geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A Riemann surface is a $1$-dimensional algebro-geometric object with good properties. The name ‘surface’ comes from the classical case, which is $1$-dimensional over the complex numbers and therefore $2$-dimensional over the real numbers.
There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities.
Classically, a Riemann surface is a connected complex-$1$-dimensional complex manifold, in the strictest sense of ‘manifold’. In other words, it’s a Hausdorff second countable space $M$ which is locally homeomorphic to the complex plane $\mathbb{C}$ via charts (i.e., homeomorphisms) $\phi_i:U_i \to V_i$ for $U_i \subset M, V_i \subset \mathbb{C}$ open and such that $\phi_j \circ \phi_i^{-1}: V_i \cap V_j \to V_i \cap V_j$ is holomorphic.
There are generalizations e.g. over local fields in rigid analytic geometry.
Evidently an open subspace of a Riemann surface is a Riemann surface. In particular, an open subset of $\mathbb{C}$ is a Riemann surface in a natural manner.
The Riemann sphere $P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}$ or $S^2$ is a Riemann sphere with the open sets $U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}$ and the charts
The transition map is $\frac{1}{z}$ and thus holomorphic on $U_1 \cap U_2 = \mathbb{C}^*$.
An important example comes from analytic continuation?, which we will briefly sketch below. A function element is a pair $(f,V)$ where $f: V \to \mathbb{C}$ is holomorphic and $V \subset \mathbb{C}$ is an open disk. Two function elements $(f,V), (g,W)$ are said to be direct analytic continuations of each other if $V \cap W \neq \emptyset$ and $f \equiv g$ on $V \cap W$. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).
Starting with a given function element $\gamma = (f,V)$, we can consider the totality $X$ of all equivalence classes of function elements that can be obtained by continuing $\gamma$ along curves in $\mathbb{C}$. Then $X$ is actually a Riemann surface.
Indeed, we must first put a topology on $X$. If $(g,W) \in X$ with $W=D_r(w_0)$ centered at $w_0$, then let a neighborhood of $g$ be given by all function elements $(g_w, W')$ for $w \in W, W' \subset W$; these form a basis for a suitable topology on $X$. Then the coordinate projections $(g,W) \to w_0$ form appropriate local coordinates. In fact, there is a globally defined map $X \to \mathbb{C}$, whose image in general will be a proper subset of $\mathbb{C}$.
Since we have local coordinates, we can define a map $f: X \to Y$ of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map $f: X \to \mathbb{C}$; for meromorphicity, this becomes $f: X \to S^2$.
Many of the usual theorems of elementary complex analysis (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a Laurent expansion, etc.
Let $f: X \to Y$ be a regular map. If $X$ is compact and $f$ is nonconstant, then $f$ is surjective and $Y$ compact.
To see this, note that $f(X)$ is compact, and an open subset by the open mapping theorem?, so the result follows by connectedness of $Y$.
Since a Riemann surface $X$ is a $2$-dimensional smooth manifold in the usual (real) sense, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle $T^*(X)$, but it is more natural to take the complexified cotangent bundle $\mathbb{C} \otimes_{\mathbb{R}} T^*(X)$, which we will in the future just abbreviate $T^*(X)$; this should not be confusing since we will only do this when we talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.
If $z = x + i y$ is a local coordinate on $X$, defined say on $U \subset X$, define the (complex) differentials
These form a basis for the complexified cotangent space at each point of $U$. There is also a dual basis
for the complexified tangent space.
We now claim that we can split the tangent space $T(X) = T^{1,0}(X) + T^{0,1}(X)$, where the former consists of multiples of $\frac{\partial}{\partial z}$ and the latter of multiples of $\frac{\partial}{\partial \bar{z}}$; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given $g: U \to \mathbb{C}$ for $U \subset \mathbb{C}$ open and $0 \in U$ (just for convenience), we can write
where $A = \frac{\partial g }{\partial z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)$, which we will often abbreviate as $g_z(0), g_{\bar{z}}(0)$. If $\psi: U' \to U$ is holomorphic and conformal sending $z_0 \in U' \to 0 \in U$, we have
in particular, $\phi$ preserves the decomposition of $T_0(\mathbb{C})$.
Given $f: X \to \mathbb{C}$ smooth, we can consider the projections of the 1-form $df$ onto $T^{1,0}(X)$ and $T^{0,1}(X)$, respectively; these will be called $\partial f, \overline{\partial} f$. Similarly, we define the corresponding operators on 1-forms: to define $\partial \omega$, first project onto $T^{0,1}(M)$ (the reversal is intentional!) and then apply $d$, and vice versa for $\overline{\partial} \omega$.
In particular, if we write in local coordinates $\omega = u d z + v d\bar{z}$, then
and
To see this, we have tacitly observed that $d v = v_z d z + v_{\bar{z}} d\bar{z}$.
In the theory of Riemann surfaces, there are several important theorems. Here are two: