Vanishing theorems, linear and quadratic normality
Suppose \(X\subseteq {\mathbb{P}}^ n_{{\mathbb{C}}}\) is a closed submanifold of the complex projective space, of dimension \(d>2/3n\). It is conjectured (by R. Hartshorne) that X is the intersection of (n-d) hypersurfaces. As a first step toward this very difficult conjecture one can study the cohomology \(H^ i(X,{\mathcal O}(m))\), and especially we may ask whether it vanishes for \(0<i<d\), as it should. The paper under review contains some results (for \(m=1)\) in this direction, and it points out a mistake in an earlier article by the reviewer [which attempted to show such theorems; see J. Reine Angew. Math. 326, 136-151 (1981; Zbl 0452.14004)].
scientific article; zbMATH DE number 4002265
scientific article