Published online by Cambridge University Press: 07 September 2011
Peter Newstead and I started working on vector bundles at about the same time. We have shared many ideas for over four decades although we never actually collaborated. I am happy to contribute this summary of some of these areas of common interest to this Festschrift on the occasion of his turning sixty-five.
Introduction
The theory of vector bundles has many ramifications. One can study it from number theoretic, algebraic geometric and differential geometric points of view. It has also proved useful to mathematical physicists interested in Conformal Field theory, String theory, etc. In this account, I will mainly deal with the geometric aspects, both algebraic and differential, and will confine myself to just a few remarks on the number theoretic point of view.
The classical theory of abelian class fields seeks to understand Galois extensions of a number field in terms of the number theoretic behaviour of the corresponding integral extensions of the ring of integers in the number field. This has a geometric analogy. Consider any compact Riemann surface. Any finite covering of the surface gives a (finite) extension of the field of meromorphic functions on it. The attempt to try and understand abelian extensions of this field in terms of geometric data on the Riemann surface leads to the theory of Jacobians of Riemann surfaces.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.