---
date: 2021-10-29
tags: [ web/quick-notes ]
---

# 2021-10-29

Tags:
#web/quick-notes 

Refs:
?

## 21:10

<https://arxiv.org/pdf/1904.06756.pdf>

Some notes on [quadratic differentials](quadratic%20differentials):

- See [central charge](central%20charge), [stability conditions](stability%20conditions) on a [triangulated category](triangulated%20category.md)?

![](figures/2021-10-29_21-11-26.png)

- Moduli space of [abelian differentials](abelian%20differentials) on a curve may be isomorphic to the moduli space f stability structures on the Fukaya category of the curve.

- These moduli spaces admit good "wall and chamber" decompositions, with [wall crossing](wall%20crossing) formulas due to Kontsevich.

- Important theorems: vanishing of cohomology for [line bundles](line%20bundles) and existence of meromorphic sections:

![](figures/2021-10-29_21-18-08.png)

- What is the [divisor associated to a section](divisor%20associated%20to%20a%20section)? Answered here:


![](figures/2021-10-29_21-21-04.png)

- A [principal divisor](principal%20divisor.md) is a divisor of a meromorphic function.
  Taking $\Div(X) / \Prin\Div(X)$ yields $\Cl(X)$ the [divisor class group](divisor%20class%20group.md) of $X$.

- There is a map $\Div: \Pic(X) \to \Cl(X)$ sending a line bundle to its divisor class.
This is an iso!

- A meromorphic function has the same number of zeros and poles, i.e. $\deg D = 0$ for $D\in \Prin\Div(X)$, so degrees are well-defined for $\Cl(X)$.


![](figures/2021-10-29_21-23-01.png)

- Computations of the cohomology of the trivial and canonical bundles:


![](figures/2021-10-29_21-23-53.png)

![](figures/2021-10-29_21-24-05.png)

![](figures/2021-10-29_21-24-22.png)