Given a closed surface $S$ of genus $g \geq 1$ there are lots of choices of complex structure, and each one singles out a subspace of the surface's first cohomology group with complex coefficients corresponding to the classes of 1-forms which are holomorphic on that Riemann surface. Given an "appropriate" subspace, when does it correspond to the $H^{1,0}$ for some choice of complex structure?
To be more precise, let $\mathcal{T}(S)$ be the Teichmüller space of $S$, whose elements are pairs equivalence classes of pairs $(X, f)$ where $X$ is a Riemann surface and $f : S \to X$ is a diffeomorphism, with $(X, f) \sim (Y, g)$ if there is a holomorphic $\Phi : X \to Y$ such that $g^{-1}\Phi f$ is isotopic to $\mathrm{id}_S$. Using the marking $f$, we can pull $H^{1,0}(X)$ back to a complex $g$-dimensional subspace of $H^1(S; \mathbb{C})$. It is not the case that every $g$-dimensional $V \subseteq H^1(S; \mathbb{C})$ corresponds to some $H^{1,0}(X)$ for some choice of complex structure on $S$. For example, the theorems of Haupt and Kapovich (independently, see https://www.math.ucdavis.edu/~kapovich/EPR/fla2019.pdf) state that a character $\chi : H_1(S; \mathbb{Z}) \to \mathbb{C}$ will correspond to a holomorphic 1-form on some choice of complex structure if two conditions are satisfied.
If I had a $g$-dimensional $V \subseteq H^1(S; \mathbb{C})$ where each element satisfied the Haupt-Kapovich conditions, is it true this subspace corresponds to some $H^{1,0}(X)$? The issue that comes to mind is if $\chi, \kappa \in V$ they correspond to holomorphic 1-forms on Riemann surfaces, but is there a reason they should be holomorphic 1-forms on the same Riemann surface?