Symplectic Structures on Moduli Spaces of Parabolic Higgs Bundles and Hilbert Scheme

Parabolic triples of the form $(E_*,\theta,\sigma)$ are considered, where $(E_*,\theta)$ is a parabolic Higgs bundle on a given compact Riemann surface $X$ with parabolic structure on a fixed divisor $S$, and $\sigma$ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle $(E_*,\theta)$ a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by $\text{d}\Omega'$. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme $\text{Hilb}^\delta(Z)$, where $Z$ denotes the total space of the line bundle $K_X\otimes{\mathcal O}_X(S)$, that sends a triple $(E_*,\theta,\sigma)$ to the divisor defined by the section $\sigma$ on the spectral curve corresponding to the parabolic Higgs bundle $(E_*,\theta)$. Using this map and a meromorphic one--form on $\text{Hilb}^\delta(Z)$, a natural two--form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form $\text{d}\Omega'$.