These are notes that develop complex analysis with a computation-first approach and use it as a pathway into Riemann surfaces and core ideas in algebraic geometry—divisors, line bundles, sheaves, and cohomology—culminating in topics such as Jacobians and the Riemann–Roch theorem.

Complex analysis essentials

Cauchy–Riemann equations for $f=u+iv$ on $\mathbb{C}$:
\[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\qquad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.\]
Cauchy integral formula (for $f$ holomorphic on and inside $\gamma$):
\[f(z_0)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{z-z_0}\,dz,\qquad f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_{\gamma}\frac{f(z)}{(z-z_0)^{n+1}}\,dz.\]
Residue theorem:
\[\int_{\gamma} f(z)\,dz = 2\pi i\sum_{a\in \mathrm{Inside}(\gamma)} \mathrm{Res}_{z=a} f(z).\]

Riemann surfaces by explicit construction

A typical two-sheeted branched cover can be written as
\[y^2 = \prod_{j=1}^{2g+2}(x-a_j),\]
whose compactification gives a genus-$g$ hyperelliptic Riemann surface.
Locally, holomorphic charts glue via biholomorphic transition maps
\[z_\alpha = \varphi_{\alpha\beta}(z_\beta),\qquad \varphi_{\alpha\beta}'\neq 0.\]

Divisors and line bundles

A divisor is a formal sum
\[D=\sum_{p\in X} n_p\,p,\quad n_p\in\mathbb{Z},\ \text{with finitely many }n_p\neq 0.\]
Its degree is $\deg D=\sum_p n_p$.
For a meromorphic function $f$, its divisor is
\[(f)=\sum_{p\in X}\mathrm{ord}_p(f)\,p.\]
The associated space of sections is
\[L(D)=\{\, f\ \text{meromorphic} : (f)+D\ge 0 \,\}\cup\{0\},\qquad \ell(D)=\dim L(D).\]

Sheaves and cohomology (standard sequences)

The exponential sequence on a complex manifold $X$:
\[0\to \mathbb{Z}\to \mathcal{O}_X \xrightarrow{\exp(2\pi i\,\cdot)} \mathcal{O}_X^\ast \to 0,\]
giving a long exact sequence in cohomology and relating $\mathrm{Pic}(X)=H^1(X,\mathcal{O}_X^\ast)$ to $H^2(X,\mathbb{Z})$.
Dolbeault cohomology:
\[H^{p,q}_{\bar\partial}(X) \cong H^q(X,\Omega_X^p).\]

Duality and Riemann–Roch

Serre duality on a compact Riemann surface $X$:
\[H^1(X,\mathcal{O}(D))^\vee \cong H^0(X, K\otimes \mathcal{O}(-D)),\]
where $K$ is the canonical bundle.
Riemann–Roch theorem (for divisor $D$):
\[\ell(D)-\ell(K-D)=\deg D + 1 - g.\]

Jacobians and Abel–Jacobi

The Jacobian of a genus-$g$ curve $X$ can be described as a complex torus
\[\mathrm{Jac}(X)=\mathbb{C}^g/\Lambda,\]
where $\Lambda$ is the period lattice coming from integrating a basis of holomorphic 1-forms over $H_1(X,\mathbb{Z})$.
The Abel–Jacobi map (schematically) is
\[\mathrm{AJ}:\mathrm{Div}^0(X)\to \mathrm{Jac}(X),\qquad D=\sum n_p p \mapsto \left(\sum n_p\int_{p_0}^{p}\omega_1,\ldots,\sum n_p\int_{p_0}^{p}\omega_g\right)\;\mod \Lambda.\]

What’s inside

Foundations of Complex Analysis: holomorphic functions, Cauchy theory, residues, contour methods.
Riemann Surfaces via branched covers: branch cuts, gluing, compactification, genus.
Differential Forms and Stokes: exterior calculus viewpoint, Gauss–Bonnet, Hodge–Weyl on compact surfaces.
Sheaves and Cohomology: line bundles/divisors, Čech cohomology, exponential sequence, de Rham & Dolbeault.
Duality and Riemann–Roch: Serre duality, computations/examples.
Jacobians and Abel–Jacobi: periods, Picard varieties, Abel’s theorem, inversion.

How to read

Analysis-first: complex analysis $\rightarrow$ residues/branch cuts $\rightarrow$ surface constructions
Geometry-first: forms $\rightarrow$ Gauss–Bonnet/Hodge $\rightarrow$ duality $\rightarrow$ Riemann–Roch
AG-first: divisors/line bundles $\rightarrow$ sheaves/cohomology $\rightarrow$ Riemann–Roch $\rightarrow$ Jacobians