Hypersurfaces of the flag variety: Deformation theory and the theorems of Kodaira-Spencer, Torelli, Lefschetz, M. Noether and Serre (Q1113969)

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 \({\mathbb{F}}\).