Submersions of CR submanifolds
A real submanifold N of a complex Banach manifold V is called a CR submanifold if \(T^ h=TN\cap J(TN)\) is a complex subbundle of the tangent bundle TV; J being the relevant complex structure. Assume V Kähler, let \(T^{\vee}N\) be the orthogonal complement of \(T^ hN\) in TN, \(T^ nN\) be the normal bundle. Assume that J interchanges \(T^{\vee}N\) and \(T^ nN\). Let \(\pi\) : \(N\to W\) be a submersion on an almost Hermitian manifold W such that \(T^{\vee}N\) is a kernel of \(\pi\) and \(\pi\) induces a complex isometry of \(T^ hN\) and TW. Under these assumptions, W is Kähler and \(H^ V=H^ W-4| B|^ 2\), where \(H^ V\) \((H^ W)\) is the holomorphic sectional curvature and B is the second fundamental form of N. As an application, the moduli space of stable holomorphic vector bundles over compact Riemann surfaces has nonnegative holomorphic sectional curvature.
scientific article; zbMATH DE number 4005174
scientific article