Pares de Higgs, grassmanniana infinita y sistemas integrables

Abstract

This work makes a detailed study of Krichever's construction for several moduli spaces, which we have chosen motivated by Hitchin's Abelianization Program. In 1988, Hitchin discovered a map from the cotangent space to the moduli space of vector bundles (over a fixed Riemann surface) to an affine space of global sections, and he has shown that it is an integrable system. He addressed then the following question to the scientific comunity: can we compute, in some concrete way, the differential equations of this integrable system? Our first aim has been to study in depth Hitchin's question using as tools the infinite Grassmannian and the Krichever map. The second goal consists of looking for an integrable system with analogue properties to that of Hitchin, and finally, to find out group schemes that uniformizes such a moduli spaces. This last goal is an important step before thinking moduli spaces as solution varieties of hierarchies of differential equations. We have explicitly computed equations characterizing the moduli space of Higgs pairs, to which we add formal trivialization data. To achieve this result, we have shown that this space is a scheme, we have characterized the image of the appropriated Krichever map (which takes values not in a single infinite Grassmannian, but in a fibration of infinite Grassmannians), and we have translated this condition into a bilinear identity in terms of Baker-Akhiezer functions. We also generalize the Krichever contruction for the following moduli spaces: vector bundles and curves, pointed and finite coverings between smooth curves, and finite coverings as before with a line bundle upstairs in addition. This study allows us to find out an integrable system which behaves in an similar way as Hitchin system does, and to formulate a relationship with Hitchin's Abelianization Program. Finally, we have shown that certain group schemes - among which it is worth to point out the group of semilinear automorphisms - play the role of local generators for such moduli spaces.