I've recently been reading Complex Analysis in One Variable by Narasimhan and Nievergelt. I've done enough work in complex analysis prior that I find the majority of the text quite understandable and beautiful; however, I was pleasantly shocked when I began reading the section entitled "The sheaf of germs of holomorphic functions". After re-reading through the section a number of times, I have found I can formally follow the argument, but lack any intuition for the definitions or the mechanics.

Unfortunately, my passion for analysis has meant the majority of my studies have been directed away from Algebra, and so I've only taken an introductory level course to Group Theory and Ring Theory. Nevertheless, I strongly desire to become more familiar with sheaf theoretic techniques in complex analysis. I could of course simply wait until I have taken more courses in Algebra to become acquainted with it enough to grasp Category Theory at an intuitive level and then begin chipping away at a Sheaf Theory text. However, I am presently only in my first year of college, and I don't wish to be forced to wait a few years until Grad School to appreciate basic applications of Sheaf Theory.

As such, I am wondering if there is a more optimized route to learn the very basic Sheaf Theory I might need to appreciate it's applications to Analysis, and in particular single variable Complex Analysis. I imagine I will not need a complete understanding of modern Sheaf Theory to merely understand its applications to my field. As such, I am wondering what the minimally required topics are that I should study in order to build up some basic intitution for the uses of Sheaf Theory in N&N's text.

Edit: Per the suggestions below, I have checked out copies of Gunning's and Forster's texts on Riemann Surfaces. My initial reaction to both texts has been quite positive; in particular I have found Gunning's text fantastic in that it provides the definition of a sheaf as soon as possible. I look forward to reading through the texts as I find time!

However, I remain completely open to other perspectives on how to best attack the necessary Sheaf theory, and so other answers are welcome!

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    $\begingroup$ this is a wonderful and important question. I hope you get a good answer. $\endgroup$ – James S. Cook May 5 '18 at 20:33
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    $\begingroup$ Another book you may find useful is Otto Forster's Lectures On Riemann Surfaces. He introduces sheaves early in the book and uses them extensively in expressing later concepts, e.g., differential forms and line bundles. $\endgroup$ – André 3000 May 6 '18 at 4:47
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    $\begingroup$ @Quasicoherent Thank you for the suggestion! I checked out a copy of Forster's text, and my initial reaction is entirely positive. I thank you and barto below for recommending the text. $\endgroup$ – Brevan Ellefsen May 6 '18 at 5:44

I learned more about how to use (a certain version of) sheaf theory in the context you mention from R. Gunning's "Riemann Surfaces" (the old Princeton Yellow Series paperback) than from sources purporting to be "about sheaf theory".

It is true that the cohomology-of-sheaves presented there is Cech cohomology, not derived-functor cohomology, but in most regards (especially for compact, connected Riemann surfaces) that is an irrelevant technical detail. That is, Grothendieck's realization in the early 1950s that sheaf cohomology can be understood as the derived functors of the global sections functor explains/imbeds sheaf cohomology in the (eminently useful) context of derived functors... but that is surely not your first concern.

EDIT: also, as I reflect on other potential sources, I am reminded of the truism that people do have irrational biases, of various origins, sometimes simply needing to have a "group" to belong to. And, sometimes, this cohesion is achieved less by positive aspects than by vilifying some other group. Similarly, although this is not at all reflected in the question here, "algebra/analysis" or "pure/applied" and such labels are all-too-often used as battle-flags rather than accurate descriptors of anything. But even if one manages to ignore the noise in the signal, there is still a significant risk that an otherwise-helpful source throws in some jabs at an opponent. In the case at hand, I recall that the introduction of an otherwise-fairly-interesting several complex variables book made a faux-historical but wildly counterfactual comment (about 1980) "now that sheaves have faded into the background...". Given the relative authority of the author, this was a bizarre and irresponsible thing to write. So, by this point, I certainly mistrust texts' or peoples' portrayals/caricatures of subject not their own, and reserve some skepticism even of self-descriptions.

  • $\begingroup$ Given that now both Hikaru and yourself have recommended Gunning, I'll definitely check out a copy of his Riemann Surfaces text. As an additional benefit, doing so will finally give me an reason to undertake a more complete study of Riemann Surfaces. Thank you! $\endgroup$ – Brevan Ellefsen May 5 '18 at 21:52
  • $\begingroup$ As an intriguing aside, it seems Gunning and Narasimhan wrote a joint paper entitled "Immersion of Open Riemann Surfaces" $\endgroup$ – Brevan Ellefsen May 5 '18 at 22:02
  • $\begingroup$ I'm very happy with Gunnings book so far! If I might ask though, I'm confused by what prompted the edit to your post. I by no means dislike the more algebraically oriented fields - I just happen to have a passion for analysis. If I didn't like algebraic methods I can assure you I wouldnt be trying to use sheaf theory! I merely haven't studied as much algebra thus far into my schooling, since I'm less than a year into my undergraduate. As such, your edit seems to me to be, at times, more of a tangential rant than an answer to me question. $\endgroup$ – Brevan Ellefsen May 6 '18 at 3:49
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    $\begingroup$ @BrevanEllefsen, you're right, it is a bit rant-ish... What prompted it was reflection on some other potential sources, and on decades-long personal exposure to unscholarly attitudes... :) I'll tighten it up to make a genuine point. $\endgroup$ – paul garrett May 6 '18 at 14:30
  • $\begingroup$ @paulgarrett I liked the first edit more. Since it highlighted a problem. OP used the word "analysis" to refer to his interests, but citing a book of one complex variable. That is already a grey area between analysis and geometry. As pointed out in the first edit, many would say "that is not analysis". As OP can notice most answers advised geometry books. That may be a hint to better understand where your interests lies. On a second note almost none of those who studied CT will refer to category theory as algebra. Paul's hint imho was to let go of these flags as soon as possible. $\endgroup$ – Lolman May 7 '18 at 7:22

Coincidentally, I'm currently reading Frank Warner's Foundations of differentiable manifolds and lie groups.

In chapter 5 of this book he presents a sheaf-theoretic proof of the de Rahm theorem. The presentation does not presuppose any knowledge of sheaves or cohomology but it is not a general treatment. In the same chapter, the author recommends Gunning for a more general approach towards the theory of Riemann surfaces.

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    $\begingroup$ Ah, yes, you're recommending the immediate sequel to the "Riemann Surfaces" book I recommended! :) $\endgroup$ – paul garrett May 5 '18 at 21:24

You can take a look at Forster's Riemann Surfaces. He introduces sheaves in section 1.6 with a (to me) nice and elementary application in Theorem 10.5, but he starts using them really only in Chapter 2, where Cech cohomology comes in.

  • $\begingroup$ Thank you for the suggestion! I'll definitely take a look at Foster's! $\endgroup$ – Brevan Ellefsen May 6 '18 at 4:46

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