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Farkas and kra

Springer GTM 71.

Complex Analysis

"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

$ \zeta = f(\tilde{z}) = \sum_{k=n}^\infty a_k \tilde{z}^k $

with n positive and $ a_n $ nonzero. Pulling out a factor of $ \tilde{z}^n $, we can rewrite this as

$ \zeta = \tilde{z}^n\sum_{k=0}^\infty a_k \tilde{z}^k =: \tilde{z}^n g(\tilde{z}) $

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.

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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:

Farkas and kra canonical homology basis

    • 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 \} $.

Section III.5 Applications of the Riemann-Roch TheoremEdit

Section III.6 Abel's Theorem and the Jacobi Inversion ProblemEdit

Section III.7 Hyperelliptic Riemann SurfacesEdit

Section III.8 Special Divisors on Compact SurfacesEdit

Section III.9 Multivalued FunctionsEdit

Chapter IV: UniformizationEdit

Chapter V: Automorphisms of Compact Surfaces: Elementary TheoryEdit

Chapter VI: Theta FunctionsEdit

Chapter VII: ExamplesEdit