Vector Bundles

Let X be a smooth projective curve defined over an algebraically closed field k, and let FX denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually glob- ally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if (F m X ) ∗ E ∼ Ea ⊕ E f for some m, where Ea is some ample vector bundle and E f is some finite vector bundle over X (either of Ea and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

The author describes a canonical structure on a Grassmannian fibration whose fiber is a Grassmann manifold of the tangent spaces of a smooth manifold. This structure generalizes the symplectic structure on the cotangent bundle. This symplectic form takes its values in a vector space or even in a vector bundle. This structure is canonical; it is uniquely defined by a smooth manifold. Bibliography: 5 titles.

A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that J○J=−Id TM . The composition J○J is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kahler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundles and Einstein–Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kahler manifold admits an Einstein–Hermitian connection if and only if it is polystable.

Heat kernel expansions in vector bundles

The aim of this note is to prove some bounds on the global sections of vector bundles over a smooth, complete and connected curve C . Just by an application of the Clifford theorem, the authors prove (Proposition 2) (*) h 0 (E)≤deg(E)/2+2 for a semistable rank 2 vector bundle E and discuss when (*) is sharp. They propose a sharper bound for an indecomposable bundle (which is shown to be correct for a hyperelliptic curve) but, as added in proof, this bound is overoptimistic in the general case (see Proposition IV.7 of a paper by the reviewer [Duke Math. J. 64 (1991), no. 2, 333–347] or forthcoming work of Tan). By a dimension count the authors prove (Corollary 6) h 0 (E)≤deg(E)/2+rank(E) for every globally generated semistable bundle E . In this set-up, they give a Martens-type theorem (Proposition 9).

We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of a curvature of a superconnection comes from the study of the singularities of smooth sections of vector bundles (Catastrophe Theory). We demonstrate that the classical examples of obstructions to a local equivalence: exterior differential for two-forms, Riemannian tensor, Weil tensor, curvature of a linear connection and Nijenhuis tensor can be treated in terms of some general approach. This approach, applied to the superconnection leads to a new notion of a curvature (proposed in the paper) of a superconnection. 1. A Brief Review of the Notion of a Superconnection The notions of a superconnection and of the corresponding supercurvature were introduced by Quillen in 1985 [7]. In this section we give a brief review of the matter and introduce the basic notations. By ξ = (E, p,M) we denote a vector bundle over the manifold M (dimM = m, dim(ξ) = n), by ξ∗ – the dual bundle and byC∞(ξ) – the space of the vector fields, i.e., the space of the smooth sections of the bundle ξ. Respectively Ωk(M) = C∞(ΛkT ∗(M)) is the the space of the differential k-forms on the manifoldM and

Chapter Six - The Energy of Sections

A vector bundle flow $({\Phi ^t},{\phi ^t})$ on the vector bundle $E$ over a compact metric space $M$ induces a one-parameter group $\{ \Phi _t^\# \}$ of bounded operators acting on the continuous sections of $E$, with infinitesimal generator $L$. An example is given by the tangent flow $(T{\phi ^t},{\phi ^t})$, if ${\phi ^t}$ is a flow on a smooth manifold. In this article, the spectrum of the generator $L$ is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of $M$. Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.

A vector bundle flow ( Φ t , ϕ t ) ({\Phi ^t},{\phi ^t}) on the vector bundle E E over a compact metric space M M induces a one-parameter group { Φ t # } \{ \Phi _t^\# \} of bounded operators acting on the continuous sections of E E , with infinitesimal generator L L . An example is given by the tangent flow ( T ϕ t , ϕ t ) (T{\phi ^t},{\phi ^t}) , if ϕ t {\phi ^t} is a flow on a smooth manifold. In this article, the spectrum of the generator L L is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of M M . Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.

Existence of natural and projectively equivariant quantizations

In this paper, by considering a bundle of algebras on a manifold, we construct a vector bundle that naturally contains the tangent bundle of that manifold and has the most important properties of tangent bundles. This vector bundle can be considered as an extended tangent bundle associated to that algebra bundle. To this end, we consider the algebra of the sections of that algebra bundle and find all derivations on this algebra. These derivations are related to the sections of a vector bundle that is the desired bundle. We also investigate the concepts of connection and metric on this extended tangent bundle and prove the existence of the Levi-Civita connection of a metric on the extended tangent bundle.

We define a Lie algebroid structure for a class of vector bundles of rank k over a k-dimensional smooth manifold W, which are isomorphic to the tangent bundle TW. We construct an exciting example of these types of vector bundles. This example is constructed based on partial Caputo fractional derivatives. We call this vector bundle a fractional vector bundle and denote it by {mathscr {F}}^{nu } {mathscr {W}}.

A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentially algebraic theory.

Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress toward a long-standing question. Along this line we prove that Chern numbers on nef vector bundles obey reverse dominance ordering, which improves upon some classical and recent results. We propose a simultaneous positivity question on (signed) Chern numbers of compact complex or Kähler manifolds whose (co)tangent bundles are semipositive in various senses, and show that it holds true for compact homogeneous complex manifolds.

Let $\Omega\subset M$ be an open subset of a Riemannian manifold $M$ and let $V:M\to \IR$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the natural form domain of the Schr\"odinger operator $-\Delta+V$ in $L^2(M)$, in this paper we study systematically the following question: Under which assumption on $\Omega$ is the statement $$ \text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus \Omega$ one has $f|_\Omega\in W^{1,2}_{0}(\Omega;V)$} $$ true for every such $V$? We prove that without any further assumptions on $V$, the above property is satisfied, if $\Omega$ is Kac regular, a probabilistic property which means that the first exit time of Brownian motion on $M$ from $\Omega$ is equal to its first penetration time to $M\setminus \Omega$. In fact, we treat more general covariant Schr\"odinger operators acting on sections in metric vector bundles, allowing new results concerning the harmonicity of Dirac spinors on singular subsets. Finally, we prove that locally Lipschitz regular $\Omega$'s are Kac regular.