Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin^c manifolds in ?adek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.     

Sur la géométrie systolique des variétés de Bieberbach

We review the theory of quaternionic Kähler and hyperkähler structures. Then we consider the tangent bundle of a Riemannian manifold M endowed with a metric connection D, with torsion, and with its well estabilished canonical complex structure. With an almost Hermitian structure on M it is possible to find a quaternionic Hermitian structure on TM, which is quaternionic Kähler if, and only if, D is flat and torsion free. We also review the symplectic nature of TM, in the wider context of geometry with torsion. Finally we discover an S ³-bundle of complex structures, which expands to TM the well known S ²-twistor bundle of a quaternionic Hermitian manifold M.     

A note on the construction of complex and quaternionic vector fields on spheres

A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard (4n ? 1)-sphere is shown to be twice the number of linearly independent quaternionic vector fields plus d, where d = 1 or 3.     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space .     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space . (Received 27 August 1999; in revised form 18 November 1999)     

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C¹-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.     

The denominators of Lagrangian surfaces in complex Euclidean plane

A quotient of two linearly independent quaternionic holomorphic sections of a quaternionic holomorphic line bundle over a Riemann surface is a conformal branched immersion from a Riemann surface to four-dimensional Euclidean space. On the assumption that a quaternionic holomorphic line bundle is associated with a Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane, we shall classify the denominators of Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane.      

On exotic characteristic classes for spherical fibrations

We define for every so-called admissible relation r in the Steenrod algebra $\text{A}$ and for every oriented spherical fibration ξ over a CW-space an exotic characteristic class (mod 2) ε(r)(ξ), which is primitive and vanishes for sphere bundles. The set of exotic classes associated with the universal spherical fibration and the admissible Adem relations are compared with the algebra generators of H¹(BSG;$\text{Z}$₂) due to Milgram. Moreover, their behaviour under the action of $\text{A}$ is computed. Finally, we give a secondary Wu formula for exotic classes of special Poincaré duality spaces.     

Characteristic classes and transversality

Let ξ be a smooth vector bundle over a differentiable manifold M. Let be a generic bundle morphism from the trivial bundle of rank n−i+1 to ξ. We give a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold and a map whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by ?, of the fundamental class of the manifold . We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.     

Congruence Criteria for Finite Subsets of Quaternionic Elliptic and Quaternionic Hyperbolic Spaces

We investigate congruence classes of m-tuples of points in the quaternionic elliptic space ?P ⁿ . We establish a canonical bijection between the set of congruence classes of m-tuples of points in ?P ⁿ and the set of equivalence classes of positive semidefinite Hermitian m×m matrices of rank at most n+1 with the 1's on the diagonal. We show that with each m-tuple of points in ?P ⁿ is associated a tuple of points on the real unit sphere S ². Then we get that the congruence class of an m-tuple of points in ?P ⁿ is determined by the congruence classes of all its triangles and by the direct congruence class of the associated tuple on the sphere S ² provided that no pair of points of the m-tuple has distance π/2. Finally we carry out the same kind of investigation for the quaternionic hyperbolic space ?H ⁿ . Most of the results are completely analogous, although there are also some interesting differences.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

Projections of Veronese surface and morphisms from projective plane to Grassmannian

In this note, we describe the image of ?² in Gr(2, ?⁴) under a morphism given by a rank two vector bundle on ?² with Chern classes (2, 2).     

Semistability vs. nefness for (Higgs) vector bundles

Generalizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs, λs in the Grassmannians Grs(E) of locally-free quotients of E and in the projective bundles PQs, respectively (here 0<s<rkE and Qs is the universal quotient bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class .     

On the singularities of foliations and of vector bundle maps

LetF be a (smooth) Γ _q ^∞ -stucture (often called a codimension-q Haefliger structure) on a compact manifoldX ⁿ . Cohomological invariants associated to the singularities ofF are defined whose vanishing is shown to be a necessary condition for deformingF to a codimension-q foliation onX ⁿ . An analagous approach to vector bundle maps is then utilized to prove a general theorem concerning the possibility of embedding a vector bundle in the tangent bundle ofX ⁿ , and applications to the planefield problem are given. In the final section geometric realizations of the singularity classes associated toF are constructed.     

A criterion for ample vector bundles over a curve in positive characteristic

Let X be a smooth projective curve defined over an algebraically closed field of positive characteristic. We give a necessary and sufficient condition for a vector bundle over X to be ample. This generalizes a criterion given by Lange in [Math. Ann. 238 (1978) 193-202] for a rank two vector bundle over X to be ample.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

Syzygies of projective bundles

In this paper we study defining equations and syzygies among them of projective bundles. We prove that for a given p≥0, if a vector bundle on a smooth complex projective variety is sufficiently ample, then the embedding given by the tautological line bundle satisfies property N_p.     

Nonorientable Manifolds, Complex Structures, and Holomorphic Vector Bundles

A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that JJ=–Id _TM . The composition JJ is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kähler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundles and Einstein–Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kähler manifold admits an Einstein–Hermitian connection if and only if it is polystable.