# Abelian vs Holomorphic Differentials vs Quadratic Differentials

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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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».
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. –  Matt E Jan 17 '11 at 20:00