Tags: #web/quick-notes
Refs: ?
21:10
https://arxiv.org/pdf/1904.06756.pdf
Some notes on quadratic differentials:
- See central charge, stability conditions on a triangulated category?
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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.
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These moduli spaces admit good “wall and chamber” decompositions, with wall crossing formulas due to Kontsevich.
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Important theorems: vanishing of cohomology for line bundles and existence of meromorphic sections:
- What is the divisor associated to a section? Answered here:
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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 \(X\).
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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!
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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: