Sections Of Bundles Research Articles

GCD inequalities inspired by Vojta’s conjecture

AbstractWe prove several GCD inequalities inspired by Vojta’s deep conjecture in Diophantine geometry. These inequalities work with all algebraic numbers, so they can be thought of as attempts to extend GCD inequalities for S-units obtained by Corvaja and Zannier. Some of the inequalities we derive are weakened versions of inequalities obtained by Silverman under the assumption of Vojta’s conjecture. The upper bounds for the GCD’s in these inequalities all involve some contributions of heights and some contributions coming from the local heights outside S. The main ingredient of the proofs is the recent Diophantine approximation result of Ru and Vojta which is based on Schmidt subspace theorem and which involves a birational invariant coming from the asymptotic growth of the number of global sections of certain line bundles.

Graded jet geometry

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of Lagrangian mechanics and the calculus of variations. It is thus only natural to require their generalization in geometry of Z-graded manifolds and vector bundles.Our aim is to construct the k-th order jet bundle JEk of an arbitrary Z-graded vector bundle E over an arbitrary Z-graded manifold M. We do so by directly constructing its sheaf of sections, which allows one to quickly prove all its usual properties. It turns out that it is convenient to start with the construction of the graded vector bundle of k-th order (linear) differential operators DEk on E. In the process, we discuss (principal) symbol maps and a subclass of differential operators whose symbols correspond to completely symmetric k-vector fields, thus finding a graded version of Atiyah Lie algebroid. Necessary rudiments of geometry of Z-graded vector bundles over Z-graded manifolds are recalled.

Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Let (g,k) be a complex quaternionic symmetric pair with k having an ideal sl(2,ℂ), k=sl(2,ℂ)+mc. Consider the representation Sm(ℂ2)=ℂm+1 of k via the projection k→sl(2,ℂ) onto the ideal sl(2,ℂ). We study the finite dimensional irreducible representations V(λ) of g which contain Sm(ℂ2) under k⊆g. We give a characterization of all such representations V(λ) and find the corresponding multiplicity, the dimension of Hom(V(λ)|k,Sm(ℂ2)). We consider also the branching problem of V(λ) under l=u(1)ℂ+mc⊂k and find the multiplicities. Geometrically the Lie subalgebra l⊂k defines a twistor space over the compact symmetric space of the compact real form Gu of Gℂ, Lie(Gℂ)=g, and our results give the decomposition for the L2-spaces of sections of certain vector bundles over the symmetric space and line bundles over the twistor space. This generalizes Cartan–Helgason’s theorem for symmetric spaces (g,k) and Schlichtkrull’s theorem for Hermitian symmetric spaces where one-dimensional representations of k are considered.

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.

Regularity of sections of CR vector bundles

In this note, we show that every generalized section σ \sigma of a CR vector bundle E E over a CR manifold M M has the property that near most points of its singular support, there exists a proper abstract CR subbundle F ⊂ E F \subset E which has the property that every real subbundle of E E which contains the image of σ \sigma also contains F F .

Localization Theorem for Homological Vector Fields

We present a general theorem which computes the cohomology of a homological vector field on global sections of vector bundles over smooth affine supervarieties. The hypotheses and results have the clear flavor of a localization theorem.

Holomorphic sections of line bundles vanishing along subvarieties

Holomorphic sections of line bundles vanishing along subvarieties

Spectral asymptotics of elliptic operators on manifolds

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator [Formula: see text] directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function [Formula: see text] and the heat trace [Formula: see text]. The kernel [Formula: see text] of the heat semigroup [Formula: see text], called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as [Formula: see text], that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as [Formula: see text], that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

O razreshimosti lineynykh differentsial'nykh operatorov na vektornykh rassloeniyakh nad mnogoobraziem

Necessary and sufficient condition is established for the closedness of the rangeor surjectivity of a differential operator acting on smooth sections of vector bundles. For connectednoncompact manifolds it is shown that these conditions are derived from the regularityconditions and the unique continuation property of solutions. An application of these results toelliptic operators (more precisely, to operators with a surjective principal symbol) with analyticcoefficients, to second-order elliptic operators on line bundles with a real leading part, and to theHodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively, Dolbeault)cohomology group on a connected noncompact smooth (respectively, complex-analytic)manifold vanishes. For elliptic operators, we prove that solvability in smooth sections impliessolvability in generalized sections.

