Showing papers on "Frame bundle published in 2001"

Pseudo-effective line bundles on compact kähler manifolds

Abstract: The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf This result has several geometric applications, eg to the study of compact Kahler manifolds with pseudo-effective canonical or anti-canonical line bundles Another concern is to understand pseudo-effectivity in more algebraic terms In this direction, we introduce the concept of an "almost" nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds true This can be checked in some cases, eg for the canonical bundle of a projective 3-fold From this, we derive some geometric properties of the Albanese map of compact Kahler 3-folds

Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold

Abstract: Given a compact Kahler manifold M and a connected reductive algebraic group G over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], a prinlcipal G -bundle over M admits an Einstein-Hermitian connection if and only if the principal bundle is polystable. If M is a projective manifold, a Higgs G -bundle over M admits an Einstein-Hermitian connection if and only if the Higgs bundle is polystable.

Principal bundles over projective manifolds with parabolic structure over a divisor

Abstract: Principal $G$-bundles with parabolic structure over a normal crossing divisor are defined along the line of the interpretation of the usual principal $G$-bundles as functors from the category of representations, of the structure group $G$, into the category of vector bundles, satisfying certain axioms. Various results on principal bundles are extended to the more general context of principal bundles with parabolic structures, and also to parabolic $G$-bundles with Higgs structure. A simple construction of the moduli space of parabolic semistable $G$-bundles over a curve is given, where $G$ is a semisimple linear algebraic group over $C$.

Weak $Spin(9)$-Structures on 16-dimensional Riemannian Manifolds

Abstract: The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame bundle of a 16-dimensional compact manifold (Stiefel-Whitney and Pontrjagin classes). We compute the homotopy groups $\pi_i (X^{84})$ of the space $X^{84}= SO(16) / Spin(9)$ for $i \le 14$. Next we introduce different geometric types of $Spin(9)$-structures and derive the corresponding differential equation for the unique self-dual 8-form $\Omega^8$ assigned to any type of $Spin(9)$-structure. Finally we construct the twistor space of a 16-dimensional manifold with $Spin(9)$-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.

n-symplectic algebra of observables in covariant Lagrangian field theory

Abstract: n-symplectic geometry on the adapted frame bundle λ:LπE→E of an n=(m+k)-dimensional fiber bundle π:E→M is used to set up an algebra of observables for covariant Lagrangian field theories. Using the principle bundle ρ:LπE→J1π we lift a Lagrangian L:J1π→R to a Lagrangian L≔ρ*(L):LπE→R, and then use L to define a “modified n-symplectic potential” θL on LπE, the Cartan–Hamilton–Poincare (CHP) Rn-valued 1-form. If the lifted Lagrangian is nonzero, then (LπE,dθL) is an n-symplectic manifold. To characterize the observables we define a lifted Legendre transformation φL from LπE into LE. The image QL≔φL(LπE) is a submanifold of LE, and (QL,d(θ|QL)) is shown to be an n-symplectic manifold. We prove the theorem that θL=φL*(θ|QL), and pull back the reduced canonical n-symplectic geometry on QL to LπE to define the algebras of observables on the n-symplectic manifold (LπE,dθL). To find the reduced n-symplectic algebra on QL we set up the equations of n-symplectic reduction, and apply the general theory to the mo...

Hodge-gaussian maps

Abstract: Let X be a compact Kahler manifold, and let L be a line bundle on X. Define Ik(L) to be the kernel of the multiplication map Sym k H 0 (L) →

The Stiefel bundle of a Banach algebra

Abstract: We introduce the Stiefel bundle associated to a given Banachable algebra and study the properties of this analytic principal fiber bundle over the Grassmannian of equivalence classes of idempotents in the algebra. Our main application concerns the bounded linear operators of a Banach space. In particular, the problem of smooth parametrization of subspaces can then be reduced to one involving the smooth extension of sections.

Iteration of Fiber Product Preserving Bundle Functors

Abstract: Using the theory of Weil algebras, we describe the composition of two product preserving bundle functors on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. Then we deduce certain interesting geometric properties of the natural transformations of some of the iterated functors.

