Pedagogical notes on line bundles on complex projective manifolds 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?
 A: 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          
A: 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).
A: 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.