GENERALISED AUTOMORPHIC SHEAVES AND THE PROPORTIONALITY PRINCIPLE OF HIRZEBRUCH-MUMFORD

AbstractWe axiomatise the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalised automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery, we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.

Smooth Homotopy of Infinite-Dimensional 𝐶^{∞}-Manifolds

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C ∞ C^{\infty } -manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations. We first introduce the notion of hereditary C ∞ C^\infty -paracompactness along with the semiclassicality condition on a C ∞ C^\infty -manifold, which enables us to use local convexity in local arguments. Then, we prove that for C ∞ C^\infty -manifolds M M and N N , the smooth singular complex of the diffeological space C ∞ ( M , N ) C^\infty (M,N) is weakly equivalent to the ordinary singular complex of the topological space C 0 ( M , N ) {\mathcal {C}^0}(M,N) under the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M . We next generalize this result to sections of fiber bundles over a C ∞ C^\infty -manifold M M under the same conditions on M M . Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal G G -bundles over M M and that of continuous principal G G -bundles over M M for a Lie group G G and a C ∞ C^\infty -manifold M M under the same conditions on M M , encoding the smoothing results for principal bundles and gauge transformations. For the proofs, we fully faithfully embed the category C ∞ C^{\infty } of C ∞ C^{\infty } -manifolds into the category D {\mathcal {D}} of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category D {\mathcal {D}} and the model category C 0 {\mathcal {C}^0} of arc-generated spaces, also known as Δ \Delta -generated spaces. Then, the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M imply that M M has the smooth homotopy type of a cofibrant object in D {\mathcal {D}} . This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey, which give sufficient conditions under which an infinite-dimensional topological manifold has the homotopy type of a C W CW -complex. We also show that most of the important C ∞ C^\infty -manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily C ∞ C^\infty -paracompact and semiclassical, and hence, results can be applied to them.

The Verlinde traces for SU_{X}(2,xi ) and blow-ups

Given a compact Riemann surface X of genus at least 2 with automorphism group G, we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of G-linearized line bundles defined on certain blow-ups of projective spaces along the curve X. The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular, we can compute the traces of automorphisms of X on the Verlinde spaces corresponding to the moduli space SU_{X}(2,xi ) when xi is a line bundle G-linearized of suitable degree.

Globally pmb{+}-regular varieties and the minimal model program for threefolds in mixed characteristic

We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic.

Ramanujan's master theorem for radial sections of line bundles over the Poincaré upper half plane

AbstractWe prove an analog of the Ramanujan's Master theorem for the radial sections of line bundles over the Poincaré upper half plane.

Killing Operator for the Kerr Metric

When D: E → F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) → F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 → F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 ο D = 0 implies ad (D) ο ad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Transitive irreducible Lie superalgebras of vector fields

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.

Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces

The description of the Paley–Wiener space for compactly supported smooth functions \(C^\infty _c(G)\) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for \(G=\textbf{SL}(2,\mathbb {R})^d\) (\(d\in \mathbb {N}\)) and \(G=\textbf{SL}(2,\mathbb {C})\). Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.

Explicit pseudo-Kähler metrics on flag manifolds

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) Kähler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil theorem). Less studied are the (indefinite) invariant pseudo-Kähler structures they also admit, which can be used to realize the same representations in higher cohomology of the sections (Bott's theorem). Using “eigenflag” embeddings, we give a very explicit description of these metrics in the case of the unitary group. As a byproduct we show that Un/(Un1×⋯×Unk) has exactly k! invariant complex structures, a count which seems to have hitherto escaped attention.

Higher-power harmonic maps and sections

The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on three-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric.

A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain -principal bundle , which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an ‘odd-dimensional cousin’ of symplectic geometry but rather a part of the latter, namely ‘homogeneous symplectic geometry’. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing the contact Hamiltonian formalism to the standard symplectic picture. We develop in this language contact Hamiltonian mechanics in the autonomous, as well as the time-dependent case, and the corresponding Hamilton–Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundles of sections of line bundles L, which play in contact geometry a fundamental rôle, similar to that played by cotangent bundles in symplectic geometry.