C O M P L E X
A N A L Y S I S
I I ,
S P R I N G 2 0 2 4

#### Course Information

- NTU COOL
- Lectures: Tuesday 13:20 ~ 15:10 and Thursday 13:20 ~ 15:10 at Astro-Math 101.
- Grading scheme:
- Homework 40%

Students are asked to do the homework on the blackboard at 14:20 ~ 15:10 on Thursdays.
- Final Exam 60%

- Main Topics:

This is a continuation of Complex Analysis (I). The main topic is the theory of Riemann surfaces, including Riemann--Roch theorm, uniformization, Abel-Jacobi theorem, etc.
- References:
- Simon Donaldson, Riemann surfaces.
- Farkas and Kra, Riemann surfaces. Second edition.

#### Lecture summaries and references

- (week 1)
- holomorphic function
- implicit function theorem via Cauchy integral
- analytic continuation and complex ODE
- classification of surface: Klein bottle ≌
**RP**^{2} # **RP**^{2}, Klein bottle # **RP**^{2} ≌ **T**^{2} # **RP**^{2}

reference: Donaldson ch.1 and ch.2.
- (week 2)
- definition of Riemann surface
- basic examples
- projective space and algebraic curve
- quotients

reference: Donaldson ch.3.
- (week 3)
- holomorphic map between Riemann surfaces
- covering spaces and fundamental groups

reference: Donaldson ch.4.
- (week 4)
- monodromy representation of holomorphic map
- Riemann's existence theorem
- basic theory of differential forms on oriented surfaces

reference: Donaldson ch.4 and ch.5.
- (week 5)
- Poincare duality
- decomposition of d on Riemann surfaces
- the Laplace operator and Dirichlet energy
- Euler characteristic and Poincare-Hopf theorem

reference: Donaldson ch.5 and ch.6.
- (week 6)
- zero and poles of meromorphic 1-forms
- Riemann-Hurwitz formula
- degree-denus formula
- cohomology defined by \bar{\partial}

reference: Donaldson ch.6 and ch.8.
- (week 7)
- genus 0 Riemann surfaces
- genus 1 Riemann surfaces
- Riemann-Roch theorem (restricted version)

reference: Donaldson ch.8.
- (week 8)
- completion with respect to the Dirichlet energy and the bounded linear functional
- Poincare type inequality in 2D
- Weyl's lemma

reference: Donaldson ch.9.
- (week 9)
- solving the Laplace on non-compact surfaces
- behavior at infinity
- uniformization theorem

reference: Donaldson ch.10.
- (week 10)
- uniformization theorem (continued)
- Cech cohomology
- sheaf and sheaf cohomology
- relation to the existence of meromorphic functions

reference: Donaldson ch.10 and ch.12.
- (week 11)
- fine sheaf
- line bundles and its sheaf cohomology
- divisors and the corresponding line bundles

reference: Donaldson ch.12.
- (week 12)
- Riemann-Roch for divisors
- Serre duality
- different interpretations of Riemann-Roch

reference: Donaldson ch.12.
- (week 13)
- Jacobian variety
- Picard group
- equivalence between line bundles and divisors

reference: Donaldson ch.12.
- (week 14)
- Weierstrass points
- first Chern class in terms of Chern connection

#### Homework

See NTU COOL.

Last modified: June 3, 2024.

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