§ Elementary uses of Sheaves in complex analysis

I always wanted to see sheaves in the wild in a setting that was both elementary but 'correct': In that, it's not some perverse example created to show sheaves (DaTaBaSeS arE ShEAvEs). Ahlfors has a great example of this which I'm condensing here for future reference.

§ Sheafs: Trial 1

  • We have function elements (f:ΩC,ΩC)(f: \Omega \rightarrow \mathbb C, \Omega \subseteq \mathbb C). ff is complex analytic, Ω\Omega is an open subset of C\mathbb C.
  • Two function elements (f1,Ω1),(f2,Ω2)(f_1, \Omega_1), (f_2, \Omega_2) are said to be analytic continuations of each other iff Ω1Ω2\Omega_1 \cap \Omega_2 \neq \emptyset, and f1=f2f_1 = f_2 on the set Ω1Ω2)\Omega_1 \cap \Omega_2).
  • (f2,Ω2)(f_2, \Omega_2) can be called as the continuation of (f1,Ω1)(f_1, \Omega_1) to region Ω2\Omega_2.
  • We will have that the analytic continuation of f1f_1 to Ω2\Omega_2 is unique. If there exists a function element (g2,Ω2)(g_2, \Omega_2), (h2,Ω2)(h_2, \Omega_2) such that g2=f1=h2g_2 = f_1 = h_2 in the region Ω1Ω2\Omega_1 \cap \Omega_2, then by analyticity, this agreement will extend to all of Ω2\Omega_2.
  • Analytic continuation is therefore an equivalence relation (prove this!)
  • A chain of analytic continuations is a sequence of (fi,Ωi)(f_i, \Omega_i) such that the adjacent elements of this sequence are analytic continuations of each other. (fi,Ωi)(f_i, \Omega_i) analytically continues (fi+1,Ωi+1)(f_{i+1}, \Omega_{i+1}).
  • Every equivalence class of this equivalence relation is called as a global analytic function. Put differently, it's a family of function elements (f,U)(f, U) and (g,V)(g, V) such that we can start from (f,U)(f, U) and build analytic continuations to get to (g,V)(g, V).

§ Sheafs: Trial 2

  • We can take a different view, with (f,zC)(f, z \in \mathbb C) such that ffis analytic at some open set Ω\Omega which contains zz. So we should picture an ff sitting analytically on some open set Ω\Omega which contains zz.
  • Two pairs (f,z)(f, z), (f,z)(f', z') are considered equivalent if z=zz = z' and f=ff = f' is some neighbourhood of z(=z)z (= z').
  • This is clearly an equivalence relation. The equivalence classes are called as germs .
  • Each germ (f,z)(f, z) has a unique projection zz. We denote a germ of ff with projection zzas fzf_z.
  • A function element (f,Ω)(f, \Omega) gives rise to germs (f,z)(f, z) for each zΩz \in \Omega.
  • Conversely, every germ (f,z)(f, z) is determined by some function element (f,Ω)(f, \Omega)since we needed ff to be analytic around some open neighbourhood of zz: Call this neighbourhood Ω\Omega.
  • Let DCD \subseteq \mathbb C be an open set. The set of all germs {fz:zD}\{ f_z : z \in D \}is called as a sheaf over DD. If we are considering analytic ff then this will be known as the sheaf of germs of analytic functions over DD. This sheaf will be denoted as Sh(D)Sh(D).
  • There is a projection π:Sh(D)D;(f,z)z\pi: Sh(D) \rightarrow D; (f, z) \mapsto z. For a fixed z0Dz0 \in D, the inverse-image π1(z0)\pi^{-1}(z0) is called as the stalk over z0z0. It is denoted by Sh(z)Sh(z).
  • ShSh carries both topological and algebraic structure. We can give the sheaf a topology to talk about about continuous mappings in and out of ShSh. It also carries a pointwise algebraic structure at each stalk: we can add and subtract functions at each stalk; This makes it an abelain group.

§ Sheaf: Trial 3

A sheaf over DD is a topological space ShSh and a mapping π:ShD\pi: Sh \rightarrow D with the properties:
  • π\pi is a local homeomorphism. Each sSs \in S has an open neighbourhood DDsuch that π(D)\pi(D) is open, and the restriction of π\pi to DD is a homeomorphism.
  • For each point zDz \in D, the stalk π1(z)Sz\pi^{-1}(z) \equiv S_z has the structre of an abelian group.
  • The group operations are continuous with respect to the topology of ShSh.
We will pick DD to be an open set in the complex plane; Really, DD can be arbitrary.

§ Germs of analytic functions satisfy (Sheaf: Trial 3)