Leibniz 2-Algebras and Twisted Courant Algebroids

Let Y be an integral projective curve whose singularities are of type A_k, i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= p_a(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned degree d line bundles, L, with h⁰(Y, L) ≥ r + 1 even if the Brill - Noether number ρ(g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane section of a simple K3 surface S ? P ^g.     

Lie Algebroids, Holonomy and Characteristic Classes

We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid.     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

On Almost Complex Lie Algebroids

The almost complex Lie algebroids over smooth manifolds are considered in the paper. In the first part, we give some examples and we extend some basic results from almost complex manifolds to almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied.     

Vanishing Theorems on Holomorphic Lie Algebroids

The paper describes a Bochner-type study for holomorphic horizontal vector fields defined on a holomorphic Finsler algebroid E. We obtain in this setting a vanishing theorem for horizontal fields with compact support on E.     

Lie and Courant algebroids on foliated manifolds

This is an exposition of the subject, which was developed in the author’s papers [19, 20]. Various results from the theory of foliations (cohomology, characteristic classes, deformations, etc.) are extended to subalgebroids of Lie algebroids that generalize the tangent integrable distributions. We also suggest a definition of foliated Courant algebroids and give some corresponding results and constructions.     

Tangent and Cotangent Lifts and Graded Lie Algebras Associated with Lie Algebroids

Generalized Schouten, Frölicher–Nijenhuis and Frölicher–Richardson brackets are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie algebroids are introduced and discussed and the behaviour of the related graded Lie brackets under these lifts is studied. In the case of the canonical Lie algebroid on the tangent bundle, a new common generalization of the Frölicher–Nijenhuis and the symmetric Schouten brackets, as well as embeddings of the Nijenhuis–Richardson and the Frölicher–Nijenhuis bracket into the Schouten bracket, are obtained.     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Almost Analyticity on Almost (Para) Complex Lie Algebroids

Results in Mathematics - The goal of this paper is to generalize the notion of almost analyticity in the almost (para)complex Lie algebroids framework. We use a global formalism which yields to...     

Foliated g-structures and riemannian foliations

Abstract Using the properties of the commuting sheaf of aG-foliation of finite type we prove that some of theseG-foliations must be Riemannian.     

Duality Features of Left Hopf Algebroids

We explore special features of the pair (U ^?,U _?) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S ^? from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phùng’s categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.     

Foliated control theory I

The main result is a control theorem for the space of stable pseudo-isotopies on E with control near the leaves of in M, where : E M is a fiber bundle over a Riemannian manifold M having a compact manifold for fiber and is a smooth foliation of M such that each leaf inherits a flat Riemannian geometry from M. A similar result has been proved by Farrell and Jones under the assumption that each leaf of inherits a geometry with strictly negative curvature from M.Both authors were supported in part by the National Science Foundation.     

Foliated control with hyperbolic leaves

The main result is a control theorem for the space of stable pseudo-isotopies on E with control near the leaves of F in M, where : E M is a fiber bundle over the compact closed Riemannian manifold M having a compact manifold for fiber and F is a smooth foliation of M. The proof of this foliated control result combines the (unfoliated) control results of [14] with the long and thin cell structures and the asymptotic transfer of [6] and [7].Both authors were supported in part by the NSF.     

Foliated Manifolds and Conformal Heat Morphisms

The objects under study, in this article, are Riemannian manifoldsfoliated by hypersurfaces. Looking at the transverse direction as time,we construct the generalised heat operator and, in the spirit of atime-space extension of harmonic morphisms, we introduce the conformal heat morphisms. Concrete examples of these concepts arepresented, as well as, a characterisation of conformal heat morphisms.Lastly, we calculate the heat Lie algebra of the generalised heatoperator.