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To get a sheaf of sets, we must impose what are called the “gluing” conditions:
1. For any pair of intersecting open sets U, V , and sections σ ∈ S(U), τ ∈ S(V ) such
that σ|U∩V = τU∩V
there exists a section ρ ∈ S(U ∪ V ) such that ρ|U = σ and ρ|V = τ . In other words, sections glue together in the obvious way.

2. If σ ∈ S(U ∪ V ) and σ|U = σ|V = 0 then σ = 0.
One of the most basic examples of a sheaf is the sheaf of sections of a vector bundle.
Given a vector bundle on a space, we can associate to any open set U the group of sections of
the bundle over that open set. In fact, technically there are several ways to get a sheaf from
a vector bundle ? we could associate smooth sections, or we could associate holomorphic
sections if the vector bundle has a complex structure, or instead of creating a sheaf of sets,
we could create a sheaf of modules, which is the more usual construction. For the purposes
of these physics lectures, these distinctions will largely be irrelevant. Technically, we will
almost always be interested in sheaves of modules of holomorphic sections, but will speak
loosely of other cases.
Sheaves have a property known as being “locally free” if they come from holomorphic
vector bundles, in the fashion above. For most of these notes, we shall ignore the distinction between locally-free sheaves and holomorphic vector bundles, and will use the terms interchangeably.
We should also mention some notation that will be used throughout these notes.
A holomorphic line bundle with first Chern class c1 on a given space will typically be denoted O(c1).
This notation makes most sense on projective spaces, where the first Chern class is
simply an integer, so that O(n) denotes a holomorphic line bundle of first Chern class n.

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