A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions
Let \((E,J^E)\) be a complex vector bundle over a complex manifold \((M,J)\). A complex connection \(\nabla\) in \(E\) whose curvature component of type \((0,2)\) vanishes is called \textit{admissible} (by Abe and Kurosu) and defines on \(E\) a holomorphic structure. Holomorphic connections and connections with curvature of type \((1,1)\) are the principal examples of admissible connections. The main geometric data associated with a decomposition of a holomorphic vector bundle endowed with an admissible connection into a direct sum of two complex vector subbundles are the two induced connections and second fundamental forms. The authors study the relations between the following properties of the data: holomorphicity of one or both of the subbundles, Hodge type of the curvature of the induced connections and second fundamental forms. In the presence of a Hermitian metric in \(E\), they recover well known facts about Hermitian vector bundles. Then they apply the analysis to the case of complex affine immersions, i.e.\ holomorphic immersions between two complex manifolds \(f: M \rightarrow \widetilde{M}\) endowed with torsion-free complex connections \(\nabla, \widetilde{\nabla}\), respectively, such that \(\nabla\) is induced by \(\widetilde{\nabla}\) with respect to some transversal complex subbundle \(N \subset f^*T\widetilde{M}\). As a result, they obtain characterizations of holomorphic affine immersions (the case when \(N\subset f^*T\widetilde{M}\) is a \textit{holomorphic} subbundle) and complex affine immersions with induced or normal curvature of type \((1,1)\) in terms of equations involving the second fundamental form and the shape tensor. This generalizes results of Dillen and Nomizu-Pinkall-Podesta, cited in the paper under review, which were obtained in the case of complex hypersurfaces in complex affine space. Finally they prove the fundamental theorem of affine differential geometry for complex affine immersions into affine space, which states that given abstract geometric data satisfying the equations of Gauß, Codazzi and Ricci on a simply connected manifold, there exists a complex affine immersion inducing those data, which is unique up to a complex affine transformation.
scientific article; zbMATH DE number 2061995
scientific article