Kodaira dimension of moduli space of vector bundles on surfaces (Q1319395)

Let \(H\) be an ample line bundle on a smooth algebraic complex surface \(X\) and let \(M_ H\) be the moduli space of rank-2 \(H\)-semistable sheaves \(E\) on \(X\) with fixed determinant and given second Chern class. The dimension of \(M_ H\), its singularities, its normality, and its Kodaira dimension are studied in this paper, in line with the work done in understanding the geometry of \(M_ H\) by \textit{Gieseker}, \textit{Maruyama}, \textit{Donaldson}, \textit{Friedman} and others. As to the Kodaira dimension of \(M_ H\), known results strongly suggest that it should always be closely connected with the Kodaira dimension of \(X\). More precisely these dimensions are the same if \(X = \mathbb{P}^ 2\) (\textit{Barth}, \textit{Hulek} and others), for some ruled surfaces (\textit{Quin}) as well as for K3 surfaces (\textit{Mukai}), whereas \(\kappa (M_ H) \geq 0\) for some surfaces of general type (\textit{O'Grady}). In this paper the author shows \(\kappa (M_ H) = 2\) for a class of minimal surfaces of general type.