Hypersurfaces of the flag variety: Deformation theory and the theorems of Kodaira-Spencer, Torelli, Lefschetz, M. Noether and Serre
type: Item
instance of: scholarly article
title: Hypersurfaces of the flag variety: Deformation theory and the theorems of Kodaira-Spencer, Torelli, Lefschetz, M. Noether and Serre
zbMATH Open document ID: 0662.14029
DOI: 10.1007/BF01183036
author: Joachim Wehler
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173907
review text: 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 \({\mathbb{F}}\).
reviewed by: Henning Haahr Andersen
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 4081730
zbMATH Keywords: smooth hypersurface in the full flag manifold
zbMATH Keywords: small deformation
zbMATH Keywords: Lefschetz theorem
zbMATH Keywords: Picard number
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4081730