Riemann-Roch variations

Abstract: We construct a mirror-type correspondence that assigns variations (that is, local systems, -modules or -adic sheaves) to pairs , where is a variety and is a complex of densely filtered vector bundles over . We consider Calabi-Yau complete intersections in projective spaces. In the particular case when the complex is quasi-isomorphic to the tangent bundle on a generic Calabi-Yau complete intersection, this construction yields the variation that arises in the relative cohomology of the mirror-dual pencil. We call it the Riemann-Roch variation. The Riemann-Roch data of the divisorial sublattice in the -group can be read off the Riemann-Roch local system since it encodes the information about the Euler characteristics of all sheaves (in an essentially non-commutative way).

Non-abelian analogues of Abel's theorem

Abstract: Vafa?[29] extended a?version of mirror symmetry to pairs consisting of a?Calabi-Yau manifold and a?fixed vector bundle on it. In?[30] he considered the mathematical meaning of this extension. In this paper we prove the main facts concerning the geometry of vector bundles on Calabi-Yau manifolds and describe all constructions that enable us to embed them in the general context of modern physical concepts.

Nonholonomic mechanics and connections over a bundle map

Abstract: A general notion of connections over a vector bundle map is considered, and applied to the study of mechanical systems with linear nonholonomic constraints and a Lagrangian of kinetic energy type. In particular, it is shown that the description of the dynamics of such a system in terms of the geodesics of an appropriate connection can be easily recovered within the framework of connections over a vector bundle map. Also the reduction theory of these systems in the presence of symmetry is discussed from this perspective.

Fibre bundle formulation of nonrelativistic quantum mechanics: II. Equations of motion and observables

Abstract: We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced by an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrodinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. In the second part of this investigation are derived several forms of the bundle (analogue of the) Schrodinger equation governing the time evolution of state liftings of paths or sections along paths. We prove that up to a constant the matrix-bundle Hamiltonian, entering the bundle analogue of the matrix form of the conventional Schrodinger equation, coincides with the matrix of coefficients of the evolution transport. This allows us to interpret the Hamiltonian as a gauge field. We apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian lifting of paths or morphism along paths in corresponding bundles.

Surfaces with flat normal bundle: an explicit construction

Abstract: An explicit construction of surfaces with flat normal bundle in the Euclidean space E n (unit hypersphere S n ) in terms of solutions of certain linear system is proposed. In the case of E 3 our formulae can be viewed as a direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in affine 3-space. Parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences of spheres and surfaces with flat normal bundle in the Lie quadric in P 5 , this gives an explicit representation of surfaces with flat normal bundle in the 4-dimensional space form of the Lorentzian signature. Direct projective analog of this construction is the known parametrization of W -congruences by three solutions of the Moutard equation. Under the Plucker embedding W -congruences give rise to surfaces with flat normal bundle in the Plucker quadric. Integrable evolutions of surfaces with flat normal bundle and parallels with the theory of nonlocal Hamiltonian operators of hydrodynamic type are discussed in the conclusion.

Gauge theory of Riemann ellipsoids

Abstract: The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle . The structure group G = SO(3) is the vorticity group, and the bundle = GL+(3,) is the connected component of the general linear group. The base manifold is the space of positive-definite real 3×3 symmetric matrices, identified geometrically with the space of inertia ellipsoids. The angular momentum is not the only conserved quantity. The Kelvin circulation is also conserved as a consequence of gauge invariance. The bundle is a Riemannian manifold whose metric is determined by the kinetic energy. Nonholonomic constraints determine connexions on the bundle. In particular, the trivial connexion corresponds to rigid body motion, the natural Riemannian connexion to irrotational flow, and the invariant connexion to the falling cat.

Product preserving gauge bundle functors on vector bundles

Abstract: A complete description is given of all product preserving gauge bundle functorsF on vector bundles in terms of pairs (A,V ) consisting of a Weil algebraA and anA-moduleV with dimR(V )<1. Some applications of this result are presented.

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

Abstract: The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K X , since Y is a curve on K X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve.

Reduction in Path Integrals on a Riemannian Manifold with Group Action

Abstract: A new method for the factorization of the path-integral measure in path integrals for a particle motion on a compact Riemannian manifold with a free isometric unimodular group action is proposed. It is shown that path-integral measure is not invariant under the factorization. An integral relation between the path integral given on the total space of the principal fiber bundle and the path integral on the base space of this bundle (the orbit space of the group action) is obtained.

