K3 surfaces with interesting groups of automorphisms

1997/01/24 by Viacheslav V. Nikulin, Nikulin, Viacheslav V. · 1 citation
Mathematics · Physics and Astronomy · #10D20 #14D20 #17B65 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.alg-geom/9701011

openalex publication_date 1997/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By the fundamental result of I.I. Piatetsky-Shapiro and I.R. Shafarevich (1971), the automorphism group Aut(X) of a K3 surface X over C and its action on the Picard lattice S_X are prescribed by the Picard lattice S_X. We use this result and our method (1980) to show finiteness of the set of Picard lattices S_X of rank ≥ 3 such that the automorphism group Aut(X) of the K3 surface X has a non-trivial invariant sublattice S₀ in S_X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic lattices S₀ it has been proved by the author before (1980, 1995). Thus we extend this results to negative sublattices S₀. We give several examples of Picard lattices S_X with parabolic and negative S₀. We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. These results are important for the theory of Lorentzian Kac--Moody algebras and Mirror Symmetry.

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• Virtually abelian symmetry groups of hyperbolic lattices

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