I am studying Riemann surfaces now. How about Donaldson's book "Riemann surfaces"? Could you recommend some references, and point out the required mathematical knowledge? Many thanks in advance!
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$\begingroup$The answer depends on the topics you are interested in.
Unfortunately I do not know Donaldson's textbook, but I have found this reference
It is nice to see the chapter on "Elliptic Functions and integrals".
I personally recommend Farkas & Kra's "Riemann Surfaces" and Forster's "Lectures on Riemann Surfaces". The first book may have a bit old fashioned notation when it explains divisors, but it is a great text, with a lot of explicit computations and a great section on theta functions.
Both textbooks introduce the minimum amount of complex analysis which is needed to understand the theory of Riemann surfaces.
I would even have a look at the first chapters of Griffiths & Harris "Principles of Algebraic Geometry": they deal with complex geometry.
$\endgroup$ 1 $\begingroup$I loved Donaldson. He rapidly connects various disparate undergrad concepts as he builds his theory, so I felt these connections must be how he thinks as well. His approach is step by step, as opposed to sudden incomprehensible insights. He makes sure to explain every important step at the level of undergrad maths though he often uses quick chatty English not symbols. For example, he makes sure to mention that Theorem 1 in Ch.1 and Lemma 2 in Ch.4 are basically implicit and inverse function theorems, though this being complex analysis, you get an explicit characterization of these functions, which he lays out in Ch.1.
The buildup from Ch.4 to Ch.11 (which is the converse of Ch.4) is incredible. Ch.11 switches to algebra, so he reminds you of simple algebra concepts like Gauss Lemma and Euclidean division, again in English. This is exactly what he did for analysis in Ch.1 and topology in Ch.2.
I thought that this book was a phenomenal unifier of three undergrad textbooks: Stein-Shakarchi analysis series, Artin algebra, and Munkres topology. But most amazing was witnessing the clarity of a great mind and discovering how it thinks at the undergrad level.
Warning: the online version is buggy. You should get the published version.
I found the Open University video lecture series on topology to be useful. They have the same tone as the book and gave me much-needed visual insight into various proofs in Ch. 4 and 6 of Donaldson’s book.
The videos are also a transfixing reminder of England in the 80’s.
Teleman’s Riemann Surfaces notes online at Berkeley has great pictures in Chapter 1 that complement Donaldson Ch.2. You can extend Teleman’s pictures to polynomials such as $𝑤^2=$ third degree polynomial in $z$, in which case $+k = \infty$ and so gluing gives a donut with one missing hole at the gluing point of $+k$.
Teleman’s construction also shows why expressing $z$ as cube root of a quadratic polynomial in $w$ for $w^2 - 1 = z^3$ and gluing 3 sheets won’t work. In the chart at infinity, $z^3 = w^2$ (Lemma 3 in Ch. 4 of Donaldson), and so $z = t^2, w = t^3, w/z = t$. We should therefore have one $t$ sheet at infinity, not 3.
Another good visual complement to Ch. 2 of Donaldson is Intuitive Topology by Prasalov.
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