Vector bundles on algebraic curves.
(Fibrés vectoriels sur les courbes algébriques.)

*(French)*Zbl 0842.14025
Publications Mathématiques de l’Université Paris VII. 35. Paris: Univ. Paris 7, U.F.R. de Math.-Publ. 164 p. (1995).

This book is the outcome of a course given by the author at the University Paris VII in 1991 and is intended to be an introduction to the theory of vector bundles on smooth projective curves over the field of complex numbers. Some elementary knowledge from algebraic geometry is assumed. This material is presented in the first two chapters (without proofs). In chapter four the author studies the properties connected to flatness and direct images of coherent sheaves, and then he gives the construction of the Hilbert-Grothendieck scheme of all coherent sheaves which are quotient of a fixed vector bundle and having a fixed Hilbert polynomial. In chapter three the author undertakes the classification of vector bundles from the topological point of view. Then he is led to study the family of all vector bundles on a curve of genus \(g\), of rank \(r\) and degree \(d\). Since this family is in general unbounded, he introduces and studies the notions of stability and semistability for vector bundles. Using Mumford’s geometric invariant theory, one constructs a quotient \(M(r,d)\) of a certain Hilbert-Grothendieck scheme via the action of a special linear group. The points of \(M(r, d)\) are in one-to-one correspondence with the set of certain equivalence classes of semistable vector bundles (the \(S\)-equivalence), and \(M(r, d)\) is a projective algebraic variety. For stable vector bundles these equivalence classes reduce to isomorphism classes. The author is able to simplify significantly the presentation of this theory due to Mumford and Seshadri in several technical points. The last chapter deals with existence theorems, irreducibility and smoothness of the open subset of \(M(r, d)\) corresponding to stable bundles. All in all this book is an excellent introduction to the classification of vector bundles on curves. It can be used as a valuable textbook for students, Ph.D. students in algebraic geometry and all mathematicians interested in questions related to vector bundles.

Reviewer: L.Bădescu (Bucureşti)