Lefschetz operators and the existence of projective equations (Q805691)

Let j: \(X\hookrightarrow {\mathbb{P}}_ N({\mathbb{C}})\) be an embedding of a projective manifold X. The induced normal bundle carries information on geometric properties of j. The author analyzes the relationship of various maps between certain cohomology groups. As an application he obtains e.g. the following result: Let X be a closed positive-dimensional submanifold of a projective manifold W with codimension r, and i: \(W\hookrightarrow {\mathbb{P}}_ N({\mathbb{C}})=:P\) an embedding which induces an arithmetically normal embedding j: \(X\hookrightarrow P.\) Equivalent are: (i) The exact sequence of the normal bundles \(O\to N_{X/W}\to N_{X/P}\to N_{W/P}\to O\) splits, and \(N_{X/W}\simeq {\mathcal O}_ X(m_ 1)\oplus...\oplus {\mathcal O}_ X(m_ r).\) (ii) There exist hypersurfaces \(S_ 1,...,S_ r\) of degree \(m_ 1,...,m_ r\) respectively such that \(j(X)=i(W)\cap S_ 1\cap...\cap S_ r\) (transversal).