Thesis--Göttingen. / Vita. Includes bibliographical references (p. 53-54).

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 63-65).

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaf 42).

A vector bundle E over a CW-complex X is said to be stably trivial of type (n,k) if E, © ke = ne, where e denotes the trivial line bundle. Let V[sub n,k] , be the Stiefel manifold of orthonormal k- frame in euclidian n-space R[sup n] and let n[sub n,k] , be the real (n-k)- dimensional vector bundle over V[sub n,l] , whose fiber over a k—frame x is the subspace of R[sup n] orthogonal to the span of the vectors in x . The vector bundle n[sub n,k], is "weakly universal" for stably trivial vector bundles of type (n,k), i.e. for any stably trivial vector bundle of type (n,k), there is a map f: X -> V[sub n,k] , not necessarily unique up to homotopy, such that f *n[sub n,k] = E, . We study the following questions: (a) for which values of r is the r-fold Whitney sum rn[sub n,k] , trivial, and (b) what is the maximum number of linearly independent cross-sections of n[sub n,k] ©[sup se] (0 < s < k - 1) . Among the results obtained are: (1) 2n[sub n,2] is trivial iff n is even or n=3; (2) 3n[sub n,2] is trivial if n is even; (3) rn[sub n,k], is not trivial if r is odd and < (n-2)/(n-k): (4) n[sub n,k] © (k-l)c is not trivial if n ≄ 2,4,8 and 1 < k < n - 3; (5) n[sub n,k] © [sup se] admits exactly s linearly independent cross-sections if n and k are odd; (6) n[sub n,k] © (k-2)e admits at most (k-1) linearly independent sections if 2<_k<_n-3. These results are used to construct examples of stably free modules and unimodular matrices over commutative noetherian rings. The techniques used are those of homotopy theory, including Postnikov systems, K-theory and, specially, Spin operations on vector bundles. A chapter of the thesis is devoted to defining the Spin operations formally as a type of K-theoretic characteristic classes for a certain type of real vector bundles. Formulae to compute the Spin operations on a Whitney sum of vector bundles are given. / Science, Faculty of / Mathematics, Department of / Graduate

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This thesis is comprised of three logically independent parts. As the title suggests, each part is related to vector bundles on abelian varieties. We first use Brill-Noether theory to study the geometry of a general curve in its canonical embedding. We prove that there is no $g$ for which the canonical embedding of a general curve of genus $g$ lies on the Segre embedding of any product of three or more projective spaces. We then consider non-degenerate line bundles on abelian varieties. Central to our work is Mumford's index theorem. We give an interpretation of this theorem, and then prove that non-degenerate line bundles, with nonzero index, exhibit positivity analogous to ample line bundles. As an application, we determine the asymptotic behaviour of families of cup-product maps. Using this result, we prove that vector bundles, which are associated to these families, are asymptotically globally generated. To illustrate our results, we consider explicit examples. We also prove that simple abelian varieties, for which our results apply in all possible instances, exist. This is achieved by considering a particular class of abelian varieties with real multiplication. The final part of this thesis concerns the theory of theta and adelic theta groups. We extend and refine work of Mumford, Umemura, and Mukai. For example, we determine the structure and representation theory of theta groups associated to a class of vector bundles which we call simple semi-homogeneous vector bundles of separable type. We also construct, and clarify functorial properties enjoyed by, adelic theta groups associated to line bundles. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-06-24 17:14:21.687

In this thesis we study the moduli of holomorphic vector bundles over a non-compact complex space X, which will mainly be of dimension 2 or 3 and which contains a distinguished rational curve ℓ ⊂ X. We will consider the situation in which X is the total space of a holomorphic vector bundle on CP1 and ℓ is the zero section. While the treatment of the problem in this full generality requires the study of complex analytic spaces, it soon turns out that a large part of it reduces to algebraic geometry. In particular, we prove that in certain cases holomorphic vector bundles on X are algebraic. A key ingredient in the description of themoduli are numerical invariants that we associate to each holomorphic vector bundle. Moreover, these invariants provide a local version of the second Chern class. We obtain sharp bounds and existence results for these numbers. Furthermore, we find a new stability condition which is expressed in terms of these numbers and show that the space of stable bundles forms a smooth, quasi-projective variety.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 116-117).

This document presents a systematic investigation of the geometric index theory of Dirac operators coupled superconnections. A local version of the index theorem for Dirac operators coupled to superconnection is proved, and extended to families. An [eta]-invariant is defined, and it is shown to satisfy an APS-like theorem. A geometric determinant line bundle with section, metric, and connection is associated to a family of Dirac operators coupled to superconnections, and its holonomy is calculated in terms of the [eta]-invariant. / text

published_or_final_version / Mathematics / Master / Master of Philosophy