Automorphism groups of normal CR 3-manifolds

分裂 Nakamura 流形及其自同构

Split Nakamura manifolds and their automorphisms

本文研究分裂Nakamura流形，探讨其de Rham和Dolbeault上同调，分析Frölicher谱序列退化与∂¯∂-引理，并讨论其变形，最终详述自同构群。

On two theorems about local automorphisms of geometric structures

This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemannian manifolds. We also revisit a strong result of Gromov which describes the structure of the orbits of local automorphisms of manifolds endowed with $A$-rigid structures, and give a statement of this result in the setting of Cartan geometries.

Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.

Hyperbolic $n$-dimensional manifolds with automorphism group of dimension $n^2$

We obtain a complete classification of complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$, for which the dimension of the group of holomorphic automorphisms is equal to $n^2$.

Homogeneous Kobayashi-hyperbolic manifolds with automorphism group of subcritical dimension

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.

Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group

In this paper we determine all Kobayashi-hyperbolic 2-dimensional complex manifolds for which the group of holomorphic automorphisms has dimension 3. This work concludes a recent series of papers by the author on the classification of hyperbolic $n$-dimensional manifolds, with automorphism group of dimension at least $n^2-1$, where $n\ge 2$.

Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism groups

See math.CV/0509030 which replaces this paper.

Automorphism Supergroups Of Supermanifolds

A classical theorem states that the group of automorphisms of a manifold $M$ preserving a $G$-structure of finite type is a Lie group. We generalize this statement to the category of $cs$ manifolds and give some examples, some of which being generalizations of classical notions, others being particular to the super case. Notably, we have to introduce a new notion of supermanifolds which we call mixed supermanifolds.

复切触流形的切触线丛的浸入性及其在自同构群中的应用

Immersivity of the contact line bundle of a complex-contact manifold and an application to the automorphism group

本文研究复切触流形切触线丛的浸入性，将既有极丰沛、一般有限加约化自同构群的条件放宽为浸入性，证明具浸入切触线丛的连通紧复切触流形自同构群同构于对应秩>1单复李代数的自同构群，并给出其不连通的型的条件。

On the automorphism groups of hyperbolic manifolds

We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel space. These results complement earlier theorems of the authors on the possible dimensions of automorphism groups of domains in comlex space. The paper also contains a proof of our earlier result on characterizing $n$-dimensional hyperbolic complex manifolds with automorphism grou…

Continuing the classification of homogeneous Kobayashi-hyperbolic manifolds with high-dimensional automorphism group

We classify all homogeneous Kobayashi-hyperbolic manifolds of dimension $n \ge 2$ whose group of holomorphic automorphisms has dimension either $n^2 - 7$ or $n^2 - 8.$ This paper continues the work of A. Isaev, who classified all such manifolds with automorphism group dimension $n^2 - 6$ and greater.

Hyperbolic manifolds of dimension $n$ with automorphism group of dimension $n^2-1$

We consider complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ for which the dimension of the group of holomorphic automorphisms is equal to $n^2-1$. We give a complete classification of such manifolds for $n\ge 3$ and discuss several examples for $n=2$.

Further steps towards classifying homogeneous Kobayashi-hyperbolic manifolds with high-dimensional automorphism group

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 4$ whose group of holomorphic automorphisms has dimension either $n^2-4$, or $n^2-5$, or $n^2-6$. This paper continues a series of articles that achieve classifications for automorphism group dimension $n^2-3$ and greater.

Geometric automorphism groups of symplectic 4-manifolds

Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $Γ$, $A(Γ)$ the automorphism group of $Γ$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when $D(M)$ is of infinite index in $A(Γ)$ for a symplectic 4-manifold.

Lie group structures on automorphism groups of real-analytic CR manifolds

Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms is a Lie group (in an appropriate topology). Our conditions are in particular satisfied when M is an arbitrary compact real-analytic hypersurface embedded in some Stein manifold.

A note on vanishing of equivariant differentiable cohomology of proper actions and application to CR-automorphism and conformal groups

We establish that for any proper action of a Lie group on a manifold the associated equivariant differentiable cohomology groups with coefficients in modules of $\mathcal{C}^\infty$-functions vanish in all degrees except than zero. Furthermore let $G$ be a Lie group of $CR$-automorphisms of a strictly pseudo-convex $CR$-manifold $M$. We associate to $G$ a canonical class in the first differential cohomology of $G$ with coefficients in the $\mathcal{C}^\infty$-functions on $M$. This class is non-zero if and only if $G$ is essential in the sense that there does not exist a $CR$-compatible strict…

具有传递约化自同构群的闭G$_2$-结构

Closed G$_2$-structures with a transitive reductive group of automorphisms

本文完整分类了容许传递约化自同构群且带闭非平行G$_2$-结构的七维流形，证明群中心一维，并揭示此类流形为平坦因子与带不变严格辛半平坦SU(3)-结构的非紧齐性六维流形的黎曼乘积。

Developing Maps and Engel Automorphisms

A completely nonintegrable $2$-dimensional distribution on a $4$-manifold is called an Engel structure. A $4$-manifold with an Engel structure is called an Engel manifold. The developing map for an Engel manifold is very important tool to determine the Engel structure. Montgomery used it to prove that an Engel automorphism is determined by the values on a global slice. Moreover, Montgomery constructed Engel manifolds whose automorphism group is small. In this paper, we prove that the automorphism group of an Engel manifold is embedded into the automorphism group of the Cartan prolongation of a…

Transversal infinitesimal automorphisms on Kähler foliations

Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field

The Automorphism Group of a Banach Principal Bundle with {1}-structure

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly on P provided the Lie algebra of infinitesimal automorphisms consists of complete vector fields. As a consequence we show that the automorphism group of a connected geodesically complete affine Banach manifold M can be turned into a Banach-Lie group acting smoothly on M.

