On the cohomologically trivial automorphisms of elliptic surfaces I: χ(S)=0

2024/08/29 by Fabrizio Catanese, Catanese, Fabrizio, Davide Frapporti +6
Mathematics · #14F99 #14H30 #14J27 #14J50 #14J80 #32L05 #32M99 #32Q15 #32Q55 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2408.16936

openalex publication_date 2024/08/29 · openalex created_date 2024/10/19 · openalex updated_date 2026/07/28

Abstract

In this first part we describe the group Aut_ℤ(S) of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface S with Kodaira dimension κ(S)=1), in the initial case χ(O_S) =0. In particular, in the case where Aut_ℤ(S) is finite, we give the upper bound 4 for its cardinality, showing more precisely that if Aut_ℤ(S) is nontrivial, it is one of the following groups: ℤ/2, ℤ/3, (ℤ/2)². We also show with easy examples that the groups ℤ/2, ℤ/3 do effectively occur. Respectively, in the case where Aut_ℤ(S) is infinite, we give the sharp upper bound 2 for the number of its connected components.

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