I have two questions:
1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of $H_1(X,\mathbb{C})$ such that every period $\int_{\delta_i}\omega$ has module 1?
My guess would be yes: given any base $\{\delta_i\}_{i=1,\dots 2g}$ all I have to do is renormalize every element as $\frac{\delta_i}{|\int_{\delta_i}\omega|}$. Am I right?
2) Given $\omega$ as before and given $(\theta_1,\dots,\theta_{2g})\in (S^1)^{2g}$, is it always possible to find a base $\{\tau_i\}_{i=1,\dots 2g}$ of $H_1(X,\mathbb{C})$ such that $\int_{\delta_i}\omega$ has argument $\theta_i$ for every $i$?
My guess is no, but I can not prove it, but I think $(\theta_1,\dots,\theta_{2g})$ should at least respect some conditions.. If I'm right do you know which are the conditions on $(\theta_1,\dots,\theta_{2g})$?
Thank you