89 lines
4.0 KiB
Markdown
89 lines
4.0 KiB
Markdown
# Math 401, Fall 2025: Thesis notes, S4, Complex manifolds
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## Complex Manifolds
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> This section extends from our previous discussion of smooth manifolds in Math 401, R2.
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>
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> For this week [10/21/2025], our goal is to understand the Riemann-Roch theorem and its applications.
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>
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> References:
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>
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> - [Introduction to Complex Manifolds](https://bookstore.ams.org/gsm-244)
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### Holomorphic vector bundles
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#### Definition of real vector bundle
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Let $M$ be a topological space, A **real vector bundle** over $M$ is a topological space $E$ together with a surjective continuous map $\pi:E\to M$ such that:
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1. For each $p\in M$, the fiber $E_p=\pi^{-1}(p)$ over $p$ is endowed with the structure of a $k$-dimensional real vector space.
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2. For each $p\in M$, there exists an open neighborhood $U$ of $p$ and a homeomorphism $\Phi: \pi^{-1}(U)\to U\times \mathbb{R}^k$ called a **local trivialization** such that:
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- $\pi^{-1}(U)=\pi$(where $\pi_U:U\times \mathbb{R}^k\to \pi^{-1}(U)$ is the projection map)
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- For each $q\in U$, the map $\Phi_q: E_q\to \mathbb{R}^k$ is isomorphism from $E_q$ to $\{q\}\times \mathbb{R}^k\cong \mathbb{R}^k$.
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#### Definition of complex vector bundle
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Let $M$ be a topological space, A **complex vector bundle** over $M$ is a real vector bundle $E$ together with a complex structure on each fiber $E_p$ that is compatible with the complex vector space structure.
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1. For each $p\in M$, the fiber $E_p=\pi^{-1}(p)$ over $p$ is endowed with the structure of a $k$-dimensional complex vector space.
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2. For each $p\in M$, there exists an open neighborhood $U$ of $p$ and a homeomorphism $\Phi: \pi^{-1}(U)\to U\times \mathbb{C}^k$ called a **local trivialization** such that:
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- $\pi^{-1}(U)=\pi$(where $\pi_U:U\times \mathbb{C}^k\to \pi^{-1}(U)$ is the projection map)
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- For each $q\in U$, the map $\Phi_q: E_q\to \mathbb{C}^k$ is isomorphism from $E_q$ to $\{q\}\times \mathbb{C}^k\cong \mathbb{C}^k$.
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#### Definition of smooth complex vector bundle
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If above $M$ and $E$ are smooth manifolds, $\pi$ is a smooth map, and the local trivializations can be chosen to be diffeomorphisms (smooth bijections with smooth inverses), then the vector bundle is called a **smooth complex vector bundle**.
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#### Definition of holomorphic vector bundle
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If above $M$ and $E$ are complex manifolds, $\pi$ is a holomorphic map, and the local trivializations can be chosen to be biholomorphic maps (holomorphic bijections with holomorphic inverses), then the vector bundle is called a **holomorphic vector bundle**.
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### Holomorphic line bundles
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A **holomorphic line bundle** is a holomorphic vector bundle with rank 1.
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> Intuitively, a holomorphic line bundle is a complex vector bundle with a complex structure on each fiber.
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### Simplicial, Sheafs, Cohomology and homology
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What is homology and cohomology?
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> This section is based on extension for conversation with Professor Feres on [11/05/2025].
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#### Definition of meromorphic function
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Let $Y$ be an open subset of $X$. A function $f$ is called meromorphic function on $Y$, if there exists a non-empty open subset $Y'\subset Y$ such that
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1. $f:Y'\to \mathbb{C}$ is a holomorphic function.
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2. $A=Y\setminus Y'$ is a set of isolated points (called the set of poles)
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3. $\lim_{x\to p}|f(x)|=+\infty$ for all $p\in A$
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> Basically, a local holomorphic function on $Y$.
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#### De Rham Theorem
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This is analogous to the Stoke's Theorem on chains, $\int_c d\omega=\int_{\partial c} \omega$.
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$$
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H_k(X)\cong H^k(X)
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$$
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Where $H_k(X)$ is the $k$-th homology of $X$, and $H^k(X)$ is the $k$-th cohomology of $X$.
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#### Simplicial Cohomology
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Riemann surfaces admit triangulations. The triangle are 2 simplices. The edges are 1 simplices. the vertices are 0 simplices.
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Our goal is to build global description of Riemann surfaces using local description on each triangulation.
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#### Singular Cohomology
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### Riemann-Roch Theorem (Theorem 9.64)
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Suppose $M$ is a connected compact Riemann surface of genus $g$, and $L\to M$ is a holomorphic line bundle. Then
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$$
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\dim \mathcal{O}(M;L)=\deg L+1-g+\dim \mathcal{O}(M;K\otimes L^*)
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$$
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