Pedagogical notes on line bundles on complex projective manifolds - MathOverflow

Pedagogical notes on line bundles on complex projective manifolds

2012-07-23T15:10:04Z

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?

Answer by Francesco Polizzi for Pedagogical notes on line bundles on complex projective manifolds

2012-07-23T15:23:38Z

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.

Answer by Margaret Friedland for Pedagogical notes on line bundles on complex projective manifolds

2012-07-23T17:30:29Z

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

Answer by David Corwin for Pedagogical notes on line bundles on complex projective manifolds

2012-07-24T01:29:07Z

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).