Linear bound for abelian automorphisms of varieties of general type (

The aim of this paper is to prove the following Theorem 1. Let G be a finite abelian group acting faithfully on a complex smooth project variety X of general type with numerically effective (nef) canonical divisor, of dimension n. Then |G| ≤ C(n)K n X , where C(n) depends only on n. We refer to the Introduction of [Ca-Sch] for a nice account of the history for the study of bounds of automorphism groups of varieties of general type. The authors of that paper have also shown a polynomial bound for abelian automorphism groups. To prove Theorem 1, the only major obstacle to a generalisation of our argument for surfaces [X] is the lack of a theorem of minimal models in higher dimension: the basic idea is to find a pencil on X , whose general fibres are invariant under the action of G, then use induction on n. To do so one needs bounded globally generatedness of pluricanonical sheaves, and vanishing theorems. Unfortunately, these theorems currently exist only for varieties with extra conditions which are not preserved by fibres. Therefore we consider the problem for varieties in a more general category, as is done in [Ca-Sch]. Our main observation in Theorem 2 is that in the polynomial bound of Theorem 0.1 of [Ca-Sch], most copies of d may be compensated by the ambient dimension N , leading thus to a linear bound. — 1 —