§ 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 analyticcontinuations of each other iff Ω1Ω2\Omega_1 \cap \Omega_2 \neq \emptyset, andf1=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) toregion 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 thatg2=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 thatthe 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 globalanalytic 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 buildanalytic 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 shouldpicture 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' andf=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: Callthis 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 thenthis will be known as the sheaf of germs of analytic functions over DD. Thissheaf 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 isdenoted by Sh(z)Sh(z).
  • ShSh carries both topological and algebraic structure. We can give the sheafa topology to talk about about continuous mappings in and out of ShSh.It also carries a pointwise algebraic structure at each stalk: we canadd 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 abeliangroup.
  • 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)