Symmetries of quaternionic Kähler manifolds with $S^1$-symmetry

We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-Kähler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the $c$-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of …

Automorphisms of the Weyl manifold

Assume that $M$ is a smooth manifold with a symplectic structure $ω$. Then Weyl manifolds on the symplectic manifold $M$ are Weyl algebra bundles endowed with suitable transition functions. From the geometrical point of view, Weyl manifolds can be regarded as geometrizations of star products attached to $(M,ω)$. In the present paper, we are concerned with the automorphisms of the Weyl manifold corresponding to Poincaré-Cartan class ($c_0$ is a $\check{\rm C}$ech cocycle corresponding to the symplectic structure $ω$.) $[c_0+\sum_{\ell=1}^\infty c_{\ell} ν^{2\ell}]\in \check{H}^2 (M)[[ν^2]]$. We…

Automorphisms of multiplicity free Hamiltonian manifolds

We compute the sheaf of automorphisms of a multiplicity free Hamiltonian manifold over its momentum polytope and show that its higher cohomology groups vanish. Together with a theorem of Losev, arXiv:math/0612561, this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determined by its momentum polytope and its principal isotropy group.

Automorphisms of locally conformally Kahler manifolds

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.

复 $b^k$-流形的自同构

On Automorphisms of Complex $b^k$-Manifolds

本文推广Mendoza的复$b$-流形定义至高阶退化情形，引入复$b^k$-流形，研究其局部与全局自同构，并为经典全纯函数空间提出$b^k$-模拟。

关于辛半平坦6-流形的自同构群

On the automorphism group of a symplectic half-flat 6-manifold

本文证明带辛半平坦SU(3)-结构的紧致6-流形自同构群李代数为阿贝尔且维数受min{5,b_1(M)}限制，并给出T S^3上SO(4)余齐性一作用不变的完备例子。

超 Kähler 流形的抛物自同构

Parabolic automorphisms of hyperkahler manifolds

本文研究超 Kähler 流形上的抛物自同构，证明保持 Lagrangian 纤维化者在纤维上遍历作用，并将 K3 曲面的相关结果推广至一般超 Kähler 情形。

On the Kobayashi pseudometric, complex automorphisms and hyperkaehler manifolds

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperkähler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphi…

容许紧的全纯自同构可迁群的紧连通复流形

Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms

本文提出一种仅基于李理论标准结论的李代数方法，为紧复齐性流形相关经典结论给出新证明，还提供了将典范阿贝尔李代数与紧李代数上可积复结构关联的方式，拓展了前人研究。

Automorphisms of normal quasi-circular domains

It was shown by Kaup that every origin-preserving automorphism of quasi-circular domains is a polynomial mapping. In this paper, we study how the weight of quasi-circular domains and the degree of such automorphisms are related. By using the Bergman mapping, we prove that every origin-preserving automorphism of normal quasi-circular domains in $\mathbb C^2$ is linear.

Group manifolds and homogeneous spaces with HKT geometry: the role of automorphisms

We present a new simple proof of the fact that certain group manifolds as well as certain homogeneous spaces G/H of dimension 4n admit a quaternionic triple of integrable complex structures that are covariantly constant with respect to the same torsionful Bismut connection, i.e. exhibit the HKT geometry. The key observation is that different complex structures are interrelated by automorphisms of the Lie algebra. To construct the quaternion triples, one only needs to construct the proper automorphisms, which is a more simple problem.

关于抛物结构与闭非球面流形的自同构群

On the automorphism group of parabolic structures and closed aspherical manifolds

本文研究闭非球面抛物 ${\sfG}$-流形的自同构群，证明某些抛物结构会强烈限制流形拓扑，并刻画了具有几乎可解基本群的标准 $CR$-流形与四元数接触流形的商结构。

Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type

In Kähler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized Kähler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized Kähler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that…

$π_1$ of symplectic automorphism groups and invertibles in quantum homology rings

We define a homomorphism from (a certain extension of) the fundamental group of the Hamiltonian automorphism group of a symplectic manifold to the group of invertibles in its quantum cohomology ring. The manifold must satify a technical condition similar to weak monotonicity. This revised version has been entirely rewritten. It is more general and contains new examples and applications.

极值流形上的等变CM最小化

Equivariant CM minimization for extremal manifolds

本文提出在极值凯勒流形中证明等变CM最小化猜想，通过扩展Székelyhidi的结果，允许自同构和极值度量，推广了Hattori对cscK流形的研究。

Linear \infty-Harmonic maps between Rienmannian manifolds

In this paper, we give complete classifications of linear $\infty$-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all $\infty$-harmonic linear endomorphisms of Sol space and show that there is a subgroup of $\infty$-harmonic linear automorphisms in the group of linear automorphisms of Sol space.

超Kähler自同构的Kummer刚性

Kummer Rigidity for Hyperkähler Automorphisms

本研究针对射影超Kähler流形上具有正拓扑熵且以体积测度为最大熵测度的全纯自同构，通过Jensen不等式证明分布的一致伸缩性及流形平坦性，结合已有结果推出其为Kummer型例子，部分推广了复曲面对应结论。