Simple vector bundles over symplectic Kähler manifolds
A holomorphic symplectic structure on a complex manifold is given by a closed holomorphic 2-form \(\omega\) which is non-degenerate in the sense that if \(\omega(u,v)=0\) for all tangent vectors v, then \(u=0\). Let M be a complex Kähler manifold of dimension n and E a \(C^{\infty}\) complex vector bundle of rank r over M. Let D''(E) denote the space of semi- connections in E. A semi-connection D'' is called a holomorphic structure if \(N(D'')=D''\circ D''=0.\) Let H''(E) be the set of holomorphic structures in E. The holomorphic vector bundle defined by D'' is denoted by \(E^{D''}\). We call \(E^{D''}\) simple if its endomorphisms are all of the form \(cI_ E\), where \(c\in {\mathbb{C}}\). Let \({\mathcal S}''(E)\) denote the set of simple holomorphic structures in E and GL(E) the group of \(C^{\infty}\) automorphisms of the bundle E. Then the moduli space \({\mathcal S}''(E)/GL(E)\) is a complex analytic space with \(C^{\infty}\) topology. In a forthcoming paper in Math. Z. (see the preview Zbl 0602.53042), \textit{H.-J. Kim} shows that it is a non-singular complex manifold in a neighborhood of \([D'']\in {\mathcal S}''(E)/GL(E)\) if \(H^ 2(M, End^ 0(E^{D''}))=0\), where \(End^ 0(E^{D''})=\{u\in End(E^{D''});Tr(u)=0\}\). In the present paper, the author shows the following: Let M be a complex Kähler manifold with a holomorphic symplectic structure \(\omega_ M\). Let E be a \(C^{\infty}\) complex vector bundle over M. Let \({\mathcal M}(E)=\{[D'']\in {\mathcal S}''(E)/GL(E);H^ 2(M, End^ 0(E^{D''}))=0\}\) so that \({\mathcal M}(E)\) is a non-singular (possibly non-Hausdorff) complex manifold. Then \(\omega_ M\) induces in a natural way a holomorphic symplectic structure on \({\mathcal M}(E)\).
scientific article; zbMATH DE number 3971677