Existence of sections of line bundles over a toroidal group and its applications

Dedicated to Professor Joji Kajiwara on the occasion of his sixtieth birthday     

Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds

We consider a two-parameter generalization $D_{ab}$ of the Riemann Dirac operator $D$ on a closed Sasakian spin manifold, focusing attention on eigenvalue estimates for $D_{ab}$ . We investigate a Sasakian version of twistor spinors and Killing spinors, applying it to establish a new connection deformation technique that is adapted to fit with the Sasakian structure. Using the technique and the fact that there are two types of eigenspinors of $D_{ab}$ , we prove several eigenvalue estimates for $D_{ab}$ which improve Friedrich’s estimate (Friedrich, Math Nachr 97, 117–146, 1980).     

On tangent bundles with Sasakian metrics of Finslerian and Riemannian manifolds

On the basis of the so-called phase completion the notion of vertical, horizontal and complete objects is defined in the tangent bundles over Finslerian and Riemannian manifold. Such a tangent bundle is made into a manifold of almost Kaehlerian structure by endowing it with Sasakian metric. The components of curvature tensors with respect to the adapted frame are presented. This having been done it is shown possible to study the differential geometry of Finslerian spaces by dealing with that of their own tangent bundles. This work was supported by National Research Coundil of Canada A-4037 (1960–70). Entrata in Redazione l'8 marzo 1970.     

Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds: An addendum

We define a Gaussian measure on the space of almost holomorphic sections of powers of an ample line bundle over a symplectic manifold , and calculate the joint probability densities of sections taking prescribed values and covariant derivatives at a finite number of points. We prove that they have a universal scaling limit as . This result will be used in another paper to extend our previous work on universality of scaling limits of correlations between zeros to the almost-holomorphic setting.

     

Holomorphic sections of line bundles over (H,C)-groups

Using an analogue of the refined Chern class, we study holomorphic sections of line bundles over an (H,C)-group. As an application we get that every meromorphically separable (H,C)-group is quasi-abelian.     

Characteristic rank of vector bundles over Stiefel manifolds

The characteristic rank of a vector bundle ξ over a finite connected CW-complex X is by definition the largest integer ${k, 0 \leq k \leq \mathrm{dim}(X)}$ , such that every cohomology class ${x \in H^{j}(X;\mathbb{Z}_2), 0 \leq j \leq k}$ , is a polynomial in the Stiefel–Whitney classes w _i (ξ). In this note we compute the characteristic rank of vector bundles over the Stiefel manifold ${V_k(\mathbb{F}^n), \mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}}$ .     

Orientability of vector bundles over real flag manifolds

We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, which include the tangent bundle and also stable bundles of certain gradient flows. Closed formulas, in terms of roots, are provided.     

Homotopy connectedness theorems for submanifolds of Sasakian manifolds

The homotopy connectedness theorem for invariant immersions in Sasakian manifolds with nonnegative transversal q-bisectional curvature is proved. Some Barth-Lefschetz type theorems for minimal submanifolds and (k, ?)-saddle submanifolds in Sasakian manifolds with positive transversal q-Ricci curvature are proved by using the weak (?-)asymptotic index. As a corollary, the Frankel type theorem is proved.     

Holomorphic sections of pre-quantum line bundles on

Let be the Iwasawa decomposition of a complex connected semi-simple Lie group . Let be a parabolic subgroup containing , and let be its commutator subgroup. In this paper, we characterize the -invariant Kähler structures on , and study the holomorphic sections of their corresponding pre-quantum line bundles.

     

Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds

Let be the total space of a fibre bundle with base a simply connected manifold whose loop space homology grows exponentially for a given coefficient field. Then we show that for any Riemannian metric on , the topological entropy of the geodesic flow of is positive. It follows then, that there exist closed manifolds with arbitrary fundamental group, for which the geodesic flow of any Riemannian metric on has positive topological entropy.

     

On the stability of flat complex vector bundles over parallelizable manifolds

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G/\mathrm{\Gamma }$, where G is a complex connected Lie group and Γ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_{\rho }$ associated with any irreducible representation $\rho :\mathrm{\Gamma }?\text{GL}(r,\mathbb{C})$. More precisely, we prove that $E_{\rho }$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where E is a stable vector bundle. All the rational Chern classes of E vanish, in particular, its degree is zero.We deduce a stability result for flat holomorphic vector bundles $E_{\rho }$ of rank 2 over $G/\mathrm{\Gamma }$. If an irreducible representation $\rho :\mathrm{\Gamma }?\text{GL}(2,\mathbb{C})$ satisfies the condition that the induced homomorphism $\mathrm{\Gamma }?\mathrm{PGL}(2,\mathbb{C})$ does not extend to a homomorphism from G, then $E_{\rho }$ is proved to be stable.     

On multiply of sections of line bundles on compact Riemann surfaces

In this paper,we give some conditions on the surjective of multiply maps H~0(R,L)×H~0(R,K)→H~0(R,L(?)K).Here R is a compact Riemann surface,L a line bundle on R and K is the canonical line bundle.     

Core versus graded core, and global sections of line bundles

We find formulas for the graded core of certain -primary ideals in a graded ring. In particular, if is the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under a suitable hypothesis, the core and graded core of the ideal of generated by all elements of degrees at least (for some, equivalently every, large ) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary characteristic. Several open problems are posed whose solutions would lead to progress on a non-vanishing conjecture of Kawamata.

     

The Cartan model of the canonical vector bundles over the Grassmannian manifolds

We give a representation of the canonical vector bundles

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over the Grassmannian manifolds G(n, p) as noncompact symmetric affine spaces together with their Cartan model in the group of the Euclidean motions SE(n).