Tags: #web/quick-notes

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Some notes on quadratic differentials:

  • 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: