Principal bundles over projective varieties

Abstract

During his PhD, the candidate studied the moduli space of (Higgs) principal bundles on algebraic curves using, in particular, the theory of decorated bundles. He gave a compactication of the moduli space of principal Higgs bundles on nodal curves. For higher dimensional variety not much is known even in the simplest case of the projective space. It is reasonable to expect that one can also obtain a good understanding of the moduli spaces of certain decorated bundles on projective spaces, like principal bundles, using monads. One of the goals of this project is to study the construction of principal bundles on projective spaces, especially P2 and P3, using monads.In the final part of his PhD, the candidate studied Higgs bundles on high dimensional varieties and their restriction to curves. In particular, he obtained a classification of those stable Higgs bundles on a variety X which remain semistable when pulled-back to any smooth curve at least when X has nef tangent bundle. The general case is still an open problem. The aim now is to extend certain results on Higgs bundles on curves to Higgs bundle on projective varietie,s and consequently to obtain a classification like the one mentioned above to arbitrary varieties.