Springer GTM 71.
"In this book we present the theory of Riemann surfaces and its many different facets. We begin from the most elementary aspects and try to bring the reader up to the frontier of present-day research."
Chapter 0: An OverviewEdit
Chapter I: Riemann SurfacesEdit
Section I.1: Definitions and ExamplesEdit
- I.1.6: Thus, we also have (since a non-vanishing holomorphic function on a disc has a logarithm) that:
From above we have
with n positive and $ a_n $ nonzero. Pulling out a factor of $ \tilde{z}^n $, we can rewrite this as
Now g converges wherever f does -- at $ \tilde{z}=0 $ it converges to $ a_n $ and elsewhere we can just divide $ f(\tilde{z}) $ by $ \tilde{z}^n $. Since $ g(\tilde{z}) $ is nonzero at $ \tilde{z}=0 $ and holomorphic, it's nonzero on some disc of positive radius, and consequently has a logarithm on this disc. Write $ h(\tilde{z}) := \exp \left\{\frac{1}{n} \log h(\tilde{z})\right\} $ on this disc; then $ g(\tilde{z}) = h(\tilde{z})^n $, so $ \zeta = \tilde{z}^n h(\tilde{z})^n $ as claimed.
- I.1.6, Proposition: The "normal form" of the mapping f given by (1.6.1) shows that this is open in N.
Section I.2: Topology of Riemann SurfacesEdit
Section I.3: Differential FormsEdit
Section I.4: Integration FormulasEdit
Chapter II: Existence TheoremsEdit
Section II.1 Hilbert Space Theory - A Quick ReviewEdit
Section II.2 Weyl's LemmaEdit
Section II.3 The Hilbert Space of Square Integrable FormsEdit
Section II.4 Harmonic DifferentialsEdit
Section II.5 Meromorphic Functions and DifferentialsEdit
Chapter III: Compact Riemann SurfacesEdit
Section III.1 Intersection Theory on Compact SurfacesEdit
Section III.2 Harmonic and Analytic Differentials on Compact SurfacesEdit
Section III.3 Bilinear RelationsEdit
Section III.4 Divisors and the Riemann-Roch TheoremEdit
- III.4.8 Theorem (Riemann-Roch)
There's way too much stuff in this proof. It should be broken up into multiple pieces, and the notation should be pulled out.
Notation:
- Let M be a compact Riemann surface with canonical homology basis $ \{a_1, \ldots, a_g, b_1, \ldots, b_g $, as in this detail from figure I.4:
- If $ A = \sum_{j=1}^m n_j P_j \in \operatorname{Div} M $ for some Riemann surface M, write $ A' := \sum_{j=1}^m (n_j+1) P_j $.
- Write $ \Omega_0(A) $ for the meromorphic 1-forms with no 'a'-periods and no residues:
$ \Omega_0(A) := \{ \omega \; | \; \omega \text{ a mero 1-form}, \forall j \int_{a_j} \omega = 0, \forall P \in M \operatorname{res}_P \omega = 0, (\omega) \geq A \} $.