MY-ASSIGNMENTEXPERT™可以为您提供sites.google MA307 Riemann surface黎曼曲面课程的代写代考和辅导服务！
这是印度科學理工學院黎曼曲面课程的代写成功案例。
MA307课程简介
Course description: Riemann surfaces are one-dimensional complex manifolds, obtained by gluing together pieces of the complex plane by holomorphic maps. This course will be an introduction to the the theory of Riemann surfaces, with an emphasis on analytical and topological aspects. After describing examples and constructions of Riemann surfaces, the topics covered would include branched coverings and the Riemann-Hurwitz formula, holomorphic 1-forms and periods, the Weyl’s Lemma and existence theorems, the Hodge decomposition theorem, Riemann’s bilinear relations, Divisors, the Riemann-Roch theorem, theorems of Abel and Jacobi, the Uniformization theorem, Fuchsian groups and hyperbolic surfaces.
Prerequisites
There will be no class on January 5th (Tuesday).
There will be no class on December 17th (Thursday).
There will be no class on November 19th (Thursday). Also, no lectures would be held in the midterm week of Nov 23 -27.
A new Teams code has been emailed to the mailing list.
There will be weekly homework also.
There will be no lecture on Tuesday, October 6th.
Lectures will be held on Tuesdays and Thursdays, 3:30-5pm.
Our first meeting will be held on Thursday October 1st , 3:30-5pm.
Classes will be online, at the beginning over Microsoft Teams. The first meeting link will be put up here, and on the IISc Intranet Bulletin board, before the meeting time.
MA307 Riemann surface HELP（EXAM HELP， ONLINE TUTOR）
Show that the affine algebraic curve $S=\left{(z, w) \in \mathbb{C}^2 \mid w^2-z=0\right}$ is conformally equivalent to the complex plane $\mathbb{C}$.
Show that $X=\left{[x: y: z] \in \mathbb{C P}^2 \mid x^3 y+y^3 z+z^3 x=0\right}$ is a smooth projective curve.
Let $\Psi:(0, \infty) \times(0, \infty) \rightarrow \mathbb{R}^3$ be an immersed surface given by $\Psi(x, y)=$ $(x \cos y, x \sin y, y)$.
(a) Show that the induced Riemannian metric has the expression $d s^2=d x^2+\left(1+x^2\right) d y^2$.
(b) Find a change of coordinates $(x, y) \mapsto(u, v)$ such that $(u, v)$ are isothermal coordinates for the surface.
Let $\Gamma<\operatorname{Aut}(\mathbb{C})$ be the infinite cyclic subgroup generated by the translation $z \mapsto z+1$. Show that $\mathbb{C} / \Gamma$ is biholomorphic to $\mathbb{C}^*$.
Consider the action of the finite group $\mathbb{Z}_2$ on $\mathbb{C P}^1$ as follows: the trivial element of $\mathbb{Z}_2$ acts by the identity map, and the non-trivial element acts by the map $[x: y] \mapsto[y: x]$.
(a) Show that the quotient space $X=\mathbb{C P}^1 /[x: y] \sim[y: x]$ is a Riemann surface.
(b) Show that $X$ is biholomorphic to $\mathbb{C P}^1$.
Let $X$ be a compact Riemann surface, and $S \subset X$ a finite set of points. Assume that $f: X \backslash S \rightarrow \mathbb{C}$ is a non-constant holomorphic function. Show that the image of $f$ is dense in $\mathbb{C}$.
Show that the smooth projective curve $X=\left{[x: y: z] \in \mathbb{C P}^2 \mid x^n+\right.$ $\left.y^n+z^n=0\right}$ is a surface of genus $g=(n-1)(n-2) / 2$.
(This is an instance of the “degree-genus formula”: any smooth projective curve defined by a non-singular homogeneous polynomial $F(x, y, z)$ of degree $n$ has genus $g$ given by the formula above!)
Let $X$ be the elliptic curve defined by the zero set ${[x: y: z] \in$ $\left.\mathbb{C P}^2 \mid y^2 z-x z^2+x^3=0\right}$
(a) Show that any holomorphic $\operatorname{map} F: \mathbb{C P}^1 \rightarrow X$ is a constant map.
(b) Show that there is a non-constant holomorphic map $G: X \rightarrow \mathbb{C P}^1$ that is branched over the four points $-1,0, i$ and $-i$. (Correction: the points should be $-1,1, i$ and $-i$ !)
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