A complex analysis professor once told me that "sheaves are all over the place" in complex analysis. Of course one can define the sheaf of holomorphic functions: if $U\subset \mathbf{C}$ (or $\mathbf{C}^n$) is a nonempty open set, let $\mathcal{O}(U)$ denote the $\mathbf{C}$-vector space of holomorphic functions $f:U\to\mathbf{C}$, and we let $\mathcal{O}(\varnothing)=\{0\}$. The restriction maps are given by restriction holomorphic functions to open subsets. This defines a sheaf on $\mathbf{C}$ with respect to its usual topology.
Here are my questions:
- Are there interesting re-interpretations of well-known results in basic complex analysis in the language of sheaf theory (just to get one thinking about how things might translate)?
- Are there interesting new geometric insights that one gains by introducing this structure? (Feel free to reformulate the context of the question if 2 doesn't make sense).
I guess I find it counter-intuitive that sheaves should say anything interesting about complex analysis, while it seems natural that they should say things about the geometry of the space on which they're defined.
Best Answer
As a complement to Matt's very interesting answer, let me add a few words on the historical context of Leray's discoveries.
Leray was an officer in the French army and after Frances's defeat in 1940, he was sent to Oflag XVII in Edelsbach, Austria (Oflag=Offizierslager=POW camp): look here .
The prisoners founded a university in captivity, of which Leray was the recteur (dean).
Leray was a brilliant specialist in fluid dynamics (he joked that he was un mécanicien, a mechanic!), but he feared that if the Germans learned that he gave a course on that subject, they would force him to work for them and help them in their war machine (planes, submarines,...).
So he decided to teach a harmless subject: algebraic topology!
So doing he recreated the basics on a subject in which he was a neophyte and invented sheaves, sheaf cohomology and spectral sequences.
After the war his work was examined, clarified and amplified by Henri Cartan (who introduced the definition of sheaves in terms of étalé spaces) and his student Koszul.
Serre (another Cartan student) and Cartan then dazzled the world with the overwhelming power of these new tools applied to algebraic topology, complex analysis in several variables and algebraic geometry.
I find it quite interesting and moving that the patriotism of one courageous man (officers had the option to be freed if they agreed to work for the Nazis) changed the course of 20th century mathematics.
Here, finally, is Haynes Miller's fascinating article on Leray's contributions.