## References

#resources

- Survey Paper: Anton Zorich, [Flat Surfaces](https://arxiv.org/abs/math/0609392)
- Alex Eskin, Andrei Okounkov, [Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials](https://arxiv.org/abs/math/0006171)
- Alex Eskin, Howard Masur, Anton Zorich, [Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel--Veech Constants
](https://arxiv.org/abs/math/0202134)
- Alex Eskin, Andrei Okounkov, [Pillowcases and quasimodular forms](https://arxiv.org/abs/math/0505545)
- Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich, [Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes](https://arxiv.org/abs/1903.10904)
  - See Phil for appendix! 
- Engel, [Hurwitz Theory of Elliptic Orbifolds, I](https://arxiv.org/abs/1706.06738)
- Engel, [Hurwitz Theory of Elliptic Orbifolds, II](https://arxiv.org/abs/1809.07434)

## Notes

**Definition:**
A map $\pi: \Sigma \to \Sigma'$ of Riemann surfaces is said to be [[ramified]] at a point $p\in \Sigma'$ iff in local charts $\pi$ has the form $z\mapsto z^n$ for some $n>1$.

> I.e. all points in a punctured neighborhood of $\pi(p)$ have $n$ preimages.

**Definition:**
If $\pi$ is ramified at $p$, the number of preimages $n$ is referred to as $e_p$, the [ramification index](ramification%20index.md) of $p$*.

*Fact:*
$\vector{\beta}(\Sigma_g) = [1, 2g, 1, 0, \cdots]$ and $\chi(\Sigma) = 2-2g$.

**Theorem:**
If $\pi$ is an unramified covering map of degree $n$, then $\chi(\Sigma') = n\chi(\Sigma)$.

**Theorem ([Riemann-Hurwitz](Riemann-Hurwitz)):**
If $\pi: \Sigma \to \Sigma'$ is a ramified covering map of degree $N$, then

\[
\chi(\Sigma') = N \chi(\Sigma) - \sum (e_p - 1) \quad\text{ i.e. } 2 g(\Sigma') - 2=  N(2g(\Sigma) - 2)  + \sum (e_p - 1)
.\]

*Another useful form:*
Let $r \in \Sigma'$ be the number of ramification points, and $b$ the number of [branch points](branch%20points), i.e. their images in $\Sigma$.
Then

\[
\chi(\Sigma') = N(\chi(\Sigma) - b) + r
.\]


[Holomorphic Form](Holomorphic%20Form)
A holomorphic $p\dash$form on $X$ is a section of $\Lambda^p T\dual X$, the $p$th exterior power of the holomorphic cotangent bundle of $X$.

For $n = \dim_\CC X$, the $n\dash$forms are an important special case.
Any such form $w$ is given in local coordinates $(z_1, \cdots, z_n)$ by 

\[
w = w(z_1, \cdots, z_n) dz_1 \wedge \cdots \wedge dz_n
\]

for some holomorphic function $w: \CC^n \to ?$.

[Canonical Bundle](Canonical%20Bundle.md) : 
Given a complex manifold $M$, we can define the tangent bundle $\CC^n \to TM \to M$ and the cotangent bundle $\CC^n \to T\dual M \to M$, which we'll just denote $T\dual M$.
Then the canonical bundle is the bundle $\CC\to \Lambda^n T\dual M \to M?$, denoted by $\omega$, obtained by taking the $n$th exterior power.

It is a theorem that the fibers are in fact complex lines $\CC^1$.
For vector bundles, this is referred to as the [determinant bundle](determinant%20bundle).
If $M$ is a smooth manifold, then $\omega$ has a global section.

> Note: a holomorphic $n\dash$form is exactly the same as a section of the canonical bundle.

Interesting aside: a [Calabi-Yau](Calabi-Yau.md) is a manifold with a nowhere vanishing holomorphic $n\dash$form, which implies that the [canonical bundle](canonical%20bundle.md) admits a map to a trivial [line bundle](line%20bundle.md) that is an isomorphism, i.e. the canonical bundle is trivial.

*Exercise:*
For $\Sigma_g$ a compact [Riemann surface](Riemann%20surface) of genus $g$, the dimension of the space of holomorphic sections of the canonical bundle, i.e. the space of [holomorphic differentials](holomorphic%20differentials) on $\Sigma_G$, is given by $\dim H^0(X; \Omega) = g$ (the genus of the surface).
Proof: use [Riemann-Roch](Riemann-Roch.md).

Classification of elliptic [orbifolds](orbifolds) of dimension 2:
Define $(n_1, \cdots; m_1, \cdots)$ as the *profile*, where $n_i$ are *elliptic* points (locally look like quotient by $\ZZ/n\ZZ$), and $m_i$ are *corner reflectors* (locally look like quotient by a dihedral group):

![](Archive/Tilings/sections/figures/2020-01-29-20_44.png)

Conformal (or equivalently complex) [structures](complex%20structure) on a genus $g$ surface form a [moduli space](moduli%20space.md) $\MM_g$ of dimension $3g-3$ for $g > 1$.

Let $\alpha$ be any [integer partition](integer%20partition) of $2g-2$, and $\mch(\alpha)$ the moduli space of pairs $(\Sigma_g, \omega)$ where $\Sigma_g$ is a Riemann surface of genus $g$ and $\omega$ is a holomorphic 1-form (Abelian differential) on $M$ with the orders of its zeros given by $\alpha$.
Letting $\mch$ be the moduli space of all abelian differentials on Riemann surfaces of genus $g$ is stratified by $\mch(\alpha)$ as $\alpha$ ranges over all partitions.
For flat tori, $\mch = \GL_+(2, \RR)/\SL(2, \ZZ)$.

For $\Sigma_g$ a Riemann surface, there is a formula ([Gauss-Bonnet](Gauss-Bonnet) in the flat metric) relating the degrees of the zeros of a holomorphic 1-form to the genus:

\[
\sum d_j = 2g-2
.\]