Brill-Noether Theory For Rank Two Vector Bundles

In the research of compact Riemann surfaces, it is necessary to discuss the problems on the imbeddings from a compact Riemann surface to complex projective spaces and even Grassmann manifolds. The imbeddings are given by the sections of the holomorphic vector bundles on the compact Riemann surface, and therefore, in order to discuss the problems on the existence and classification of these imbeddings, we need to determine the dimension of the section space of a vector bundle and to discuss what vector bundles there are in the case of giving the dimension of a section space. The study of these problems is called Brill-Noether theory for vector bundles. For line bundles, this theory is very complete. Our purpose is the discussion about the Brill-Noether theory for rank two vector bundles on the basis of the classical theory for line bundles. Since the discussion for the decomposable vector bundles can be transferred to the line bundles involved, we primarily deal with the indecomposable ones, and further reduce to discuss the indecomposable special rank two vector bundles generated by their sections. In this dissertation, we give and prove the vanishing theorem, the Clifford theorems and the existence theorem for rank two vector bundles generated by their sections, and partially give the classification of these vector bundles. The main difficulties in the discussion are finding the least degree in the vanishing theorem, determining the least upper bound in the Clifford theorems, the condition of the existence theorem and the criterion of classifying the indecomposable special rank two vector bundles generated by their sections. To do this, we use the cohomology theory of sheaves, Riemann-Roch theorem and P. Griffiths and J. Harris' famous result as the primary basis to study and discuss the problems, and obtain good results. Our methods and conclusions may lay some foundation for the study of the Brill-Noether theory for vector bundles of higher dimensions.