Complex Analysis and Riemann Surfaces
Read this topicExplanations 6
- Uniformization and quotients MSE for complex plane Donaldson p. 84 Theorem 3: Holomorphic 1-form α\alphaα on XXX compact Riemann surface with no zeros allows us to construct iso…
By Victor Wang
- Curve theory references Standard references: Hartshorne, Qing Liu, Vakil. Stichtenoth is a good reference for the function field perspective, including a simple argument for asy…
By Victor Wang, Victor Wang, Alex Federspiel
- Theta functions: See Donaldson p. 88, Section 6.3 nice Fourier perspective (everything is analytic, should be OK to Fourier expand) after starting from holomorphic functions wit…
By Victor Wang
- Collect uses of properness in Donaldson and elsewhere, e.g. proof that degree is constant, p. 44-45, Proposition 7. Compare with algebraic geometry uses and analogs.
By Victor Wang
- Understand this comment of KConrad: For a finite extension of number fields, the story is similar: ramification indices of all prime ideals over the same prime must be the same.…
By Victor Wang
- Sheaves of differentials Nice to have different perspectives. See even Neukirch for "arithmetic Riemann-Roch theory" (including a notion of genus of number field, and so on). Wa…
By Victor Wang
Problems 7
- Which of the following can be used to prove that a conformal map (holomorphic function f :C→Cf\colon\mathbb{C}\to\mathbb{C}f:C→C with nonzero derivative) is locally injective?
- What properties of ggg are used in the proof of Hartogs's extension theorem (ggg is a holomorphic extension of fff)? Select all that apply.
- Let f(z)=z−1f(z) = z^{-1}f(z)=z−1 for z≠0z\neq0z=0. Describe the function g(z)=12πi∮∣ζ∣=1dζf(ζ)ζ−zg(z) = \frac{1}{2\pi i}\oint_{\lvert \zeta\rvert = 1} d\zeta\frac{f(\zeta)}{\z…
- When and how are fff and ggg related? Give the most powerful (general) correct answer.
- Which Möbius transformation sends distinct points a,b,ca,b,ca,b,c in P1\mathbb{P}^1P1 to 0,1,∞0,1,\infty0,1,∞?
- Describe the stabilizer of i∈Hi\in \mathbb{H}i∈H within Aut(H)=PSL2(R)\operatorname{Aut}(\mathbb{H}) = \operatorname{PSL}_2(\mathbb{R})Aut(H)=PSL2(R). Denote an element of PS…
- Which conformal mapping identifies the upper half plane H\mathbb{H}H with the unit disk D\mathbb{D}D?