scholarly journals Negative Vector Bundles and Complex Finsler Structures

1975 ◽  
Vol 57 ◽  
pp. 153-166 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Complex Manifold ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Finsler Structure

A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)

scholarly journals OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

2019 ◽  
Vol 7 ◽  
Author(s):  
A. ASOK ◽  
J. FASEL ◽  
M. J. HOPKINS
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Algebraic Variety ◽  
Vector Bundles ◽  
Necessary Condition ◽  
Chern Classes ◽  
Smooth Complex ◽  
Steenrod Operations ◽  
Complex Algebraic Variety ◽  
Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

scholarly journals Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

2006 ◽  
Vol 13 (1) ◽  
pp. 7-10
Author(s):  
Edoardo Ballico
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Open Neighborhood ◽  
Holomorphic Vector Bundle ◽  
Stein Manifold ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles

Abstract Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂(𝑈).

Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula

1973 ◽  
Vol 25 (4) ◽  
pp. 765-771
Author(s):  
Hansklaus Rummler
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Result ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Group ◽  
Parallel Translation ◽  
Result Theorem

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

scholarly journals LIE ALGEBROIDS AS SPACES

2018 ◽  
Vol 19 (2) ◽  
pp. 487-535 ◽  
Author(s):  
Ryan Grady ◽  
Owen Gwilliam
Keyword(s):  
Tangent Bundle ◽  
Symplectic Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Faithful Functor ◽  
Derived Geometry

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

HORIZONTAL LAPLACIAN ON TANGENT BUNDLE OF FINSLER MANIFOLD WITH g-NATURAL METRIC

2012 ◽  
Vol 09 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI ◽  
CHUNPING ZHONG
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Vector Bundles ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Order Differential Operator ◽  
Finsler Manifolds ◽  
Scalar And Vector Fields ◽  
Matsumoto Metric ◽  
Weitzenböck Formula

Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenböck formula of the horizontal Laplacian associated to Ga,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to Ga,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to Ga,b are investigated.

scholarly journals PARABOLIC VECTOR BUNDLES AND EQUIVARIANT VECTOR BUNDLES

2002 ◽  
Vol 13 (09) ◽  
pp. 907-957 ◽  
Author(s):  
IGNASI MUNDET I RIERA
Keyword(s):  
Slope Stability ◽  
Irreducible Component ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Full Subcategory ◽  
Natural Numbers ◽  
Normal Crossing ◽  
Irreducible Components

Given a complex manifold X, a normal crossing divisor D ⊂ X whose irreducible components D1, …, Ds are smooth, and a choice of natural numbers [Formula: see text], we construct a manifold [Formula: see text] with an action of a torus Γ and we prove that some full subcategory of the category of Γ-equivariant vector bundles on [Formula: see text] is equivalent to the category of parabolic vector bundles on (X, D) in which the lengths of the filtrations over each irreducible component of D are given by [Formula: see text]. When X is Kaehler, we study the Kaehler cone of [Formula: see text] and the relation between the corresponding notions of slope-stability.

Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles

2006 ◽  
Vol 49 (1) ◽  
pp. 36-40 ◽  
Author(s):  
Georgios D. Daskalopoulos ◽  
Richard A. Wentworth
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Convergent Sequence ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Hölder Class ◽  
Holomorphic Vector Bundles ◽  
Holder Class

AbstractUsing a modification of Webster's proof of the Newlander–Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

scholarly journals Homogeneous vector bundles and stability

1986 ◽  
Vol 101 ◽  
pp. 37-54 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Matrix Function ◽  
Vector Bundles ◽  
Hermitian Matrix ◽  
Holomorphic Vector Bundle ◽  
Positive Definite ◽  
Frame Field ◽  
Hermitian Vector Bundle ◽  
Homogeneous Vector

In [5, 6, 7] I introduced the concept of Einstein-Hermitian vector bundle. Let E be a holomorphic vector bundle of rank r over a complex manifold M. An Hermitian structure h in E can be expressed, in terms of a local holomorphic frame field s1, …, sr of E, by a positive-definite Hermitian matrix function (hij) defined by

The Tangent Bundle of an Almost Complex Manifold

2001 ◽  
Vol 44 (1) ◽  
pp. 70-79 ◽  
Author(s):  
László Lempert ◽  
Róbert Szőke
Keyword(s):  
Complex Manifold ◽  
Tangent Bundle ◽  
Deformation Theory ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Holomorphic Maps ◽  
Almost Complex Manifolds ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Natural Way

AbstractMotivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complexmanifold with an almost complex structure. We describe various properties of this structure.

scholarly journals Dirac Structures on Banach Lie Algebroids

2014 ◽  
Vol 22 (3) ◽  
pp. 219-228
Author(s):  
Vlad-Augustin Vulcu
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Differential Calculus ◽  
Vector Bundles ◽  
Dirac Structure ◽  
Lie Algebroid ◽  
Symmetric Form ◽  
Courant Bracket ◽  
Dirac Structures ◽  
Original Definition

Abstract In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle T M ⊕ T* M that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle E ⊕ E*, where E is a Banach Lie algebroid and E* its dual. Recall that E* is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle E ⊕ E* and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.

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