I am studying a particular class of real Riemann surfaces; these are branched covers of the complex plane or Riemann sphere where all of the branch points lie on the real line. I am wondering if there are any known restrictions on the period matrices of such surfaces. For example, are they pure imaginary in this case? I am quite sure that they are for 2 branches (hyperelliptic curves).
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1$\begingroup$ What do you mean by period matrix? If $C$ is a compact Riemann surface of genus $g$, you can always normalize with respect to a symplectic basis for $H_1(C,\mathbb{Z})$ so that the period matrix takes the form $\Omega = (I\ \ Z)$ where $I$ is the $g\times g$ identity matrix and $Z$ is symmetric with positivedefinite imaginary part. If you are referring just to the matrix $Z$, then you might want to look at the paper "Period Matrices of Real Riemann Surfaces and Fundamental Domains" by Giavedoni. $\endgroup$ – Kevin Jul 7 '12 at 15:58

1$\begingroup$ Isn't the condition that $Z$ is purely imaginary equivalent to the existence of a direct sum decomposition $H_1(C,\mathbf{Z})=H_1^+(C,\mathbf{Z}) \oplus H_1^(C,\mathbf{Z})$ ? $\endgroup$ – François Brunault Jul 7 '12 at 21:25

$\begingroup$ You might then be able to construct explicitly such an adapted basis in the case of curves $y^2=f(x)$ with $f$ having only real roots. $\endgroup$ – François Brunault Jul 7 '12 at 21:26

$\begingroup$ Kevin  Indeed I mean the matrix $Z$. Thanks for that reference (I have also found an old paper by Gross and Harris, "Real Algebraic Curves", which has some relevant results). Francois  What do the two factors of that decomposition refer to explicitly? (For the curve $y^2 = f(z)$, I can indeed show that $Z$ is pure imaginary by using a standard basis for holomorphic differentials and appropriate canonical homology basis). $\endgroup$ – Albion Lawrence Jul 9 '12 at 21:05

$\begingroup$ @Albion Lawrence : $H_1^\pm$ refers to the $\pm 1$eigenspace with respect to complex conjugation acting on $C$. $\endgroup$ – François Brunault Jul 10 '12 at 6:22