Translation surface

Remarks

Let $\Sigma$ be the finite set of zeros of $\omega$. Near a point $p \notin \Sigma$, the function $z(q) = \int_p^q \omega$ is a local coordinate in which $\omega = dz$, and two such coordinates differ by a translation $z \mapsto z + c$. So $X \setminus \Sigma$ gets a flat metric $\abs{\omega}$, and a zero of order $k$ is a cone point of angle $2\pi(k + 1)$. If $X$ has genus $g$ and the zeros have orders $k_1, \dots, k_n$, then $k_1 + \cdots + k_n = 2g - 2$.

Equivalently, a translation surface is given by finitely many polygons in $\C$ whose sides are glued in pairs by translations. Each side is glued to a parallel side of the same length with opposite outward normal. The form $\omega$ is $dz$ on each polygon.

Example: the unit square with opposite sides glued is the torus $X = \C / (\Z + i\Z)$ with $\omega = dz$. The form $dz$ is invariant under the translations by $\Z + i \Z$, so it descends to $X$, and it has no zeros, in line with $2g - 2 = 0$.

Example: the regular octagon with opposite sides glued has all eight vertices glued to one point, with total angle $8 \cdot \frac{3\pi}{4} = 6\pi$. So $\omega$ has one zero of order $2$, and $2 = 2g - 2$ gives genus $g = 2$.

Non-example: $(X, 0)$ is not a translation surface, since the zero form gives no charts.