On lifts of linear tensor fields to Weil bundles

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles—this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors—and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of G G -bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.

We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of $k$-fold vector bundles consisting of symmetric $k$-fold vector bundles equipped with a family of morphisms indexed by the symmetric group ${\mathbb S}_k$. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising $N$-manifolds, and how one can use the full linearisation functor to "superise" a graded bundle.

The description of the product preserving bundle functors on Mf in terms of Weil algebras reflects their general properties in a rather complete way. In the present chapter we use some other procedures to deduce the basic geometric properties of arbitrary bundle functors on Mf. Hence the basic subject of this theory is a bundle functor on Mf that does not preserve products. Sometimes we also contrast certain properties of the product-preserving and non-productpreserving bundle functors on Mf. First we study the bundle functors with the so-called point property, i.e. the image of a one-point set is a one-point set. In particular, we deduce that their fibers are numerical spaces and that they preserve products if and only if the dimensions of their values behave well. Then we show that an arbitrary bundle functor on manifolds is, in a certain sense, a ‘bundle’ of functors with the point property. For an arbitrary vector bundle functor F on Mf with the point property we also derive a canonical Lie group structure on the prolongation FG of a Lie group G.

In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable maps to the relative Grassmannian. We establish an essentially optimal upper bound on the dimension of the two compactifications. Based on that, we prove that for an arbitrary vector bundle, the Quot schemes of quotients of large degree are irreducible and generically smooth. We precisely describe all the vector bundles for which the same thing holds in the case of the moduli spaces of stable maps. We show that there are in general no natural morphisms between the two compactifications. Finally, as an application, we obtain new cases of a conjecture on effective base point freeness for pluritheta linear series on moduli spaces of vector bundles.

Graded jet geometry

The geometrization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi mathvariant="double-struck">N</mml:mi></mml:math>-manifolds of degree 2

Let ${\mathcal E}$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{{\mathbb C}}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $\psi_{2n}$ of the algebra ${\rm Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{{\mathcal E},\psi_{2n}}(\mathcal D)$ from any holomorphic differential operator ${\mathcal D}$ on ${\mathcal E}$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that $\int_X f_{{\mathcal E},\psi_{2n}}({\mathcal D})$ gives the Lefschetz number of $\mathcal D$ upto a constant independent of $X$ and ${\mathcal E}$. In addition, we obtain a "local" result generalizing the above statement. When $\psi_{2n}$ is the cocycle from [Duke Math. J. 127 (2005), 487-517], we obtain a new proof as well as a generalization of the Lefschetz number theorem of Engeli-Felder. We also obtain an analogous "local" result pertaining to B. Shoikhet's construction of the holomorphic noncommutative residue of a differential operator for trivial vector bundles on complex parallelizable manifolds. This enables us to give a rigorous construction of the holomorphic noncommutative residue of $\mathcal D$ defined by B. Shoikhet when ${\mathcal E}$ is an arbitrary vector bundle on an arbitrary compact complex manifold $X$. Our local result immediately yields a proof of a generalization of Conjecture 3.3 of [Geom. Funct. Anal. 11 (2001), 1096-1124].

Given two arbitrary vector bundles on the Fargues–Fontaine curve, we completely classify all vector bundles which arise as their extensions.

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stressing the functorial aspects of the standard Laplace operator $$\Delta $$ with respect to the category of geometric vector bundles we show that the standard Laplace operator commutes not only with all homomorphisms, but also with a large class of natural first order differential operators between geometric vector bundles. Several examples are included to highlight the conclusions of this article.

Given a holomorphic vector bundle \mathcal E on a connected compact complex manifold X , in [FLS] a ℂ -linear functional I_{\mathcal E} on H^{2n}(X, ℂ) is constructed. This is done by producing a linear functional on the 0-th completed Hochschild homology \widehat{\mathrm{HH}}_0(\mathcal{D}\mathrm{iff}(\mathcal E)) of the sheaf of holomorphic differential operators on \mathcal E using topological quantum mechanics. It is shown in [FLS] that this functional is \int_X if \mathcal E has non-zero Euler characteristic, and the conjecture is that it is \int_X for all \mathcal E . In a subsequent work [Ram] the author proved that the linear functional I_{\mathcal E} is independent of the vector bundle \mathcal E . This article builds upon the work in [Ram] to prove that I_{\mathcal E} = \int_X for an arbitrary holomorphic vector bundle \mathcal E on an arbitrary connected compact complex manifold X . This is done using an argument that is very natural from the geometric point of view. Moreover, this argument enables one to make the approach to this conjecture developed first in [FLS] and subsequently in [Ram] independent of the Riemann–Roch–Hirzebruch theorem. This argument allows us to extend the construction in [FLS] to a construction of a linear functional I_{\mathcal E} on H^{2n}_c(Y, ℂ) for a holomorphic vector bundle \mathcal E with bounded geometry on an arbitrary connected complex manifold Y with bounded geometry, and to prove that I_{\mathcal E} = \int_Y . We also generalize a result of [Ram] pertaining to “cyclic homology analogs” of I_{\mathcal E} .

The notion of {\it conjugate connections}, discussed in \cite{be:c} for a given manifold $M$ and its tangent bundle, is extended here to covariant derivatives on an arbitrary vector bundle $E$ endowed with quadratic endomorphisms. The main property of pairs of such covariant derivatives, namely the duality, is pointed out. As generalization, the case of anchored (particularly Lie algebroid) covariant derivatives on $E$ is considered. As applications we study the Finsler bundle of $M$ as well as the Finsler connections on the slit tangent bundle of a Finsler geometry.

Double principal bundles

A note on actions of some monoids

We recall the basic theory of double vector bundles and the canonical pairing of their duals, introduced by the author and by Konieczna and Urbanski. We then show that the relationship between a double vector bundle and its two duals can be understood simply in terms of an associated cotangent triple vector bundle structure. In particular, we show that the dihedral group of the triangle acts on this triple via forms of the isomorphisms R, introduced by the author and Ping Xu. We then consider the three duals of a general triple vector bundle and show that the corresponding group is neither the dihedral group of the square nor the symmetry group on four symbols.

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie's sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie's original definition; we show how it fits into a bigger picture of equivalent structures on `neighbor' double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.