arXiv research

Proves strong Tits alternative for 3D automorphisms over zero char fields.

Study automorphism groups of Inoue surfaces using quadratic number fields.

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

New proof shows smallest non-cyclic automorphism group quotients are linear 2-groups.

The paper studies automorphisms of 2-step nilpotent Lie groups, showing continuity up to center and field automorphisms.

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…

Cone structures in quantum field theory linked to information geometry.

Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.

Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the g…

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplec…

This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(α,β)$-metrics. Let $F$ be an $(α,β)$-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group $G$. We consider the automorphism and isometry g…

In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector …

A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give an alternative proof of the (surprising…

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups a…

We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that …

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…

Researchers classify lattices in a specific four-dimensional group.

We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism f, we assign an AF-algebra A(f) (an operator algebra). It is proved that the assignment is functorial, i.e. every f', conjugate to f, maps to an AF-algebra A(f'), which is stably isomorphic to A…

We compute the automorphism group of OT manifolds of simple type. We show that the graded pieces under a natural filtration are related to a certain ray class group of the underlying number field. This does not solve the open question whether the geometry of the OT manifold sees the class number directly, but brings us…

Introduces a theorem for groups acting on trees.

The paper develops a Galois theory for cluster algebras and Riemann surfaces.

Classifies flat projective structures with specific symmetries.

The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.

An automorphism on a complex supermanifold $\mathcal M$ is called unipotent if it reduces to the identity on the associated graded supermanifold $gr(\mathcal M)$. These automorphisms are close to be complementary to those responsible for homogeneity of a supermanifold. In analogy, their study yields results on the clas…

A vector field is called a Beltrami vector field, if $B\times(\nabla\times B)=0$. In this paper we construct two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and such that both have an orientation-preserving …

VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singul…

Study on symmetries of quaternionic Kähler manifolds with S^1-symmetry.

It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL_2 over S-integers.

Homological stability fails for Cremona groups, rational varieties, and function fields.

WZW models are abstract conformal field theories with an infinite dimensional symmetry which accounts for their integrability, and at the same time they have a sigma model description of closed string propagation on group manifolds which, in turn, endows the models with an intuitive geometric meaning. We exploit this d…

In this work one shows that given a connected $C^\infty$-manifold $M$ of dimension $\geq 2$ and a finite subgroup $G\subset \Diff(M)$, there exists a complete vector field $X$ on $M$ such that its automorphism group equals $G\times \mathbb{R}$ where the factor $\mathbb{R}$ comes from the flow of $X$.

We use the general theory developed in our article arXiv:1208.5510 in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.

In the paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists …

We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…

Extends classification of hyperbolic Dehn fillings in quadratic fields.

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting Cartan geometries. We prove that a compact Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus.

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.

In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.

Extends K-energy to complexified Kähler classes for scalar curvature study.

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of …

We study nilmanifolds admitting Anosov automorphisms by applying elementary properties of algebraic units in number fields to the associated Anosov Lie algebras. We identify obstructions to the existence of Anosov Lie algebras. The case of 13-dimensional Anosov Lie algebras is worked out as an illustration of the techn…

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) …

The paper constructs contact-hyperbolic manifolds with large automorphism groups.