Higgs bundles ondel Pezzo fibrations.

摘要

The objective of this thesis is to study principal bundles on del Pezzo fibrations. Our main result classifies a class of principal bundles on a del Pezzo fibration Y/X, for which the base X is an integral curve and the generic fiber is smooth. This classification is in terms of spectral data on the base X together with the information on how to lift the objects described by the spectral data on X to the total space Y.; A del Pezzo surface is roughly the blow-up of the projective plane P2 at 1 ≤ r ≤ 8 points. A del Pezzo fibration p : Y → X is a projective flat map whose fibers are del Pezzo surfaces. It is well-known that Del Pezzo surfaces are closely linked to (semi)simple Lie algebras of type Er. This relation is pursued in [12] to show that there is a natural isomorphism class of bundles on a given del Pezzo surface. The elements of this isomorphism class will be called almost regular. By the same token, a bundle on a fibration whose restriction to each fiber is almost regular is also called almost regular.; An almost regular bundle on a given fibration Y/X determines a so-called regularization on an open subscheme Y0 of Y. Then the restriction of the bundle to Y0 together with its regularization can be studied by the Donagi-Gaitsgory approach [7]. Indeed, we prove that after fixing some data, almost regular bundles on Y are in one-to-one correspondence with the regularized bundles on Y0. On the other hand, [7] provides a description of the regularized bundles in terms of spectral objects on X together with the lift information to Y. We slightly extend their results to work for non-projective fibrations, such as Y0/X. To solve the inverse problem of extending a regularized bundle on Y 0 to Y, we extend the techniques of Friedman-Morgan [12] to work for families. Doing so, we take the first step in the direction of studying principal bundles on del Pezzo surfaces---a question of Friedman-Morgan (ibid.)---as well as presenting another application of [7].