Showing papers on "Frame bundle published in 2011"

Strict and nonstrict positivity of direct image bundles

Abstract: This paper is a sequel to (Berndtsson in Ann Math 169:531-560, 2009). In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kahler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we address the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. We prove that this is so provided the twisting line bundle is strictly positive along fibers, but not in general.

Algebraic Constructions in the Category of Lie Algebroids

Abstract: A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are characterized by o horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented.

The Stokes Structure of a good meromorphic flat bundle

Abstract: We give a survey on the Stokes structure of a good meromorphic flat bundle. We also show that a meromorphic flat bundle has the good formal structure if and only if it has a good lattice.

String Structures and Trivialisations of a Pfaffian Line Bundle

Abstract: The present paper is a contribution to categorial index theory. Its main result is the calculation of the Pfaffian line bundle of a certain family of real Dirac operators as an object in the category of line bundles. Furthermore, it is shown how string structures give rise to trivialisations of that Pfaffian.

On the geometry of the (1, 1)-tensor bundle with Sasaki type metric

Abstract: Curvature properties are studied for the Sasaki metric on the (1, 1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-Kahler-Nordenian B-metrics are constructed on the (1, 1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.

The partition bundle of type AN-1 (2, 0) theory

Abstract: Six-dimensional (2, 0) theory can be defined on a large class of six-manifolds endowed with some additional topological and geometric data (i.e. an orientation, a spin structure, a conformal structure, and an R-symmetry bundle with connection). We discuss the nature of the object that generalizes the partition function of a more conventional quantum theory. This object takes its values in a certain complex vector space, which fits together into the total space of a complex vector bundle (the ‘partition bundle’) as the data on the six-manifold is varied in its infinite-dimensional parameter space. In this context, an important role is played by the middle-dimensional intermediate Jacobian of the six-manifold endowed with some additional data (i.e. a symplectic structure, a quadratic form, and a complex structure). We define a certain hermitian vector bundle over this finite-dimensional parameter space. The partition bundle is then given by the pullback of the latter bundle by the map from the parameter space related to the six-manifold to the parameter space related to the intermediate Jacobian.

The limit of the Yang-Mills flow on semi-stable bundles

Abstract: By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kahler manifold X, the Yang-Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is Hermitian-Einstein and will decompose the limiting bundle into a direct sum of stable bundles. Bando and Siu prove this limiting bundle can be extended to a reflexive sheaf E' on all of X. In this paper, we construct an isomorphism between E' and the double dual of the stable quotients of the graded Seshadri filtration of E.

The index bundle for Fredholm morphisms

Abstract: We extend the index bundle construction for families of bounded Fredholm operators to morphisms between Banach bundles.

Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties

Abstract: We study aspects of the A^1-homotopy classification problem in dimensions >= 3 and, to this end, we investigate the problem of computing A^1-homotopy groups of some A^1-connected smooth varieties of dimension >=. Using these computations, we construct pairs of A^1-connected smooth proper varieties all of whose A^1-homotopy groups are abstractly isomorphic, yet which are not A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension. Projectivizations of vector bundles give rise to A^1-fiber sequences, and when the base of the fibration is an A^1-connected smooth variety, the associated long exact sequence of A^1-homotopy groups can be analyzed in detail. In the case of the projectivization of a rank 2 vector bundle, the structure of the A^1-fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank >=, the A^1-fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher A^1-homotopy groups is influenced by an appropriately defined higher obstruction class.

On the topology and the geometry of SO(3)-manifolds

Abstract: Consider the non-standard embedding of SO(3) into SO(5) given by the five-dimensional irreducible representation of SO(3), henceforth called SO(3)ir. In this note, we study the topology and the differential geometry of five-dimensional Riemannian manifolds carrying such an SO(3)ir structure, i.e., with a reduction of the frame bundle to SO(3)ir.

The anomaly line bundle of the self-dual field theory

Abstract: In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.

Gerbes and Lie Groups

Abstract: Compact Lie groups do not only carry the structure of a Riemannian manifold, but also canonical families of bundle gerbes. We discuss the construction of these bundle gerbes and their relation to loop groups. We present several algebraic structures for bundle gerbes with connection, such as Jandl structures, gerbe modules and gerbe bimodules, and indicate their applications to Wess–Zumino terms in two-dimensional field theories.

On projective space bundle with nef normalized tautological line bundle

Abstract: In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite étale covering. Introduction For a morphism between smooth projective varieties π : Y → X the relative anticanonical divisor −Kπ on Y is defined by the difference of anticanonical divisors −Kπ := −KY − π(−KX). J. Kollár, Y. Miyaoka and S. Mori proved that the relative anticanonical divisor of a non-constant generically smooth morphism cannot be ample in arbitrary characteristic [7], [11]. In the case where π : Y = PX(E) → X is a projectivization of vector bundle on X, we know that the relative anti-canonical divisor is positive proportion of the normalized tautological divisor. Miyaoka studied the case where Y is a curve and showed that the nefness of the normalized tautological divisor is equal to the semistability of vector bundle [10]. Nakayama generalized this to the arbitrary dimension in [13]. In this paper we study the more explicit structure of vector bundles with nef normalized tautological divisor. In Section 1, we review the definition and some known results. In Section 2, we treat semiample cases and show that a pullback of such a bundle by some finite unramified covering is trivial up to twist by some line bundle. In Section 3, we treat the case where X is a blow-up of a smooth variety Z along smooth subvariety or a projective bundle over a smooth variety Z. In these cases we show that the vector bundle with nef normalized tautological divisor on X is isomorphic to the pullback of vector bundle on K having the same property up to twist by the exceptional divisor. In Section 4 we study manifolds whose tangent bundle have a nef normalized tautological divisor. We prove such surfaces are isomorphic to a quotient of abelian surface by some finite étale morphism. Moreover under the assumption that such a divisor is semiample, we can show that finite étale covering of abelian varieties are all varieties satisfying this property. 2000 Mathematics Subject Classification. Primary 14J40; Secondary 14J10, 14J60.

Stability of tangent bundle on the moduli space of stable bundles on a curve

Abstract: In this paper, we prove that the tangent bundle of the moduli space $\cSU_C(r,d)$ of stable bundles of rank $r>2$ and of fixed determinant of degree $d$ (such that $(r,d)=1$), on a smooth projective curve $C$ is always stable, in the sense of Mumford-Takemoto. This verifies a well-known conjecture, and is related to a conjectural existence of a Kahler-Einstein metric on Fano varieties with Picard number one.

Isometric embeddings of banach bundles

Abstract: We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.

Semistability criterion for parabolic vector bundles on curves

Abstract: We give a cohomological criterion for a parabolic vec- tor bundle on a curve to be semistable. It says that a parabolic vector bundle E∗ with rational parabolic weights is semistable if and only if there is another parabolic vector bundle F∗ with rational parabolic weights such that the cohomologies of the vector bundle underlying the parabolic tensor product E∗ ⊗F ∗ vanish. This criterion general- izes the known semistability criterion of Faltings for vector bundles on curves and significantly improves the result in (Bis07).

Curvature function and coarse graining

Abstract: A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to "local details" ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function $F_S (A)$ defined below. It covers the holonomy function in the sense that $\exp{F_S (A)} = {\rm Hol}(l= \partial S, A)$.

(k, s)-POSITIVITY AND VANISHING THEOREMS FOR COMPACT KÄHLER MANIFOLDS

Abstract: We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kahler manifolds.

An Adapted Frame on Indicatrix Bundle of a Finsler Manifold and its Geometric Properties

Abstract: In this paper, a frame is introduced on tangent bundle of a Finsler manifold in a manner that it makes some simplicity to study the properties of the natural foliations in tangent bundle. Moreover, we show that the indicatrix bundle of a Finsler manifold with lifted sasaki metric and natural almost complex structure on tangent bundle cannot be a sasakian manifold.

Metric Non-Linear Connections on the Prolongation of a Lie Algebroid to its Dual Bundle

Abstract: Metric Non-Linear Connections on the Prolongation of a Lie Algebroid to its Dual Bundle In the present paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied. The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found. We prove that the nonlinear connection induced by a regular Hamiltonian on a Lie algebroid is the unique connection which is compatible with the metric and symplectic structures.

Causal geometries, null geodesics, and gravity

Abstract: The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle, the Raychaudhuri--Sachs equations of general relativity continue to hold, and give rise to the same phenomenon of covergence of null geodesics in regions of positive energy that underlies the theory of gravitation.

Determinant line bundle on moduli space of parabolic bundles

Abstract: In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian–Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings’ criterion of semistability plays a crucial role.

A compactification of the moduli space of principal Higgs bundles over singular curves

Abstract: A principal Higgs bundle $(P,\phi)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $\phi:X\to\text{Ad}P \otimes \Omega^1_X$ We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles More precisely, given a faithful representation $\rho:G\to Sl(V)$ of $G$, we consider principal Higgs bundles as triples $(E,q,\phi)$ where $E$ is a vector bundle with $\rk{E}=\dim V$ over the normalization $\xtilde$ of $X$, $q$ is a parabolic structure on $E$ and $\phi:E\ab{}\to L$ is a morphism of bundles, being $L$ a line bundle and $E\ab{}\doteqdot (E^{\otimes a})^{\oplus b}$ a vector bundle depending on the Higgs field $\phi$ and on the principal bundle structure Moreover we show that this moduli space for suitable integers $a,b$ is related to the space of framed modules

Construction of Bundles on Simplicial Sets and Transition Functions

Abstract: We give the definition of a bundle over a simplicial set in a similar way to that in general bundle theory. We define certain transition functions, called compatible transition functions, for the given bundle, as well as so-called admissible trivializations. Conversely, we give the description of an associated bundle over the realization of the simplicial set and trivializations for a given set of transition functions.

Vector bundles of rank four and A_3 = D_3

Abstract: Over a scheme with 2 invertible, we show that a vector bundle of rank four has a sub or quotient line bundle if and only if the canonical symmetric bilinear form on its exterior square has a lagrangian subspace. For this, we exploit a version of "Pascal's rule" for vector bundles that provides an explicit isomorphism between the moduli functors represented by projective homogeneous bundles for reductive group schemes of type A_3 and D_3. Under additional hypotheses on the scheme (e.g. proper over a field), we show that the existence of sub or quotient line bundles of a rank four vector bundle is equivalent to the vanishing of its Witt-theoretic Euler class.

PARABOLIC k-AMPLE BUNDLES

Abstract: We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle E* is defined to be k-ample if the tautological line bundle is k-ample. We establish some properties of parabolic k-ample bundles.