Let $\alpha_1, \dots, \alpha_n$ be complex numbers whose sum is zero and $u_1, \dots, u_{2g-2+n}$ be pariwise distinct nonzero complex numbers. Consider the the set of smooth genus g curves with n+1 marked points $P_0, x_1\dots, x_n$ and a marked canonical homology basis, with the following properties:
- Take the unique Abelian differential of the third kind $\omega$ on the curve having a simple pole at $x_i$ with residue $\alpha_i$, $i=1, \dots, n$ and zero a-periods. Then all b-periods of $\omega$ are zero.
- Let $P_1, \dots, P_{2g-2+n}$ be the zeroes of $\omega$. Then there exist paths $\gamma_i$ on the curve connecting $P_0$ and $P_i$ such that $\exp(\int_{\gamma_i}\omega)=u_i, i=1, \dots, 2g-2+n$.
I would like to show that this set is discrete, and possibly to find some explicit description of it. For integers values of $\alpha_i$ this is essentially Riemann's existence theorem.
Thanks
