Vector Bundles Of Rank Research Articles

Theta divisors and the geometry of tautological model

Let E be a stable vector bundle of rank r and slope \(2g-1\) on a smooth irreducible complex projective curve C of genus \(g \ge 3\). In this paper we show a relation between theta divisor \(\Theta _E\) and the geometry of the tautological model \(P_E\) of E. In particular, we prove that for \(r > g-1\), if C is a Petri curve and E is general in its moduli space then \(\Theta _E\) defines an irreducible component of the variety parametrizing \((g-2)\)-linear spaces which are g-secant to the tautological model \(P_E\). Conversely, for a stable, \((g-2)\)-very ample vector bundle E, the existence of an irreducible non special component of dimension \(g-1\) of the above variety implies that E admits theta divisor.

Construction of a linear filtration for bundles of rank on

We obtain an algorithm for the construction of a filtration with linear factors for vector bundles of rank 2 over the surface , where is a Euclidean domain. In other words, we produce an algorithm that, for an invertible -matrix over the ring , constructs matrices over and over for which is an upper triangular matrix. Bibliography: 13 titles.

Hilbert curves of 3-dimensional scrolls over surfaces

Let (X,L) be a 3-dimensional scroll over a smooth surface Y. Its Hilbert curve is an affine plane cubic consisting of a given line and a conic. This conic turns out to be the Hilbert curve of the Q-polarized surface (Y,12det⁡E), where E is the rank-2 vector bundle obtained by pushing down L via the scroll projection, if and only if E is properly semistable in the sense of Bogomolov.

English

Let D be a Cdq-convex intersection, d ≥ 2, 0≤ q ≤ n − 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, Ck-estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \)-equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n − q ≤ s ≤ n. In addition, we solve the \(\bar \partial \)-equation with a support condition in Ck-spaces. More precisely, we prove that for a \(\bar \partial \)-closed form f in \(C_{0{,_q}}^k\left( {X\backslash D,E} \right),{\kern 1pt} 1 \leqslant q \leqslant n - 2,{\kern 1pt} n \geqslant 3\), with compact support and for e with 0 < e < 1 there exists a form u in \(C_{0{,_{q - 1}}}^k\left( {X\backslash D,E} \right)\) with compact support such that \(\bar \partial u = f{\kern 1pt} in{\kern 1pt} X\backslash \bar D\). Applications are given for a separation theorem of Andreotti-Vesentini type in Ck-setting and for the solvability of the \(\bar \partial \)-equation for currents.

Derived categories of moduli spaces of vector bundles on curves

Let X be a smooth projective curve of genus g≥4 over C and M the moduli space of stable vector bundles of rank 2 and determinant isomorphic to a fixed line bundle of degree 1. We prove in this paper that Db(X) is embedded (via ΦE) in Db(M) as a full subcategory (if g≥4).

An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications

In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent tangent bundles on 22-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal E$ be a vector bundle of rank $n$ over $M^n$, and $\varphi:TM^n\to \mathcal E$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss-Bonnet formulas can be generalized to an index formula for the bundle homomorphism $\varphi$ under the assumption that $\varphi$ admits only certain kinds of generic singularities. We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.

Vector bundles on proper toric 3-folds and certain other schemes

We show that a proper algebraic n n -dimensional scheme Y Y admits non-trivial vector bundles of rank n n , even if Y Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional locus in only finitely many points. Moreover, there are such vector bundles with arbitrarily large top Chern number. Applying this to toric varieties, we infer that every proper toric threefold admits such vector bundles of rank three. Furthermore, we describe a class of higher-dimensional toric varieties for which the result applies, in terms of convexity properties around rays.

The Calderón problem for connections

In this paper we consider the problem of identifying a connection $\nabla$ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian $\nabla^*\nabla$ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of the transversal manifold being simple -- we generalise this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of \cite{CTA}. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.

