proposition (2.5)]). σ is called U(2) ⊥ U(2)-type, or Kummer type, in the former case. In this paper, except the first appendix, we assume that NR ≅ U(2) ⊥ U(2) and classify the numerically trivial involutions of Kummer type using their periods, that is, the Hodge structures on NR (cf. Remark 21). There exist a pair of elliptic curves E ′ and E ′′ and an isomorphism φ between S̃ and the Kummer surface of the product abelian 2000 Mathematics Subject Classification. 14J28, 14C34, 14K10. Supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 17340006, (S) 19104001 and for Exploratory Research 20654004.