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I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems to me that in a "usual" algebro-geometric approach line bundles come quite late, only after one defines what is a sheaf, ect. I wonder if this can be explained in "quicker" way?

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Sure, there are many places that discuss vector bundles without sheaves. "Characteristic Classes" by Milnor and Stasheff is one excellent book. There's also "Principles of Algebraic Geometry" by Griffiths and Harris. –  Artie Prendergast-Smith Jul 23 '12 at 15:21
    
Artie, I find Griffiths-Harris too difficult. Milnor and Stasheff, of course is great, but this is only topology :) –  aglearner Jul 23 '12 at 15:24
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This should be community wiki, since it's basically a textbook-recommendation question –  David White Jul 23 '12 at 16:06
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aglearner: it's true that Milnor and Stasheff is "only topology", but I still think it can be very useful in understanding vector bundles in algebraic geometry. By the way, I think Shafarevich's book Basic Algebraic Geometry has a pretty down-to-earth discussion of line bundles and vector bundles (and how they relate to sheaves), but maybe you've already seen that too. –  Artie Prendergast-Smith Jul 23 '12 at 17:13

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If you want complex vector bundles (or even complex fiber bundles) presented at a "very basic level" (without sheaf theory), try Chapter IV of

Fritzsche, Klaus; Grauert, Hans: From holomorphic functions to complex manifolds. Graduate Texts in Mathematics, 213. Springer-Verlag, New York, 2002. xvi+392 pp. ISBN: 0-387-95395-7

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For a slightly different approach, I recommend Algebraic Curves and Riemann Surfaces by Rick Miranda. This book starts out by developing everything in terms of divisors, then turns to line bundles and sheaves. I recommend it precisely because it works at developing lots of motivation for the notion of a sheaf (as well as that of sheaf cohomology).

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I think there are several good references. For instance:

R. O. Wells: Differential Analysis on complex manifolds (Springer GTM 65), Chapter III.

J. D. Moore: Lectures in Seiberg-Witten invariants (Springer LNM 1629), Chapter 1.

D. Huybrechts: Complex Geometry - An introduction (Springer Universitext), Chapter 2.

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This should be community wiki, since it's basically a textbook-recommendation question –  David White Jul 23 '12 at 16:06
    
Actually the question itself should be community wiki, since its goal is to produce a sorted list of answers, rather than a single complete answer (see the FAQ) –  Francesco Polizzi Jul 23 '12 at 16:26

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