Differential forms on moduli spaces of parabolic bundles (Q618659)

Let \(X\) denote a smooth projective variety of dimension \(n\) defined over an algebraically closed field of characteristic \(0\). For any flat family \(E_*\) of parabolic bundles on \(X\) parametrized by a smooth scheme \(Y\) and for any integer \(m, 1\leq m \leq \;\text{dim} \;X\), the author constructs a closed differential \(m\)-form \(\Omega= \Omega_{E_*}\) on \(Y\) with values in \(H^m(X, {\mathcal O}_X)\). Let \(PB^{sm}\) be the locus of smooth points in the moduli space of stable parabolic bundles on \(X\). Using the previous construction, the author shows that for any \(m \leq n, 0\leq i\leq n-m\), the choice of a nonzero element \(\sigma \in H^i(X, \Omega^{i+m}_X)\) determines a closed differential \(m\)-form \(\Omega_{\sigma}\) on \(PB^{sm}\). For \(m=2\), this can be used to construct a pre-symplectic structure (i.e. possibly degenerate closed holomorphic \(2\)-form) on \(PB\), the moduli space of parabolic bundles. The paper ends with examples indicating some possible applications.