Vector bundles: from the instanton family to a new regularity

Abstract

This research project focuses on the study of vector bundles on smooth projective varieties. The first topic consists in the study of the family of orthogonal instanton bundles on projective spaces. We will generalize the already known description, called the net of quadrics in the symplectic case, for the orthogonal one. Our goal is to find new examples, non-existence results and a description of the moduli space of the latter family.The second topic involves the splitting type of instanton bundles, a long time open question, and the construction of new families of instanton.The third topic concerns the study of a possible stratification of Steiner bundles on the projective plane, using the degree of the curve related to the Steiner construction of the bundle, i.e. through the push-forward of an invertible sheaf on a covering of the plane.The fourth and last topic involves the definition of a new concept of regularity for sheaves on the the Grassmannian variety. In the related section of this project we will explain the new interpretation of the definition for projective spaces, which will lead us to the required generalization. (AU)