By Christian Okonek
These lecture notes are meant as an creation to the equipment of type of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as attainable we now have in most cases constrained ourselves to the case X = Fn. in response to Serre (GAGA) the category of holomorphic vector bundles is similar to the class of algebraic vector bundles. right here now we have used nearly completely the language of analytic geometry. The publication is meant for college students who've a easy wisdom of analytic and (or) algebraic geometry. a few funda psychological effects from those fields are summarized first and foremost. one of many authors gave a survey within the Seminaire Bourbaki 1978 at the present kingdom of the category of holomorphic vector bundles overFn. This lecture then served because the foundation for a process lectures in Gottingen within the iciness Semester 78/79. the current paintings is a longer and up-dated exposition of that path. as a result of the introductory nature of this ebook we now have needed to miss a few tough themes similar to the limit theorem of Barth. As repayment we have now appended to every sec tion a paragraph within which historic comments are made, extra effects indicated and unsolved difficulties provided. The booklet is split into chapters. every one bankruptcy is subdivided into a number of sections which in flip are made of a few paragraphs. every one part is preceeded via a quick description of iv its contents.
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Additional resources for Vector Bundles on Complex Projective Spaces
We can now proceed to a first application of the standard construction. 1. Theorem. Let E be a hoZomorphic vector bundle of rank r ®r over Pn, x E Pn a point, ElL = GL for every line L through x. Then E is triviaZ. Proof. , is trivial over every f-fibre L = f -1 (1) 1 (x 1 l). The bundle a * E 1 E G(x) 1 for p induces an isomorphism and ElL is trivial for all 1 E G(x) by assumption. Clai~ There is an r-bundle F over G(x) with a*E- f*F. 52 If we have proved this,,then a*E is trivial, for from a • s = const.
In the proof above we used the following fact about a-processes: let a : X + Y be the a-process for a 2- dimensional complex manifold Yin finitely many points. Then Cl H (Y,tSy) for q > o. This follows with the help of the Lerav spectral seauence directlv from the fact that Rqa*oX =0 for q > 0, a*(SX of the image sheaf Rqa*(SX for q > 0 = tSy· The vanishing results from the following lemma: if u c ~ 2 is an open Stein neighborhood of o and 39 a : V u + u the a-process for in q o, then H (V,~v) =o for q > One proves this as follows: Vis given as a submanifold of U x Let J o.
F* (')G (~~ :; iS:Js (~~. If however a*E is trivial, then so is E, as the equation ~ E shows (a*~(x) ~ ~ , because a is the a-process of Wn at x). It * n thus remains to show that a E is of the form To this end we consider to coherent sheaf F F is locally free of rank r by the base-change theorem, for f is flat and h 0 (L,E I L) is equal to r for all lines L through x. The canonical homomorphism of sheaves 53 * f *f *cr *E is given on each f-fibre f CJ E f- 1 (1) by the evaluation map L * f*cr* E I~L II ( Ho - * E I"' (L,cr L)~a: (SL and is thus an isomorphism, i.
Vector Bundles on Complex Projective Spaces by Christian Okonek