A characterization of semisimple plane polynomial automorphisms

Abstract

a b s t r a c tIt is well-known that an element of the linear group GL n (C) is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms.

Conjugacy classes of special automorphisms of the affine spaces

“…As observed in [FuMa10], the same holds for the group Aut(A 2 C ) of complex polynomial automorphisms of the affine plane. Here, the topology corresponds to the topology of Aut(A n k ) induced by families parametrised by algebraic varieties A, called morphisms A → Aut(A n k ) and corresponding to elements of Aut(A n k[A] ) (see §2.1).…”

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“…As observed in [Furter and Maubach 2010], the same holds for the group Aut(‫ށ‬ 2 ‫ރ‬ ) of complex polynomial automorphisms of the affine plane. Here, the topology corresponds to the topology of Aut(‫ށ‬ n k ) induced by families parametrised by algebraic varieties A, called morphisms A → Aut(‫ށ‬ n k ) and corresponding to elements of Aut(‫ށ‬ n k[A] ) (see Section 2A).…”

“…In order to finish the proof of Theorem 1.3 it remains to show that the conjugacy classes of diagonalisable elements is closed. This was shown in [Furter and Maubach 2010], for the group Aut(‫ށ‬ 2 ‫ރ‬ ), but with transcendental methods. In the case of SAut(‫ށ‬ 2 k ), we can however give a simple proof, that works for any algebraically closed field k.…”

“…Here, the topology corresponds to the topology of Aut(‫ށ‬ n k ) induced by families parametrised by algebraic varieties A, called morphisms A → Aut(‫ށ‬ n k ) and corresponding to elements of Aut(‫ށ‬ n k[A] ) (see Section 2A). In fact, there is one easy direction in the result of [Furter and Maubach 2010], which corresponds to showing that if the conjugacy class is closed, then the element is diagonalisable. This works over any algebraically closed field k and follows from the following observation: If f ∈ Aut(‫ށ‬ n k ) is an element that fixes the origin, the conjugation of f by…”

Families of Group Actions, Generic Isotriviality, and Linearization

“…Ind-varieties and ind-groups. In order to explain the next application, let us recall that the group Aut(A n ) of polynomial automorphisms of affine n-space has the structure of an ind-group (see [FM10] or [Kum02]; this notion goes back to Shafarevich who called this objects infinite dimensional varieties or groups, see [Sha66,Sha81,Sha95]).…”