K3 surfaces with interesting groups of automorphisms

Abstract

By the fundamental result of I. I. Piatetsky-Shapiro and I. R. Shafarevich (1971), the automorphism group Aut(X) of aK3 surfaceX over ℂ and its action on the Picard latticeS X are prescribed by the Picard latticeS X . We use this result and our method (1980) to show the finiteness of the set of Picard latticesS X of rank ≥ 3 such that the automorphism group Aut(X) of theK3 surfaceX has a nontrivial invariant sublatticeS 0 inS X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS 0, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS 0. We give several examples of Picard latticesS X with parabolic and negativeS 0. We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. We give many examples of reflective hyperbolic lattices of the hyperbolic type. These results are important for the theory of Lorentzian Kac-Moody algebras and mirror symmetry.