Complex Tori and Automorphism Groups of certain P1P^1P1-bundles (original) (raw)

Automorphism groups of P1P^1P1-bundles over a non-uniruled base

arXiv (Cornell University), 2021

In this survey we discuss holomorphic P 1 −bundles p : X → Y over a non-uniruled complex compact Kähler manifold Y, paying a special attention to the case when Y is a complex torus. We consider the groups Aut(X) and Bim(X) of its biholomorphic and bimeromorphic automorphisms, respectively, and discuss when these groups are bounded, Jordan, strongly Jordan, or very Jordan.

A characterization of compact complex tori via automorphism groups

Mathematische Annalen, 2013

We show that a compact Kähler manifold X is a complex torus if both the continuous part and discrete part of some automorphism group G of X are infinite groups, unless X is bimeromorphic to a non-trivial G-equivariant fibration. Some applications to dynamics are given. Theorem 1.1. Let X be a compact Kähler manifold and G ≤ Aut(X) a subgroup of automorphisms. Assume the following three conditions. (1) G 0 := G ∩ Aut 0 (X) is infinite.

Compact Kähler manifolds with automorphism groups of maximal rank

Transactions of the American Mathematical Society, 2014

For an automorphism group G on an n-dimensional (n ≥ 3) normal projective variety or a compact Kähler manifold X so that G modulo its subgroup N (G) of null entropy elements is an abelian group of maximal rank n − 1, we show that N (G) is virtually contained in Aut 0 (X), the X is a quotient of a complex torus T and G is mostly descended from the symmetries on the torus T , provided that both X and the pair (X, G) are minimal.

Compact Kähler manifolds admitting large solvable groups of automorphisms

Advances in Mathematics, 2015

Let G be a group of automorphisms of a compact Kähler manifold X of dimension n and N (G) the subset of null-entropy elements. Suppose G admits no nonabelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N (G) is a free abelian group of rank ≤ n − 2, or G/N (G) is a free abelian group of rank n − 1 and X is a complex torus, or G is a free abelian group of rank n − 1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X is a projective manifold of dimension n ≥ 3 and is not rationally connected. We also prove and use a generalization of a theorem by Fujiki and Lieberman on the structure of Aut(X).

Two remarks on Kähler homogeneous manifolds

Annales de la faculté des sciences de Toulouse Mathématiques, 2008

We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.

Complex Tori, Theta Groups and Their Jordan Properties

Proceedings of the Steklov Institute of Mathematics, 2019

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.

On semistable principal bundles over a complex projective manifold, II

Geometriae Dedicata, 2010

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and E G −→ X a holomorphic principal G-bundle, where G is a connected reductive linear algebraic group defined over C. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P ⊂ G and a holomorphic reduction of structure group E P

The automorphism groups of complex homogeneous spaces

Mathematische Annalen, 1997

If G is a (connected) complex Lie Group and Z is a generalized flag manifold for G, then the open orbits D of a (connected) real form G 0 of G form an interesting class of complex homogeneous spaces, which play an important role in the representation theory of G 0 . We find that the group of automorphisms, i.e., the holomorphic diffeomorphisms, is a finite-dimensional Lie group, except for a small number of open orbits, where it is infinite dimensional. In the finite-dimensional case, we determine its structure. Our results have some consequences in representation theory. §1. We determine the automorphism groups for a certain interesting class of complex homogeneous spaces. Denote by Z a generalized flag manifold for a connected complex semisimple Lie group G. A real form G 0 (which we assume to be connected) of G acts on Z with a finite number of orbits, thus there are always open orbits (cf. [22]). These open orbits play a key role in the representation theory of G 0 . An open G 0 -orbit D in Z has a G 0 -invariant complex structure. The identity component of the group of holomorphic diffeomorphisms of D will be denoted by Hol (D). In the main theorem below we determine Hol (D) for each measurable open orbit (see Definition 2.1). In the case where D is measurable, D carries a G 0 -invariant (usually) indefinite hermitian metric and we determine its group of hermitian isometries. Generally the open orbits D are non-compact, however, our results include the cases where G 0 is a compact real form and D is compact, so D = Z. The compact case is contained in [12], [21], [3] and [2], from various points of view.