Vector Bundles Research Articles

On pseudo-effective cones of projective bundles and volume function

In this paper, we compute the pseudo-effective cones of various projective bundles [Formula: see text] over higher dimensional varieties [Formula: see text] under some assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. We also compute the volume function on fiber product [Formula: see text] of two projective bundles over a smooth irreducible complex projective curve [Formula: see text]. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.

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.

A remark on the relative Lie algebroid connections and their moduli spaces

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over [Formula: see text]). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

Equivariant orientation of vector bundles over disconnected base spaces

AbstractIn this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group , a precise condition under which an ‐orientation of a ‐equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel–Whitney class of a “homogeneous” ‐equivariant bundle with respect to an ‐ring spectrum . As an application, we show that the 2‐fold direct sum of any homogeneous bundle is ‐orientable, where is the Burnside Mackey functor. We notice that ‐orientability is equivalent to ‐orientability when the order of is odd. When the order of is even, we show that a ‐equivariant analog of the tautological line bundle over is ‐orientable but not ‐orientable.

Logarithmic Pandharipande–Thomas spaces and the secondary polytope

Maulik and Ranganathan have recently introduced moduli spaces of logarithmic stable pairs. In the case of toric surfaces we recast this theory using three ingredients: Gelfand, Kapranov and Zelevinsky secondary polytopes, Hilbert schemes of points, and tautological vector bundles. In particular, logarithmic stable pairs spaces are expressed as the zero set of an explicit section of a vector bundle on a logarithmically smooth space, thus providing an explicit description of their virtual fundamental class. A key feature of our construction is that moduli spaces are completely canonical, unlike the existing construction, which is only well-defined up to logarithmic modifications. We calculate the Euler–Satake characteristics of our moduli spaces in a number of basic examples. These computations indicate the complexity of the spaces we construct.

Torelli theorem for moduli stacks of vector bundles and principal G-bundles

Given any irreducible smooth complex projective curve X, of genus at least 2, consider the moduli stack of vector bundles on X of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve X and the rank of the vector bundles. The case of trivial determinant, rank 2 and genus 2 is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is P3 thus independently of the curve).We also prove a Torelli theorem for moduli stacks of principal G-bundles on a curve of genus at least 3, where G is any non-abelian reductive group.

Vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety

Abstract In this paper, we establish several results related to vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffiths-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather–Jacobian multiplier ideals.

A functorial approach to the stability of vector bundles

On a normal projective variety the locus of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable bundles that remain μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.

Spinor–Vector Duality and Mirror Symmetry

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the “Swampland” program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not.

Rational Curves on Moduli Spaces of Vector Bundles

Rational Curves on Moduli Spaces of Vector Bundles

Algebraic algorithms for vector bundles over curves

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package.

Cohomologies of tautological bundles of matroids

Tautological bundles of realizations of matroids were introduced in Berget et al. (Invent Math 233:951–1039, 2023) as a unifying geometric model for studying matroids. We compute the cohomologies of exterior and symmetric powers of these vector bundles, and show that they depend only on the matroid of the realization. As an application, we show that the log canonical bundle of a wonderful compactification of a hyperplane arrangement complement, in particular the moduli space of pointed rational curves M¯0,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{{M}}_{0,n}$$\\end{document}, has vanishing higher cohomologies.

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.

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

On nonemptiness of Newton strata in the $B_{\mathrm{dR}}^{+}$-Grassmannian for $\mathrm{GL}_{n}$

We study the Newton stratification in the B_{\mathrm{dR}}^{+} -Grassmannian for \mathrm{GL}_{n} associated to an arbitrary (possibly nonbasic) element of B(\mathrm{GL}_{n}) . Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in B(\mathrm{GL}_{n}) , our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues–Fontaine curve.

On varieties with Ulrich twisted conormal bundles

We study varieties X ⊂ P r X \subset \mathbb {P}^r such that N X ∗ ( k ) N_X^*(k) is an Ulrich vector bundle for some integer k k . We first prove that such an X X must be a curve. Then we give several examples of curves with N X ∗ ( k ) N_X^*(k) an Ulrich vector bundle.

Ulrich modules over hypersurface rings via adjoint matrices

We describe an Ulrich module on the coordinate ring of the projective hypersurface X defined by the determinant of an n × n linear matrix ϕ , and show that the module defines a vector bundle on X if and only if the ideal generated by the n − 1 order minors of ϕ is zero-dimensional; this property does not force X to be smooth, while the existing literature focuses on the smooth case. If X is any smooth hypersurface, we characterize when its derivation module is Ulrich. We raise the same question in higher codimension and show the Segre threefold has this property.

Strange duality at level one for alternating vector bundles

In this paper, we show a strange duality isomorphism at level one for the space of generalized theta functions on the moduli spaces of alternating anti-invariant vector bundles in the ramified case. These anti-invariant vector bundles constitute one of the non-trivial examples of parahoric G-torsors, where G is a twisted (not generically split) parahoric group scheme.

Stable vector bundles on fibered threefolds over a surface

Stable vector bundles on fibered threefolds over a surface