I am trying to understand what a holomorphic quadratic differential is, i have read a local definition on two books: Jürgen-Jost-“Compact Riemman Surfaces” and Kurt Strebel-“Quadratic-Differentials”. The definition that they use is local:
Definition: Let ( M , g) be a Riemann surface with a conformal metric and z a local conformal coordinate, we say that φdz2 is a holomorphic quadratic differential if φ is holomorphic.
That is the definition in Jürgen-Jost book, in Kurt Strebel “Quadratic-Differentials” They only add that a transformation rule for other coordinates is needed.
I would like to understand a global definition in terms of sections, in wikipedia i found this:
“a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle”
I guess holomorphic cotangent bundle means the bundle of holomorphic 1-forms. I don’t understand the “symmetric square part”. I would like to have a global definition and a local coordinate representation. Of course I don’t trust at all about a wikipedia link. So I would like to ask, is this definition right? Is there a good reference i should read ?
Since the holomorphic cotangent bundle T⋆X is of complex rank one, we can get rid of the word “symmetric”. A quadratic differential is simply a section of the tensor product T⋆X⊗T⋆X.
From a practical point of view, a section of this bundle looks like φ(z)dz⊗dz in a given coordinate patch parametrised by a coordinate z. But if you go to a different coordinate patch parametrised by a new coordinate w, then the same section should be written as φ(z(w))(dzdw)2dw⊗dw.
Since you asked for a reference, I searched on Google and the first thing I found was this blog post. I know it’s not a textbook, but I can assure you the person who wrote it know what he’s talking about! (I used to share an office with him.)