Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 17 / 2 / a

Table of contents
- Solution a

Solution

A rank-

r

holomorphic vector bundle is a complex manifold

E

with a holomorphic projection to

X

and local holomorphic trivializations

E ∣_{U_{i}} ≅ U_{i} \times C^{r}

that are complex-linear on each fibre. Its transition maps have the form

(x, v) \mapsto (x, g_{ij} (x) v)

with holomorphic

g_{ij} : U_{i} \cap U_{j} \to G L_{r} (C)

satisfying the cocycle identities.

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as multiplication, the trivial line bundle as identity and the dual bundle as inverse. The classification on the complex projective line gives

Pic (P^{1}) ≅ Z, n ⟼ [O_{P^{1}} (n)] .

The inverse is degree;

O (1)

is the positive generator.

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