On joint moduli spaces
The object of this paper is to construct moduli spaces for pairs \((E,M)\), where \(E\) is a holomorphic vector bundle on a compact complex manifold \(M\). More precisely, suppose the \(C^\infty\) structures of \(M\) and \(E\) are fixed, and we consider pairs such that \(M\) has an ample canonical bundle and \(E\) is stable with respect to this bundle; the author proves that there exists a moduli space for such pairs with a natural structure of Hausdorff complex space. This applies in particular when \(M\) is a smooth Riemann surface of genus \(> 1\). The author studies also the existence of holomorphic vector bundle structures under small deformations of the base space. Suppose that \(M\) is a compact Kähler surface, \(E\) is simple, and either \(M\) has trivial canonical bundle or \(H^2 (\text{End}^0 (E)) = 0\). If \(M_\vartheta\) is a small deformation of \(M\), then \(E\) admits a holomorphic structure on \(M_\vartheta\) if and only if \(c_1 (E)\) is a (1,1) class on \(M_\vartheta\). The methods throughout are analytic. \{Reviewer's remarks. In the statement of the main theorem, the author says ``ample anticanonical bundle'', but it is clear from the text that ``canonical'' is intended. For the case of Riemann surfaces, natural compactifications of the author's moduli spaces have recently been constructed by \textit{R. Pandharipande} [``A compactification over \(\overline M_\vartheta\) of the universal moduli space of slope- semistable vector bundles'', J. Am. Math. Soc., to appear] using methods of geometric invariant theory\}.
scientific article; zbMATH DE number 769469
scientific article