# Hurwitz Spaces and Rauch Variational Formulas

I have read in some papers about Rauch-type variational formulas on Hurwitz spaces, and I would like to know what exactly is the theory behind them.

A Hurwitz Space $H_g^d$ is the space of coverings of the riemann sphere of degree $d$ by a surface of genus $g$. I know that this space has a complex structure where the local coordinates are the branch points of the cover $\lambda_1, \cdots , \lambda_n$.

The variational formulas I am talking about are of the type

$\frac{d}{d\lambda_k} W(P,Q) = \frac{1}{2} W(P_k,Q)W(P,P_k)$

Where $W(P,Q)$ is the fundamental differential of the third type on a riemann surface S. That is, it's a 1-form (differential) on $S^2$ satisfying the properties :

$W(P,Q) = W(Q,P)$

$W(P,Q) = (\frac{1}{(x(P)-x(Q))^2} + O(1))dx(P)dx(Q)$ near P=Q

$\int_{a_k} W(P,Q) = 0$ where $a_k$ are basis homology cycles.

More precisely, here is my question : How do I formalise taking a derivative with respect to the local parameter on Hurwitz Space? I know that $H_g^d$ has a complex structure, and that $W$ is defined on every Riemann Surface so I could see $W$ as a function on Hurwitz space, $W : H_g^d \rightarrow ?$ but I don't know what is the correct space to "move" $W$ in. It would need to be a space of differentials on surfaces where both the surface and the differential can change, and I am quite puzzled as to how to endow such a space with a complex structure.

These formulas are found for example in this paper:

Deformations of Frobenius structures on Hurwitz Spaces formulas (2.4) and (2.6).

I hope I made my question clear and will be happy to make any clarifications.

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