arXiv research

Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.

Study of generalized vector bundles and their geometric tools.

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 …

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…

Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.

The paper defines and explores vector bundles that can be turned and their properties.

In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphis…

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…

We prove the existence of extremal, non-csc, Kähler metrics on certain unstable projectivised vector bundles $¶(E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic Kähler classes. In particular, the vector bundles $E \to M$ under consideration are assumed to split as a …

Polystable bundles on Kaehler manifolds are stable if their classes are stable.

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the stat…

Study Eisenstein metrics on modular group representations.

Study automorphisms and real structures on a special super-Grassmannian.

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bund…

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary di…

We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assu…

We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra $\cl[\leq 0](M, E)$ of non-positive order classical pseudo-differential operators acting on a finite rank vector bundle $E$ over a closed manifold $M$. Mimicking the finite dimensional Chern-Weil constru…

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra o…

Study automorphism equivariant Hitchin index for Riemann surfaces.

Derives Atiyah sequence for noncommutative bundles.

The paper studies HYM connections on stable vector bundles over Kähler manifolds.

New equations for Cosserat media motions derived from bundle automorphisms.

Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular somehow unexpected fact that the algebras of linear differential operators acting on smooth section…

Study improves Simpson's estimate for cyclic harmonic bundles.

Computes infinitesimal automorphisms for $L$-valued Higgs bundles, leading to DM stacks.

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 present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological K-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a K-cocyc…

We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplect…

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field φ\in Γ(W^+) is the same as a bundle morphism φ: TM \to TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation…

A connected Fano complex-contact manifold is isomorphic to the kaehlerian C-space of Boothby type with a natural complex-contact structure corresponding to a non-abelian simple complex Lie algebra if the contact line bundle is very ample. A. Beauville relaxed the provision to two assumptions that the contact line bundl…

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism g…

Proves well-posedness of gradient solitons on bundle gerbe.

Extends Drinfel'd doubles to include twists and generalizes Lie bialgebroids.

We study the geometry and topology of (filtered) algebra-bundles ${\bfΨ}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $Ψ^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem…

Let X be a Hermitian locally symmetric space. We prove that every Chern class of X has a canonical lift to the cohomology of the Baily- Borel-Satake compactification X* of X and that the resulting Chern numbers satisfy the Hirzebruch proportionality formula with respect to the compact dual X^ of X. The same result hold…

The paper classifies metrics on Heisenberg group's cotangent bundle.

Unique optimal symplectic connections found for submersions.

A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complet…

In this paper, we introduce the notion of $E$-Courant algebroids, where $E$ is a vector bundle. It is a kind of generalized Courant algebroid and contains Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its special cases. We explore novel phenomena exhibited by $E$-Courant algebroids and provid…

Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.

We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of the 3-sphere or of a circle bundle over a Riemann surface of positive genus.…

Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of $\mathcal{M}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the gro…

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…

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more g…

The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of geometric symmetry of a linearly singular differential equation is introduced as a trans…

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of…