Sums of stably trivial vector bundles

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

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Note on the characteristic rank of vector bundles

Mathematica Slovaca ◽

10.2478/s12175-014-0289-4 ◽

2014 ◽

Vol 64 (6) ◽

Cited By ~ 5

Author(s):

Aniruddha Naolekar ◽

Ajay Thakur

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Projective Spaces ◽

Real Vector ◽

Cw Complex ◽

Moore Spaces

AbstractWe define the notion of characteristic rank, charrankX(ξ), of a real vector bundle ξ over a connected finite CW-complex X. This is a bundle-dependent version of the notion of characteristic rank introduced by Július Korbaš in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korbaš and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.

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Stable vector bundles on algebraic surfaces II

Nagoya Mathematical Journal ◽

10.1017/s0027763000015968 ◽

1973 ◽

Vol 52 ◽

pp. 173-195 ◽

Cited By ~ 16

Author(s):

Fumio Takemoto

Keyword(s):

Vector Bundle ◽

Line Bundle ◽

Vector Bundles ◽

Ample Line Bundle ◽

Algebraic Surfaces ◽

Complex Numbers ◽

Stable Vector Bundle ◽

Projective Varieties ◽

Stable Vector Bundles ◽

Direct Sum

This paper is a continuation of “Stable vector bundles on algebraic surfaces” [10]. For simplicity we deal with non-singular projective varieties over the field of complex numbers. Let W be a variety whose fundamental group is solvable, let H be an ample line bundle on W, and let f: V → W be an unramified covering. Then we show in section 1 that if E is an f*H-stable vector bundle on V then f*E is a direct sum of H-stable vector bundles. In particular f*L is a direct sum of simple vector bundles if L is a line bundle on V.

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Universal polynomials for singular curves on surfaces

Compositio Mathematica ◽

10.1112/s0010437x13007756 ◽

2014 ◽

Vol 150 (7) ◽

pp. 1169-1182 ◽

Cited By ~ 5

Author(s):

Jun Li ◽

Yu-jong Tzeng

Keyword(s):

Vector Bundle ◽

Linear System ◽

Line Bundle ◽

Vector Bundles ◽

Projective Surface ◽

Analytic Singularity ◽

Smooth Projective Surface ◽

Chern Numbers ◽

Cutting Out ◽

Curves On Surfaces

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

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Positivity of vector bundles on homogeneous varieties

International Journal of Mathematics ◽

10.1142/s0129167x20500974 ◽

2020 ◽

Vol 31 (12) ◽

pp. 2050097

Author(s):

Indranil Biswas ◽

Krishna Hanumanthu ◽

D. S. Nagaraj

Keyword(s):

Vector Bundle ◽

Line Bundle ◽

Algebraic Group ◽

Abelian Variety ◽

Maximal Torus ◽

Vector Bundles ◽

Projective Variety ◽

Closed Curve ◽

Seshadri Constant ◽

Higher Rank

We study the following question: Given a vector bundle on a projective variety [Formula: see text] such that the restriction of [Formula: see text] to every closed curve [Formula: see text] is ample, under what conditions [Formula: see text] is ample? We first consider the case of an abelian variety [Formula: see text]. If [Formula: see text] is a line bundle on [Formula: see text], then we answer the question in the affirmative. When [Formula: see text] is of higher rank, we show that the answer is affirmative under some conditions on [Formula: see text]. We then study the case of [Formula: see text], where [Formula: see text] is a reductive complex affine algebraic group, and [Formula: see text] is a parabolic subgroup of [Formula: see text]. In this case, we show that the answer to our question is affirmative if [Formula: see text] is [Formula: see text]-equivariant, where [Formula: see text] is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on [Formula: see text].

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HECKE CORRESPONDENCE FOR SYMPLECTIC BUNDLES WITH APPLICATION TO THE PICARD BUNDLES

International Journal of Mathematics ◽

10.1142/s0129167x06003357 ◽

2006 ◽

Vol 17 (01) ◽

pp. 45-63 ◽

Cited By ~ 4

Author(s):

INDRANIL BISWAS ◽

TOMÁS L. GÓMEZ

Keyword(s):

Vector Bundle ◽

Line Bundle ◽

Bilinear Form ◽

Moduli Space ◽

Vector Bundles ◽

Projective Bundle ◽

Projective Curve ◽

Smooth Projective Curve ◽

Hecke Correspondence ◽

Stable Pairs

We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let [Formula: see text] be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg (E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on [Formula: see text], which is a projective bundle whose fiber over any point [Formula: see text] is ℙ(H0(X,E)). We prove that this projective bundle is stable.

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Euler classes of vector bundles over manifolds

Mathematica Slovaca ◽

10.1515/ms-2017-0461 ◽

2021 ◽

Vol 71 (1) ◽

pp. 199-210

Author(s):

Aniruddha C. Naolekar

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Euler Class ◽

Homology Sphere ◽

Rational Homology ◽

Rational Homology Sphere ◽

Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

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WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002243 ◽

2011 ◽

Vol 84 (2) ◽

pp. 255-260

Author(s):

EDOARDO BALLICO ◽

FRANCESCO MALASPINA

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Splitting Type ◽

Multiprojective Spaces

AbstractHere we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.

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Two-Categorical Bundles and their Classifying Spaces

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is012001012jkt181 ◽

2012 ◽

Vol 10 (2) ◽

pp. 299-369 ◽

Cited By ~ 2

Author(s):

Nils A. Baas ◽

Marcel Bökstedt ◽

Tore August Kro

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Vector Spaces ◽

Bundle Theory ◽

Classifying Spaces ◽

Classifying Space ◽

Principal Bundles

AbstractFor a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

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Generic Dynamical Properties of Connections on Vector Bundles

International Mathematics Research Notices ◽

10.1093/imrn/rnab069 ◽

2021 ◽

Author(s):

Mihajlo Cekić ◽

Thibault Lefeuvre

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Differential Operators ◽

Parallel Transport ◽

Conformal Killing ◽

Generic Properties ◽

Geometric Problems ◽

Pseudo Differential Operators ◽

Divergence Type ◽

Open Question

Abstract Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla ^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted conformal Killing tensors (CKTs) are generically trivial when $\dim (M) \geq 3$, answering an open question of Guillarmou–Paternain–Salo–Uhlmann [ 14]. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations, which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla ^{\textrm{End}({\operatorname{{\mathcal{E}}}})}$ on the endomorphism bundle $\textrm{End}({\operatorname{{\mathcal{E}}}})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e., the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely that generically there are no non-trivial subbundles of $\mathcal{E}$ that are preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott–Ruelle resonances in the Anosov case).

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