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This book is intended to serve as an introduction to the theory of semistable sheaves and at the same time to provide a survey of recent research results on the geometry of moduli spaces.
The first part introduces the basic concepts in the theory: Hilbert polynomial, slope, stability, Harder-Narasimhan filtration, Grothendieck's Quot-scheme. It presents detailed proofs of the Grauert-Mülich Theorem, the Bogomolov Inequality, the semistability of tensor products, and the boundedness of the family of semistable sheaves. It also gives a self-contained account of the construction of moduli spaces of semistable sheaves on a projective variety à la Gieseker, Maruyama, and Simpson.
The second part presents some of the recent results of the geometry of moduli spaces of sheaves on an algebraic surface, following work of Mukai, O'Grady, Gieseker, Li and many others. In particular, moduli spaces of sheaves on K3 surfaces and determinant line bundles on the moduli spaces are treated in some detail. Other topics include the Serre correspondence, restriction of stable bundles to curves, symplectic structures, irreducibility and Kodaira-dimension of moduli spaces.
List of contents
Introduction - Preliminaries - Families of Sheaves - The Grauert-Mülich Theorem - Moduli Spaces - Construction Methods - Moduli Spaces on K3 Surfaces - Restriction of Sheaves to Curves - Line Bundles on the Moduli Space - Irreducibility and Smoothness - Symplectic Structures - Birational Properties
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"Ich kenne keine andere Zusammenstellung der Theorie in Buchform, welche an Präzision und Umfang mit dieser vergleichbar ist. ... ein sehr nützliches, aktuelles und gut geschriebenes Buch" (W. Barth in DMV-Jahresberichte 2/99)
