De nition 3.1. Let X be a complex manifold. A holomorphic vector bundle of rank k over E is a complex manifold E and a holomorphic map : E ! X such that makes E ! X into a complex vector bundle of rank k, and E admits holomorphic trivializations, i.e. there’s an open cover U of X trivializing E such that for each U 2 U, there’s a biholomorphic map ' : EjU ! U C commuting with projection to U th...

The anisotropic blowup along the zero section and the fiberwise compactification of a real vector bundle F → X are manifolds with boundary for which the natural structure bundle is the b-tangent bundle. We give a necessary and sufficient condition for these procedures to give a b-complex structure when applied to a holomorphic vector bundle over a complex manifold, and analyze some aspects of t...

I repeat my definition for quantization of a vector bundle. For the cases of Töplitz and geometric quantization of a compact Kähler manifold, I give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle.

the geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on tm. the metrizability of a given semispray is of special importance. in this paper, the metric associated with the semispray s is applied in order to study some types of foliations on the tangent bundle which are compatible with sode structure. indeed, suff...

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H(X, E ⊗ F ) and H(X, E ⊗ F ) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined ove...

It is shown that a strong system of vector fields on a fiber bundle in the sense of [Mo] is induced from a principal fiber bundle if and only if each vertical vector field of the system is complete.

Let G be a reductive algebraic group over e, let F be a G-module, and let B be an affine G-variety, i.e., an affine variety with an algebraic action of G. Then B x F is in a natural way a G-vector bundle over B, which we denote by F. (All vector bundles here are algebraic.) A G-vector bundle over B is called trivial if it is isomorphic to F for some G-module F . From the endomorphism ring R of ...

It is shown that a strong system of vector elds on a ber bundle in the sense of Mo] is induced from a principal ber bundle if and only if each vertical vector eld of the system is complete.