Holomorphic one-form ($\omega \in H^0(X,K_X)$)

A section $\omega$ of the canonical bundle of a Riemann surface, locally $\omega =\varphi (z)dz$ with holomorphic $\varphi$. A nonzero form on a compact genus-$g$ surface has total zero order $2g-2$. Its integral supplies flat coordinates away from zeros.

The locus of nonzero holomorphic one-forms with a fixed list of zero multiplicities $k_1,\dots ,k_r$, summing to $2g-2$. The SL2R action on differentials preserves this list. For example $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$ are distinct genus-two strata.