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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.

cactus314
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1 Answers1

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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».

Mariano Suárez-Álvarez
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  • quadratic differentials are not single-valued unless you move to the double cover. the branching occurs where the quadratic differential is 0, right? mainly i wasn't sure what "abelian" differentials were. – cactus314 Jan 17 '11 at 02:18
  • @john, as I said, I do not know what you mean by all that. – Mariano Suárez-Álvarez Jan 17 '11 at 02:25
  • John, could you give a reference? Like Mariano, I'm confused as to what you mean. Maybe it's some terminological difference. – arsmath Jan 17 '11 at 18:57
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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. – Matt E Jan 17 '11 at 20:00