On the group of holomorphic automorphisms of model surfaces

TL;DR

This paper investigates the automorphism groups of certain complex model surfaces, showing they are subgroups of the Cremona group with bounded degree, and explores their connectedness properties.

Contribution

It establishes that automorphism groups of holomorphically homogeneous model surfaces are subgroups of the Cremona group with bounded degree, providing explicit degree estimates.

Findings

Automorphism groups are subgroups of the Cremona group.

Degree of automorphisms is bounded by a function of ambient space dimension.

Connectedness of the automorphism group is analyzed.

Abstract

It is proved that the group of holomorphic automorphisms of holomorphically homogeneous nondegenerate (finite Bloom-Graham type + holomorphic nondegenaracy) model surface Q is a subgroup of the group of birational automorphisms of the ambient space (Cremona group) with uniformly bounded degree. An estimate of the degree of the automorphisms in terms of the dimension of the ambient space is given (Theorem 4). It is proved that none of the conditions of the theorem can be omitted. We consider also the question about connectedness of the automorphism group of Q (Theorem 7).