Vector Bundles Research Articles

Twistor theory of the Chen–Teo gravitational instanton**Dedicated to Nick Woodhouse on the occasion of his 75th birthday.

Abstract Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a
twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, 
for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains
a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these 
instantons take a simple rational form.

On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

Let X be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on X that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document}. Given a choice of Hermitian metric on E we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.

Cohesive self-dual Yang-Mills modules over four manifolds

This research is motivated by Atiyah, Hitchin, and Singers paper Self-duality in Four-Dimensional Riemannian Geometry , which introduced a relationship between self-dual Yang-Mills fields on smooth manifolds and holomorphic vector bundles on their twistor spaces. Here, self-duality is a specific structure in 4-dimensional manifolds and Yang-Mills fields are gauge fields that satisfy Yang-Mills equations in 4-dimensions and are corresponded to the holomorphic bundles on twistor spaces. In this paper, we extend the relationship from vector bundles to a generalization of the Yang-Mills fields. To achieve this purpose, we apply Atiyah, Hitchin, and Singers theorem to cohesive modules, which was originally introduced by Block in in studying coherent sheaves over complex manifolds and the relations between homomorphic torus and its dual non-commutative torus. We introduce the notion of cohesive self-dual Yang-Mills modules and show that the twistor correspondence actually induces the equivalence between the dg category of cohesive self-dual Yang Mills modules P_(A_SD ) and the dg category of holomorphic cohesive modules P_(A_Hol ) on the twistor spaces.

Absolutely split locally free sheaves on proper k-schemes and Brauer–Severi varieties

We classify absolutely split vector bundles on proper k-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer–Severi varieties.

On the Chow theory of Quot schemes of locally free quotients

We prove a formula for the Chow groups of Quot schemes that resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This formula simultaneously generalizes the formulas for projective bundles, Grassmannian bundles, blowups, Cayley's trick, projectivizations, flops from Springer-type resolutions, and Grassmannian-type flips and flops. We also apply the formula to study the Chow groups of (i) blowups of determinantal ideals; (ii) moduli spaces of linear series on curves; and (iii) (nested) Hilbert schemes of points on surfaces.

Stable vector bundles on a hyper-Kähler manifold with a rank 1 obstruction map are modular

Stable vector bundles on a hyper-Kähler manifold with a rank 1 obstruction map are modular

Abelian envelopes of exact categories and highest weight categories

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional sequences. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proven for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality on highest weight categories is interpreted as a duality between the right and left abelian envelopes of a thin exact category. A duality for thin exact categories compatible with Ringel duality is introduced by means of derived categories and Serre functor on them.

Extension groups of tautological bundles on punctual Quot schemes of curves

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles.

Reduced resonance schemes and Chen ranks

Abstract The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

Reductive quotients of klt singularities

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} endowed with the action of a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document} and admitting a quasi-projective good quotient X→X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X\\rightarrow X/\\!/G$\\end{document}, we can find a boundary B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$B$\\end{document} on X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/G$\\end{document} so that the pair (X//G,B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(X/\\!/G,B)$\\end{document} is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document}-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-dimensional K-polystable smooth Fano varieties of volume v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

A Wasserstein-Type Distance for Gaussian Mixtures on Vector Bundles with Applications to Shape Analysis

A Wasserstein-Type Distance for Gaussian Mixtures on Vector Bundles with Applications to Shape Analysis

On anti-ample vector bundles and nef and big vector bundles

We prove that the direct image of an anti-ample vector bundle is anti-ample under any finite flat morphism of non-singular projective varieties. In the second part we prove some properties of big and nef vector bundles. In particular it is shown that the tensor product of a nef vector bundle with a nef and big vector bundle is again nef and big. This generalizes a result of Schneider.

Irreducibility of moduli of vector bundles over a very general sextic surface

Let S be a very general smooth hypersurface of degree 6 in P3. In this paper we will prove that the moduli space of μ-stable rank 2 torsion free sheaves with respect to hyperplane section having c1=OS(1), with fixed, c2≥27 is irreducible.

Drinfel’d double of bialgebroids for string and M theories: dual calculus framework

We extend the notion of Lie bialgebroids for more general bracket structures used in string and M theories. We formalize the notions of calculus and dual calculi on algebroids. We achieve this by reinterpreting the main results of the matched pairs of Leibniz algebroids. By examining a rather general set of fundamental algebroid axioms, we present the compatibility conditions between two calculi on vector bundles which are not dual in the usual sense. Given two algebroids equipped with calculi satisfying the compatibility conditions, we construct its double on their direct sum. This generalizes the Drinfel’d double of Lie bialgebroids. We discuss several examples from the literature including exceptional Courant brackets. Using Nambu-Poisson structures, we construct an explicit example, which is important both from physical and mathematical point of views. This example can be considered as the extension of triangular Lie bialgebroids in the realm of higher Courant algebroids, that automatically satisfy the compatibility conditions. We extend the Poisson generalized geometry by defining Nambu-Poisson exceptional generalized geometry and prove some preliminary results in this framework. We also comment on the global picture in the framework of formal rackoids and we slightly extend the notion for vector bundle valued metrics.

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.

Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces

We classify equivariant harmonic maps of the complex projective spaces CPm into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when m≧2, we will obtain the rigidity results.

The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

For a complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet of a vector bundle along the submanifold associates the L2-optimal holomorphic extension of it to the ambient manifold. When the vector bundle is given by big tensor powers of a positive line bundle, we give an asymptotic formula for this extension operator.

A mean field type equation on vector bundles

A mean field type equation on vector bundles

Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras

This paper studies the class of spherical objects over any Kodaira n -cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the n -punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk (2002) and extends earlier results by Burban–Kreußler (2012) and Lekili–Polishchuk (2019). The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla (2014). The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk–Drozd–Greuel (2012) which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product, we obtain a closed formula for the cyclic sequence of any simple vector bundle on C_{n} as introduced by Burban–Drozd–Greuel (2001).

On the evaluation of the Ray-Singer torsion path integral

There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions. Here we provide such a calculation for the path integral representation of the Ray-Singer Torsion of a flat connection on a vector bundle on base manifolds that are themselves S1 bundles of any dimension. The calculation relies on a suitable algebraic choice of gauge which leads to a convenient factorisation of the path integral into horizontal and vertical parts.