stylistic:tone, stylistic:style_transfer
Technical challenges with complex manifolds

One of the main historical motivations for introducing sheaves was constructing a device which keeps track of holomorphic functions on complex manifolds. For example, on a compact complex manifold X {\displaystyle X} (like complex projective space or the vanishing locus in projective space of a homogeneous polynomial), the only holomorphic functions

f: X → C {\displaystyle f:X\to \mathbb {C} }

are the constant functions. This means there exist two compact complex manifolds X, X ′ {\displaystyle X,X'} which are not isomorphic, but nevertheless their rings of global holomorphic functions, denoted H (X), H (X ′) {\displaystyle {\mathcal {H}}(X),{\mathcal {H}}(X')}, are isomorphic. Contrast this with smooth manifolds where every manifold M {\displaystyle M} can be embedded inside some R n {\displaystyle \mathbb {R} ^{n}}, hence its ring of smooth functions C ∞ (M) {\displaystyle C^{\infty }(M)} comes from restricting the smooth functions from C ∞ (R n) {\displaystyle C^{\infty }(\mathbb {R} ^{n})}. Another complexity when considering the ring of holomorphic functions on a complex manifold X {\displaystyle X} is given a small enough open set U ⊆ X {\displaystyle U\subseteq X}, the holomorphic functions will be isomorphic to H (U) ≅ H (C n) {\displaystyle {\mathcal {H}}(U)\cong {\mathcal {H}}(\mathbb {C} ^{n})}. Sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of X {\displaystyle X} on arbitrary open subsets U ⊆ X {\displaystyle U\subseteq X}. This means as U {\displaystyle U} becomes more complex topologically, the ring H (U) {\displaystyle {\mathcal {H}}(U)} can be expressed from gluing the H (U i) {\displaystyle {\mathcal {H}}(U_{i})}. Note that sometimes this sheaf is denoted O (−) {\displaystyle {\mathcal {O}}(-)} or just O {\displaystyle {\mathcal {O}}}, or even O X {\displaystyle {\mathcal {O}}_{X}} when we want to emphasize the space the structure sheaf is associated to.