COMPLEX GEOMETRY Course notes

Lemma 4.1.1. If F is a presheaf over X then there exists a unique sheaf F sh along with a morphism
F −→ F sh that factors thorough all morphisms F −→ G to a sheaf G.
F
G
ϕ ∃!
F sh
If X is a topological space, its structure sheaf CX 0 associated to each open subset U is given by the space
of continuous functions on U. If X is an algebraic variety, its structure sheaf O X is given by the space of
regular functions on (Zariski) open subsets. A complex algebraic manifold is also a complex manifold, and
we write O alg
X
and Ohol X to distinguish the structures:
O alg
X
O hol
X
−→ Zariski topology,
−→ ordinary topology.
Definition 4.1.3. Let A be the sheaf of rings over X. A sheaf F is called an A-module if for every open
set U, F(U) is a module over A(U), compatible with the restriction maps.
Remark 4.1.1. All notions from module theory carry over: Hom, Ker, Im, CoKer, direct sums, tensor
products, exact sequences and homology groups, etc.
Definition 4.1.4. The A-module A ⊕n = A is said to be free of rank n ∈ N.
isomorphic to A ⊕n is called locally free of rank n.
A sheaf that is locally
Remark 4.1.2. There exists a bijection between vector bundles of rank n and locally free sheaves of rank
n.
If A is a sheaf of fields, the rank one sheaves (invertible sheaves with respect to A) form a group under tensor
product, called the Picard group Pic OX (X).
Definition 4.1.5. Let (f, f # ) : (X, A) −→ (Y, B) be a morphism of ringed spaces, i.e., f : X −→ Y is a
continuous function, and
f # : B(V ) −→ A(f −1 (V ))
is a morphism for every open subset V compatible with restrictions. The pullback sheaf f ∗ G of a B-module
G is defined as follows: set f (∗) G(U) = lim G(V ) and then set f ∗ G be the sheaf associated to the
−→f(U)↩→V
presheaf f (∗) G ⊗ f (∗) B A.
Example 4.1.2. The sheaf F of holomorphic sections of a holomorphic vector bundles F over X,
i X : {x} ↩→ X
(1) i (∗)
x F =: F x (the stalk of F at x and its elements are germs of sections of F at x).
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