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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.

##### MSC:
 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
##### Keywords:
vanishing theorem; non-embedding