Vector Bundles Of Rank Research Articles

Rational curves on moduli spaces of vector bundles

W describe the unobstructed components of the Hilbert Scheme of rational curves of fixed degree [Formula: see text] in the moduli space [Formula: see text] of stable vector bundles of rank [Formula: see text] and determinant [Formula: see text] on a curve [Formula: see text]. We show that for every [Formula: see text], there are [Formula: see text] such components. We construct obstructed components of the Hilbert Scheme. We also obtain an upper bound on the degree of rational connectedness of [Formula: see text] which is linear in the dimension.

Uniform (d + 1)-bundle over the Grassmannian G(d,n)

This paper is dedicated to the classification of uniform vector bundles of rank [Formula: see text] over the Grassmannian [Formula: see text] ([Formula: see text]) over an algebraically closed field in characteristic [Formula: see text]. Specifically, we show that all uniform vector bundles with rank [Formula: see text] over [Formula: see text] are homogeneous.

Towards a modular construction of OG10

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$ -type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

Algebra and geometry of irreducible toric vector bundles of rank n on [formula omitted]

We construct a presentation for the Cox ring of the projectivization PE of any rank n irreducible toric vector bundle on Pn. We use this presentation to show that PE always satisfies Fujita's freeness and ampleness conjectures.

Moore matrices and Ulrich bundles on an elliptic curve

We give normal forms of determinantal representations of a smooth projective plane cubic in terms of Moore matrices . Building on this, we exhibit matrix factorizations for all indecomposable vector bundles of rank 2 and degree 0 without nonzero sections, also called Ulrich bundles , on such curves.

Criteria for the ampleness of certain vector bundles

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to det ( E ) \det (E) ). This result is a higher-rank version of a theorem of Schneider and Tancredi for vector bundles of rank two over surfaces. We also provide counterexamples indicating that our theorem is sharp.

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g⩾2 with the explicit parametrization in terms of the Tyurin data. The ‘non-abelian’ theta divisor consists of bundles such that h1⩾1 . On the complement of this divisor we construct a non-abelian (i.e. matrix) Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which also depends holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1.

A Lie Algebroid Structure for Vector Bundles of Finite Rank Isomorphic to Tangent Bundle of Their Base Space

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}}.

Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 on S3

We construct and give a complete classification of all the differential symmetry breaking operators [Formula: see text], between the spaces of smooth sections of a vector bundle of rank [Formula: see text] over the [Formula: see text]-sphere [Formula: see text], and a line bundle over the [Formula: see text]-sphere [Formula: see text]. In particular, we give necessary and sufficient conditions on the tuple of parameters [Formula: see text] for which these operators exist.

ODD RANK VECTOR BUNDLES IN ETA-PERIODIC MOTIVIC HOMOTOPY THEORY

AbstractWe observe that, in the eta-periodic motivic stable homotopy category, odd rank vector bundles behave to some extent as if they had a nowhere vanishing section. We discuss some consequences concerning$\operatorname {\mathrm {SL}}^c$-orientations of motivic ring spectra and the étale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category.

Locally free twisted sheaves of infinite rank

We study twisted vector bundles of infinite rank on gerbes, giving a new point of view on Grothendieck’s famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme.We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of “very positive” infinite rank vector bundles on projective varieties, and show that an infinite rank vector bundle on a curve in a surface can be lifted to the surface away from finitely many points.

Quantization of symplectic fibrations and canonical metrics

We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson’s iterations toward balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson’s iterations toward balanced metrics on Kähler manifolds with constant scalar curvature.

Slope semistability and positive cones of Grassmann bundles

Let [Formula: see text] be a vector bundle of rank [Formula: see text] on a smooth complex projective variety [Formula: see text]. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle Gr[Formula: see text] parametrizing [Formula: see text]-dimensional subspaces of the fibers of [Formula: see text], where [Formula: see text] rk([Formula: see text]), under assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. In particular, we show that nef cone and the pseudoeffective cone of Gr[Formula: see text] coincide if and only if nef cone and pseudoeffective cone of [Formula: see text] coincide under the assumption that [Formula: see text] is a slope semistable bundle on [Formula: see text] with [Formula: see text](End([Formula: see text]))=0. We also discuss about the nefness and ampleness of the universal quotient bundle [Formula: see text] on Gr[Formula: see text].

Brauer and Picard groups of moduli spaces of parabolic vector bundles on a real curve

We determine the Brauer group and Picard group of the moduli space of stable parabolic vector bundles of rank r with determinant L on a real curve Y of arithmetic genus with at most nodes as singularities.

On the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces

AbstractThe aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

Enough vector bundles on orbispaces

We show that every orbispace satisfying certain mild hypotheses has ‘enough’ vector bundles. It follows that the $K$-theory of finite rank vector bundles on such orbispaces is a cohomology theory. Global presentation results for smooth orbifolds and derived smooth orbifolds also follow.

Cancellation of vector bundles of rank 3 with trivial Chern classes on smooth affine fourfolds

If n ≡ 0 , 1 m o d 4 , we prove a sum formula V θ 0 ( a 0 , a R n ) = n ⋅ V θ 0 ( a 0 , a R ) for the generalized Vaserstein symbol whenever R is a smooth affine algebra over a perfect field k with c h a r ( k ) ≠ 2 such that − 1 ∈ k × 2 . This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension d ≥ 4 over algebraically closed fields of characteristic ≠2. As a consequence, we prove that any projective module of rank 3 with trivial Chern classes over a smooth affine algebra of dimension 4 over an algebraically closed field k with c h a r ( k ) ≠ 2 , 3 is cancellative.

Topological classification of complex vector bundles over 8-dimensional spin$$^{c}$$ manifolds

In this paper, complex vector bundles of rank r over 8-dimensional spin\(^{c}\) manifolds are classified in terms of the Chern classes of the complex vector bundles, where \(r = 3\) or 4. As an application, we see that two rank 3 complex vector bundles over 4-dimensional complex projective space \(\mathbb {C}P^{4}\) are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank 3 complex vector bundles over \(\mathbb {C}P^{4}\) are determined. Together with results of Thomas and Switzer, this completes the classification of complex vector bundles of any rank over \(\mathbb {C}P^4\).

Cohomology of the moduli stack of algebraic vector bundles

Let Vectn be the moduli stack of vector bundles of rank n on derived schemes. We prove that, if E is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies projective bundle formula, then E⁎(Vectn,S) is freely generated by Chern classes c1,…,cn over E⁎(S) for any qcqs derived scheme S. Examples include all multiplicative localizing invariants.

Orthogonal exceptional collections from -Gorenstein degeneration of surfaces

We consider a surface that admits a -Gorenstein degeneration to a cyclic quotient singularity After assuming several technical conditions, we construct d exceptional vector bundles of rank n which are orthogonal to each other.