Math 401, Fall 2025: Thesis notes, S4, Complex manifolds

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

Holomorphic vector bundles

Definition of real vector bundle

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:

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.
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:
- $\pi^{-1}(U)=\pi$(where \pi_U:U\times \mathbb{R}^k\to \pi^{-1}(U) is the projection map)
- 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.

Definition of complex vector bundle

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.

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.
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:
- $\pi^{-1}(U)=\pi$(where \pi_U:U\times \mathbb{C}^k\to \pi^{-1}(U) is the projection map)
- 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.

Definition of smooth complex vector bundle

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.

Definition of holomorphic vector bundle

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.

Holomorphic line bundles

A holomorphic line bundle is a holomorphic vector bundle with rank 1.

Intuitively, a holomorphic line bundle is a complex vector bundle with a complex structure on each fiber.

Simplicial, Sheafs, Cohomology and homology

What is homology and cohomology?

This section is based on extension for conversation with Professor Feres on [11/05/2025].

Definition of meromorphic function

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

f:Y'\to \mathbb{C} is a holomorphic function.
A=Y\setminus Y' is a set of isolated points (called the set of poles)
\lim_{x\to p}|f(x)|=+\infty for all p\in A

Basically, a local holomorphic function on Y.

De Rham Theorem

This is analogous to the Stoke's Theorem on chains, \int_c d\omega=\int_{\partial c} \omega.


H_k(X)\cong H^k(X)

Where H_k(X) is the $k$-th homology of X, and H^k(X) is the $k$-th cohomology of X.

Simplicial Cohomology

Riemann surfaces admit triangulations. The triangle are 2 simplices. The edges are 1 simplices. the vertices are 0 simplices.

Our goal is to build global description of Riemann surfaces using local description on each triangulation.

Singular Cohomology

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^*)

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