Vector bundles over a real elliptic curve

Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we determine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and degree d on the curve. When r and d are coprime, we describe the topology of the real locus and give a modular interpretation of its points. We also study, for arbitrary rank and degree, the moduli space of indecomposable vector bundles of rank r and degree d, and determine its isomorphism class as a real algebraic variety.

ON A GENERALIZATION OF HIRZEBRUCH'S THEOREM TO BOTT TOWERS

The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds Bn(�1,...,�n−1,�n) and Bn(�1,...,�n−1,�') are isomorphic to each other, as Bott towers if and only if bothn � � ' mod 2 and � 2 = (� ') 2 hold in the cohomology ring of Bn−1(�1,...,�n−1) over integer coefficients. This result will complete a circle of ideas initiated in (11) by Ishida. We also give some partial affirmative remarks toward the assertion that under certain c our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.

Globally generated vector bundles on complete intersection Calabi–Yau threefolds

We investigate the globally generated vector bundles on complete intersection Calabi-Yau threefolds with the first Chern class at most 2. We classify all the globally generated vector bundles of an arbitrary rank on quintic in $\mathbb{P}^4$ and investigate the globally generated vector bundles of rank 2 on complete intersection Calabi-Yau threefolds of codimension 2.

Tango bundles on Grassmannians

The goal of this paper is to prove the existence of indecomposable rank vector bundles on the Grassmannian variety . We will call them Tango bundles since in the particular case of they correspond to the celebrated vector bundle discovered by H. Tango in 1974. We will give a geometrical description of Tango bundles and we will prove that they are μ-stable.

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces M-X (r, c(1), c(2)) parametrizing slope semistable vector bundles of rank r and fixed Chern classes c(1), c(2) on a ruled surface whose base is a rational nodal curve. We showthat under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space M-X (r, c(1), c(2)) is rational as a variety defined over R.

On Higher Rank Globally Generated Vector Bundles over a Smooth Quadric Threefold

AbstractWe give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold withc1≤ 2 and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.

Globally generated vector bundles on a smooth quadric surface

We give the classification of globally generated vector bundles of rank 2 on a smooth quadric surface with c 1 ⩻ (2, 2) in terms of the indices of the bundles, and extend the result to arbitrary higher rank case. We also investigate their indecomposability and give the sufficient and necessary condition on numeric data of vector bundles for indecomposability.

Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

Abstract In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.

Proper triangular 𝔾a-actions on 𝔸4are translations

We describe the structure of geometric quotients for proper locally triangulable additve group actions on locally trivial A^3-bundles over a noetherian normal base scheme X defined over a field of characteristic 0. In the case where dim X=1, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank 2 over X. As a consequence, every proper triangulable Ga-action on the affine four space A^4 over a field of characteristic 0 is a translation with geometric quotient isomorphic to A^3.

Irreducible vector bundles on some elliptic non-Kahler threefolds

We study rank-2 vector bundles on non-Kähler threefolds π : X → B , which are elliptic principal bundles with at least one non-zero Chern class over a complex surface B with no curves. In this case, we prove that every rank-2 irreducible vector bundle on X is a pull-back from B up to a twist by a line bundle. These 2-vector bundles are, via the Kobayashi–Hitchin correspondence, solutions of the Yang–Mills equations on the threefold X .

A cohomological classification of vector bundles on smooth affine threefolds

We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. As a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable A1-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s A1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

CONSTRUCTING CO-HIGGS BUNDLES ON CP2

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover CP^1 \times CP^1 \to CP^2, is naturally a co-Higgs bundle, with the twisted endomorphism, or "Higgs field", also descending from the double cover. Allowing the branch conic to vary, we find that Schwarzenberger bundles give rise to an 8-dimensional moduli space of co-Higgs bundles. After studying the deformation theory for co-Higgs bundles on complex manifolds, we conclude that a co-Higgs bundle arising from a Schwarzenberger bundle with nonzero Higgs field is rigid, in the sense that a nearby deformation is again Schwarzenberger.