Lectures on Vector Bundles: 54 (Cambridge Studies in Advanced Mathematics, Series Number 54)
Cambridge University Press, 3/6/1997
EAN 9780521481823, ISBN10: 0521481821
Hardcover, 260 pages, 24.1 x 15.9 x 1.9 cm
Language: English
Originally published in French, translated by Antony Macioca
This work consists of two courses on the moduli spaces of vector bundles. The first part tackles the classification of vector bundles on algebraic curves. The construction and elementary properties of the moduli spaces of stable bundles are also discussed. In particular, Hilbert-Grothendieck schemes of vector bundles are constructed, and Mumford's geometric invariant theory is succinctly treated. The second part centres on the structure of the moduli space of semi-stable sheaves on the projective plane. Existence conditions for sheaves of given rank and Chern Class and construction ideas are sketched in the general context of projective algebraic surfaces. Professor Le Potier has provided a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.
Part I. Vector Bundles On Algebraic Curves
1. Generalities
2. The Riemann-Roch formula
3. Topological
4. The Hilbert scheme
5. Semi-stability
6. Invariant geometry
7. The construction of M(r,d)
8. Study of M(r,d)
Part II. Moduli Spaces Of Semi-Stable Sheaves On The Projective Plane
9. Introduction
10. Operations on semi-stable sheaves
11. Restriction to curves
12. Bogomolov's theorem
13. Bounded families
14. The construction of the moduli space
15. Differential study of the Shatz stratification
16. The conditions for existence
17. The irreducibility
18. The Picard group
Bibliography.