By Christian Okonek
These lecture notes are meant as an advent to the tools of type of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as attainable we have now in most cases limited ourselves to the case X = Fn. in line with Serre (GAGA) the category of holomorphic vector bundles is such as the type of algebraic vector bundles. right here we now have used nearly completely the language of analytic geometry. The booklet is meant for college kids who've a simple wisdom of analytic and (or) algebraic geometry. a few funda psychological effects from those fields are summarized at first. one of many authors gave a survey within the Seminaire Bourbaki 1978 at the present nation of the type of holomorphic vector bundles overFn. This lecture then served because the foundation for a process lectures in Gottingen within the wintry weather Semester 78/79. the current paintings is a longer and up-dated exposition of that path. end result of the introductory nature of this e-book we now have needed to pass over a few tough subject matters similar to the restrict theorem of Barth. As reimbursement we have now appended to every sec tion a paragraph during which old feedback are made, extra effects indicated and unsolved difficulties provided. The ebook is split into chapters. every one bankruptcy is subdivided into a number of sections which in flip are made from a few paragraphs. every one part is preceeded through a quick description of iv its contents.
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Those lecture notes are meant as an creation to the tools of category of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as attainable we've got typically constrained ourselves to the case X = Fn. in accordance with Serre (GAGA) the type of holomorphic vector bundles is reminiscent of the class of algebraic vector bundles.
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Additional resources for Vector Bundles on Complex Projective Spaces
C] .... o, [-CJ whose restriction to C gives the Euler sequence. (ii) is defined by the section s 0 ®2 E: H (V, &v ) 1 s (u v 1 x,v) (u v1x1v, (x,y)), which vanishes exactly on C to first order. If we now show that Ext 1 ([-c]I(C]l H1 (vI :zc] l Ext 1 ((-c]lc~[c]lc ), i . e . Cl, is injective, then it follows that (i) and (ii) are equal and in particular that E' = ~V® 2 , which is 36 what we want. Because of the exact sequence 0 [C] + H 1 (VI [ 2C]) + C2C] H 1 (C,[2C] ICl + [2CJIC 0 is injective i f H 1 (V,[C]l 0.
I f EIJPn_ 1 splits, then 0 for i 1, ... ,n-2. The cohomology sequence of 0 ~ E (k-1) E(k) ElJPn-1 (k) 0 ~ then gives _,. i-1 H (JP n-1 ,E (k) J:IP n-1) _,. _,. Hi(JP ,E(k-1) n _,. Hi(JP ,E(k)) n Hi (JP n-1, E (k) I JP n-1) 43 From this one deduces that the group is independent of k for i 2, •.. ,n-2 and fori n-1 for all k. With Theorem B we thus have 0 for 2 < i < n-1 and for all k E Z. 1 In order to see that also H ~n,E(*)) = 0 we apply the same considerations to E* . With Serre duality we have Hn-1 ~n,E * (-k-n-1)) o.
If P 1 c X then every vector bundle over X splits after restriction to P 1 . The splitting behaviour depends very much on the embedding P 1 c X as we shall see in examples of 2-bundles over P 2 • Finally we shall prove a cohomological splitting criterion for vector bundles on projective spaces, from which it in particular follows that a holomorphic vector bundle E over Pn' n ~ 2, splits as a direct sum of line bundles precisely when this is the case for the restriction to some projective plane.