zbMATH — the first resource for mathematics

Symmetric differential forms as embedding obstructions and vanishing theorems. With addendum. (English) Zbl 0790.14009
The author proves the following vanishing theorem: Let \(X \subset \mathbb{P}^ n_ \mathbb{C}\) be a \(d\)-dimensional submanifold. Then \(H^ d(X,S^ d (X,S^ k \Omega^ 1_ X \otimes {\mathcal O} (m))=0\) for \(q\leq 2d-n-1\) and \(k \geq m+2\). From this he deduces non-embedding results, e.g. the following: A projective manifold \(X\) of dimension \(d\) with ample cotangent bundle \(\Omega^ 1_ X\) cannot be embedded into \(\mathbb{P}^{2d- 1}\).
In an addendum the author points out that theorem 2.1 could also be derived from a result of R. Lazarsfeld [Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale 1983, Lect. Notes Math. 1092, 29-61 (1984; Zbl 0547.14009)] and gives an even simpler proof which was found by R. Braun.

14F17 Vanishing theorems in algebraic geometry
14E25 Embeddings in algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C35 Analytic sheaves and cohomology groups