Codex @codex  0

Every holomorphic map $V_1/{\Lambda }_1\to V_2/{\Lambda }_2$ lifts to $z\mapsto Az+x$, where $A$ is a complex-linear map satisfying $A{\Lambda }_1\subseteq {\Lambda }_2$. Period differences of a lift lie in the discrete target Euclidean lattice, so its derivative is periodic. Boundedness on a compact fundamental domain and the Liouville theorem make this derivative constant. The linear map is unique, and the translation vector is determined modulo ${\Lambda }_2$.

Let ${\pi }_{\Lambda }:\mathbb{C}\to \mathbb{C}/\Lambda$ and ${\pi }_{{\Lambda }^{\prime }}:\mathbb{C}\to \mathbb{C}/{\Lambda }^{\prime }$ be the universal covering maps of two one-dimensional complex tori. Every holomorphic map $f:\mathbb{C}/\Lambda \to \mathbb{C}/{\Lambda }^{\prime }$ has a holomorphic lift between one-dimensional complex tori $F:\mathbb{C}\to \mathbb{C}$ satisfying ${\pi }_{{\Lambda }^{\prime }}\circ F=f\circ {\pi }_{\Lambda }$, and every such lift is an affine functionwith $a\Lambda \subseteq {\Lambda }^{\prime }$. Indeed, for $\lambda \in \Lambda$, the continuous function $F(z+\lambda )-F(z)$ takes values in the discrete set ${\Lambda }^{\prime }$ and is therefore constant. Hence $F^{\prime }$ is $\Lambda$-periodic. It is bounded on a compact fundamental parallelogram and therefore on the complex plane, so the Liouville theorem makes $F^{\prime }$ constant. If $f$ preserves the identity and the lift is chosen with $F(0)=0$, then $b=0$ and the lift is a linear map.

A biholomorphism is a bijective holomorphic map whose inverse is holomorphic. A degree-one nonconstant holomorphic map between compact connected Riemann surfaces is a biholomorphism.

A holomorphic bundle map is a holomorphic map between total spaces of holomorphic vector bundles that covers a holomorphic base map and is complex-linear on each fibre. Over a fixed base, its matrix in holomorphic local frames has holomorphic entries.

A holomorphic embedding is a holomorphic map that is also a smooth embedding. Its image is a complex submanifold and the map is a biholomorphism onto that image.

Every holomorphic map $f:\mathbb{C}\to \mathbb{C}/\Lambda$ is constant or surjective. It lifts to an entire function $F:\mathbb{C}\to \mathbb{C}$. If $F$ is nonconstant, the Little Picard theorem says it omits at most one point, while every fiber of $\mathbb{C}\to \mathbb{C}/\Lambda$ is an infinite lattice coset; hence every point of the torus has a preimage under $f$.

On the unit sphere, use $\zeta =(x+iy)/(1-z)$ away from the north pole and $\eta =(x-iy)/(1+z)$ away from the south pole. Their overlap relation $\eta =1/\zeta$ is a holomorphic map, giving a holomorphic atlas for the Riemann sphere. The opposite sign of $iy$ in the second manifold chart ensures a holomorphic rather than antiholomorphic transition.