The tangent bundle of an almost-complex free loopspace

Abstract: The space LV of free loops on a manifold V inherits an action of the circle group T. When V has an almost-complex structure, the tangent bundle of the free loopspace, pulled back to a certain inÞnite cyclic cover g , has an equivariant decomposition as a completion of TV a (aC(k)), where TV is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of TV along evaluation at the basepoint (and aC(k) denotes an algebra of Laurent polynomials). On a sat manifold, this analogue of Fourier analysis is classical. The purpose of this note is to show that the study of the equivariant tangent bundle of a free smooth loopspace can be reduced to the study of a certain Þnitedimensional vector bundle over that loopspace – at least, provided the underlying manifold has an almost-complex structure (e.g. it might be symplectic), and if we are willing to work over a certain interesting inÞnite-cyclic cover of the loopspace. The Þrst section below summarizes the basic facts we’ll need from equivariant differential topology and geometry, and the second is a quick account of the universal cover of a symmetric product of circles, which is used in the third section to construct the promised decomposition of the equivariant tangent bundle. It is interesting that the covering transformations and the circle act compatibly on the tangent bundle of the covering, while their action on the splitting commutes only up to a projective factor.

Cosphere bundle reduction in contact geometry

Abstract: We extend the theorems concerning the equivariant symplectic reduction of the cotangent bundle to contact geometry. The role of the cotangent bundle is tken by the cosphere bundle. We use Albert's method for reduction at zero and Willett's method for non-zero reduction.

On the correlation functions of the vector bundle generalization of the bc-system

Abstract: It is shown that the determinants of the correlation functions of the generalized bc-system introduced recently are given as pullbacks of the non-Abelian theta divisor.

Parallel connections over symmetric spaces

Abstract: Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.

Elliptic surfaces and ample vector bundles

Abstract: Let E be an ample vector bundle of rank n − 2 ≥ 2 on a complex projective manifold X of dimension n having a section whose zero locus is a smooth surface Z. We determine the structure of pairs (X, E) as above under the assumption that Z is a properly elliptic surface. This generalizes known results on threefolds containing an elliptic surface as a smooth ample divisor. Among the applications we prove a conjecture relating the Kodaira dimension of X to that of Z, and we show that if 0 ≤ κ(Z) ≤ 1, then pg(Z) > 0 unless X is a Pn−2-bundle over a smooth surface S with pg(S) = 0.

Pinching, Pontrjagin classes, and negatively curved vector bundles

Abstract: We prove several finiteness results for the class ℳa,b,π,n of n-manifolds that have fundamental groups isomorphic to π and that can be given complete Riemannian metrics of sectional curvatures within {a,b} where a≤b<0. In particular, if M is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class ℳa,b,π1(M),n are total spaces of vector bundles over M. Furthermore, given a word-hyperbolic group π and an integer n there exists a positive e=e(n,π) such that the tangent bundle of any manifold in the class ℳ-1-e, -1, π, n has zero rational Pontrjagin classes.

The Formulation of the Chern-Simons Action for General Compact Lie Groups Using Deligne Cohomology

Abstract: We formulate the Chern-Simons action for any com- pact Lie group using Deligne cohomology. This action is defined as a certain function on the space of smooth maps from the underly- ing 3-manifold to the classifying space for principal bundles. If the 3-manifold is closed, the action is a C ∗ -valued function. If the 3- manifold is not closed, then the action is a section of a Hermitian line bundle associated with the Riemann surface which appears as the boundary.

A Fourier transform for Higgs bundles

Abstract: We define a Fourier-Mukai transform for Higgs bundles on smooth curves and study its properties. It is shown that the transform of a stable zero-degree Higgs bundle is an algebraic vector bundle on the cotangent bundle of the Jacobian of the curve. We show that this transformed bundle admits a natural extension to an algebraic vector bundle over a projective compactification of the base. The main result is that the original Higgs bundle can be reconstructed from this extension. We also compute certain invariants of the transformed bundle.

A Two Point Calibration on an SP(1) Bundle Over the Three-Sphere

Abstract: Gluck and Ziller proved that Hopf vector fields on S3 have minimum volume among all unit vector fields. Thinking of S3 as a Lie group, Hopf vector fields are exactly those with unit length which are left or right invariant, and TS3 is a trivial vector bundle with a connection induced by the adjoint representation. We prove the analogue of the stated result of Gluck and Ziller for the representation given by quaternionic multiplication. The resulting vector bundle over S3, with the Sasaki metric, has as well no parallel unit sections. We provide an application of a double point calibration, proving that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes.