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In dynamics, they talk about Abelian differentials on surfaces, are they the same as holomorphic differentials?

Quadratic differentials are multiple valued and can change sign as you move around a zero.

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An Abelian differential is just a traditional name for a holomorphic or meromorphic differential on a compact Riemann surface.

A quadratic differential is just an element of $S^2(\Omega^1)$, the symmetric square of the sheaf of differentials. I do not really see what you mean when you say «[they] are multiple valued and can change sign as you move around a zero».

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    I think that what John means is that if one tries to write a quadratic differential in the form $(df)^2$ for some function $f$, then one may not be able to find such an $f$ globally, but in trying to do so, one constructs a double cover of the Riemann surface, branched at the zeroes of the quadratic differential.2011-01-17