By Dr. Daniel Huybrechts, Dr. Manfred Lehn (auth.)
Diese Monographie gibt eine Einführung in ein aktuelles Teilgebiet der Algebraischen Geometrie: Die Theorie der Modulräume für Vektorbündel und kohärente Garben hat sich in der letzten Zeit in vielen Richtungen entwickelt und findet Interesse in zahlreichen mathematischen Zentren, z. B. in Bonn oder an der Harvard collage, USA.
Introduction - Preliminaries - households of Sheaves - The Grauert-Mülich Theorem - Moduli areas - building equipment - Moduli areas on K3 Surfaces - restrict of Sheaves to Curves - Line Bundles at the Moduli house - Irreducibility and Smoothness - Symplectic constructions - Birational Properties
Diplomstudenten höherer Semester und Mathematiker an Universitäten, Institute, Bibliotheken
Über den Autor/Hrsg
Dr. Huybrechts forscht an der Humboldt Universität Berlin und Dr. Lehn an der Universität Bielefeld.
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Additional info for The Geometry of Moduli Spaces of Sheaves: A Publication of the Max-Planck-Institut für Mathematik, Bonn
Pi=: Pmin(E). Obviously, E is semistable if and only if E is pure andpmax(E) = Pmin(E). A priori, the definition of the maximal and minimal p of a sheaf E depends on the filtration. We will see in the next theorem, that the Harder-Narasimhan filtration is uniquely determined, so that there is no ambiguity in the notation. 3 -IfF and G are pure sheaves of dimension d with Pmin(F) then Hom(F, G)= 0. > Pmax(G), Proof Suppose 'lj; : F -+ G is non-trivial. Let i > 0 be minimal with 'lj;(HN;(F)) =f 0 and let j > 0 be minimal with 'lj;(HN;(F)) C HNj(G)).
Choosing a closed immersion i : X -+ lP'~ and replacing 1i by i* 1i we may reduce to the case X = lP'~. By Serre's theorem there exist presentations OpN( -m"t" ----+ OpN( -m't' ----+ 1i----+ 0. 3 any quotient of 1i can be considered as a quotient of OpN . T ( -m')n'. Conversely, a quotient F of OpN( -m')n' factors through ti, if and only if the composite T homomorphism OpN( -m")n" -+ OpN( -m')n' -+ F vanishes. ) for some k sufficiently large integer £.. f/k(O( -m')n', P). D Since Q := Quotx;s(ti, P) represents the functor Q ·- Quotx15 (1i, P), we have Mor(Sch/S) (Y, Q) = Q(Y) for any S-scheme Y.
2). 1 Flat Families and Determinants Let f : X -+ S be a morphism of finite type of Noetherian schemes. If g : T -+ Sis an Sscheme we will use the notation X T for the fibre product T x s X, and g x : X T -+ X and h: Xr-+ T for the natural projections. ForsE S the fibre f- 1 (s) = Spec(k(s)) Xs X is denoted X 8 • Similarly, ifF is a coherent Ox-module, we write Fr := g'XF and Fs = Fix•. Often, we will think ofF as a collection of sheaves Fs parametrized by s E S. 1 -A fiat family of coherent sheaves on the fibres off is a coherent 0 xmodule F which is fiat overS.