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Hypersurfaces of the flag variety: Deformation theory and the theorems of Kodaira-Spencer, Torelli, Lefschetz, M. Noether and Serre. (English) Zbl 0662.14029
Let $$X$$ denote a smooth hypersurface in the full flag manifold $${\mathbb{F}}\subseteq {\mathbb{P}}^ n_{{\mathbb{C}}}$$. The author proves that any small deformation of $$X$$ is again a hypersurface of $${\mathbb{F}}$$ provided the degree of $$X$$ is at least 2 (3 if $$n=2$$).
He also obtains a generalization of the classical Lefschetz theorem saying that $$\pi_ i({\mathbb{F}},X)=H_ i({\mathbb{F}},X)=0$$ for $$i\leq \dim(X)-p$$, where $$p$$ is computable from the degree of the embedding $$X\in {\mathbb{F}}$$. When $$n=3$$, the author shows that the Picard number of $$X$$ is 2, and as a consequence, every curve on $$X$$ is in this case the variety of a global section of a 2-bundle on $${\mathbb{F}}$$.
Reviewer: H.H.Andersen

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14D15 Formal methods and deformations in algebraic geometry 14J10 Families, moduli, classification: algebraic theory
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