Compact Kähler manifolds admitting large solvable groups of automorphisms (original) (raw)

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.

The g-periodic subvarieties for an automorphism g of positive entropy on a compact Kähler manifold

Advances in Mathematics, 2010

For a compact Kähler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most ρ(X) of g-periodic prime divisors. When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic (a result in the spirit of the Dynamic Manin-Mumford conjecture as in [17]). Remark 1.2. (1) Suppose that the X in Theorem 1.1 (1) has ρ(X) of g-periodic prime divisors, then the algebraic dimension a(X) = 0 by the proof, Theorem 3.2 and Remark 2.9. Suppose further that the irregularity q(X) := h 1 (X, O X) > 0. Then the albanese map alb X : X → Alb(X) =: Y is surjective and isomorphic outside a few points of Y , and ρ(Y) = 0. Conversely, we might realize such maximal situation by taking a complex n-torus T with ρ(T) = 0 and a matrix H ∈ SL n (Z) with trace > n so that H induces

Derived length of zero entropy groups acting on compact Kahler manifolds

arXiv: Algebraic Geometry, 2018

Let X be a compact Kahler manifold of dimension n. Let G be a group of zero entropy automorphisms of X. Let G0 be the set of elements in G which are isotopic to the identity. We prove that after replacing G by a suitable finite-index subgroup, G/G0 is a unipotent group of derived length at most n-1. This is a corollary of an optimal upper bound of length involving the Kodaira dimension of X. We also study the algebro-geometric structure of X when it admits a group action with maximal derived length n-1.

Automorphism groups of positive entropy on minimal projective varieties

2010

We determine the geometric structure of a minimal projective threefold having two 'independent and commutative' automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X, G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fitting the 'boundary case'. 1. Introduction A normal projective variety X with only Q-factorial terminal singularities is called minimal if the canonical divisor K X is nef. For G ≤ Aut(X), the representation G| NS C (X) on the complexified Néron-Severi group NS C (X) := NS(X)⊗ Z C, is Z-connected if its Zariski-closure in GL(NS C (X)) is connected in the Zariski topology. We denote by q(X) := h 1 (X, O X) the irregularity of X. A birational morphism σ : X → X ′ is crepant if K X = σ * K X ′. We refer to [9, Definition 2.34] for the definition of terminal or canonical singularity, and to [4] for the definitions of dynamical degrees and (topological) entropy. In this paper, we prove Theorem 1.1 below and its generalization in Theorem 1.5. Theorem 1.1. Let G ≤ Aut(X) be an automorphism group on a minimal projective threefold X with the representation G| NS C (X) solvable and Z-connected. Then we have: (1) The null subset N(G) := {g ∈ G | g is of null entropy} is a normal subgroup of G such that G/N(G) ∼ = Z ⊕r for some r = r(G) ≤ dim X − 1 = 2. (2) Suppose that r = 2. Then there are a G-equivariant (birational) crepant morphism X → X ′ and a G-equivariant finite Galois cover τ : A → X ′é tale in codimension 1 for a 3-dimensional abelian variety A, with deg τ ≥ 2 occurring only when q(X) = 0. (3) Suppose that r = 2 and the identity component Aut(X) 0 of Aut(X) is trivial. Then

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.

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.

Algebraic varieties with automorphism groups of maximal rank

Mathematische Annalen, 2013

We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper [28] though it is not used in the present paper.

Bimeromorphic automorphisms groups of certain conic bundles

arXiv (Cornell University), 2019

We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders of finite subgroups of the quotient G/G 0 are bounded by a constant depending on G only. Let Y be a complex torus of algebraic dimension 0. We prove that if X is a non-trivial holomorphic P 1 −bundle over Y then the group Bim(X) of its bimeromorphic automorphisms is very Jordan (contrary to the case when Y has positive algebraic dimension). This assertion remains true if Y is any connected compact complex Kähler manifold of algebraic dimension 0 without rational curves or analytic subsets of codimension 1.

Constraints on automorphism groups of higher dimensional manifolds

2012

In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from P k (C) by a finite sequence of blow-ups along smooth centers of dimension at most r with k > 2r + 2 has finite image in GL(H * (X, Z)). In particular, every holomorphic automorphism f : X → X has zero topological entropy.