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Zheyuan Wu
2025-10-24 00:16:11 -05:00
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content/Math401/Extending_thesis/Math401_R2.md
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@@ -488,7 +488,6 @@ A Riemannian manifold is a smooth manifold equipped with a **Riemannian metric**
More formally, a **Riemannian manifold** is a pair $(M,g)$, where $M$ is a smooth manifold and $g$ is a specific choice of Riemannian metric on $M$. More formally, a **Riemannian manifold** is a pair $(M,g)$, where $M$ is a smooth manifold and $g$ is a specific choice of Riemannian metric on $M$.
An example of Riemannian manifold is the sphere $\mathbb{C}P^n$. An example of Riemannian manifold is the sphere $\mathbb{C}P^n$.
### Notion of Connection ### Notion of Connection

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content/Math401/Extending_thesis/Math401_S4.md
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@@ -268,3 +268,21 @@ $$
This operator is a vector field. This operator is a vector field.
## Complex Manifolds
> This section extends from our previous discussion of smooth manifolds in Math 401, R2.
>
> For this week [10/21/2025], our goal is to understand the Riemann-Roch theorem and its applications.
>
> References:
>
> - [Introduction to Complex Manifolds](https://bookstore.ams.org/gsm-244)
### Riemann-Roch Theorem (Theorem 9.64)
Suppose $M$ is a connected compact Riemann surface of genus $g$, and $L\to M$ is a holomorphic line bundle. Then
$$
\dim \mathcal{O}(M;L)=\deg L+1-g+\dim \mathcal{O}(M;K\otimes L^*)
$$