# 2021-10-29

Tags: #web/quick-notes

Refs: ?

## 21:10

• See central charge, stability conditions on a triangulated category?

• Moduli space of abelian differentials 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 formulas due to Kontsevich.

• Important theorems: vanishing of cohomology for line bundles and existence of meromorphic sections:

• What is the divisor associated to a section? Answered here:

• A principal divisor is a divisor of a meromorphic function. Taking $$\operatorname{Div}(X) / \mathop{\mathrm{Prin}}\operatorname{Div}(X)$$ yields $${ \operatorname{Cl}} (X)$$ the divisor class group of $$X$$.

• There is a map $$\operatorname{Div}: {\operatorname{Pic}}(X) \to { \operatorname{Cl}} (X)$$ 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 $$D\in \mathop{\mathrm{Prin}}\operatorname{Div}(X)$$, so degrees are well-defined for $${ \operatorname{Cl}} (X)$$.

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

#web/quick-notes