Realizing connected Lie groups as automorphism groups of complex manifolds

Abstract

Abstract. We show that every connected real Lie group can be realized as the full automorphism group of a Stein hyperbolic complex manifold. Mathematics Subject Classification (2000). Primary 32M05; Secondary 22E15, 32Q28, 32Q45.

On Automorphisms and Endomorphisms of Projective Varieties

“…By a result of Winkelmann (see [22]), every connected real Lie group G can be realized as the automorphism group of some complex Stein manifold X, which may be chosen complete, and hyperbolic in the sense of Kobayashi. Subsequently, Kan showed in [12] that we may further assume dim C (X) = dim R (G).…”

Automorphisms and Dynamics: A List of Open Problems

“…The first problem has been solved by Brion in the following strong sense: any connected algebraic group G over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety X; if the characteristic of the field is 0, one can moreover assume that X is smooth of dimension dim(X) = 2 dim(G) (see [4]; see also [45] for Kobayashi hyperbolic manifolds).…”

Some structure theorems for algebraic groups

“…It is known that every finite group is the automorphism group of a smooth projective complex curve (see [28]); moreover, every compact connected real Lie group is the automorphism group of a bounded domain (satisfying additional conditions), see [5,51]. Also, every connected real Lie group of dimension n is the automorphism group of a Stein complete hyperbolic manifold of dimension 2n (see [61,30]). Theorem 7.3.1, obtained in [11,Thm.…”