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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

14M15 Grassmannians, Schubert varieties, flag manifolds
14D15 Formal methods and deformations in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
Full Text: DOI EuDML
[1] Bott, R.: Homogeneous vector bundles. Ann. Math.66, 203-248 (1957) · Zbl 0094.35701 · doi:10.2307/1969996
[2] Carlson, J., Green, M., Griffiths, P., Harris, J.: Infinitesimal variations of Hodge structure (I). Comp. Math.50, 109-205 (1983) · Zbl 0531.14006
[3] Demazure, M.: A very simple proof of Bott’s theorem. Invent. Math.33, 271-272 (1976) · Zbl 0383.14017 · doi:10.1007/BF01404206
[4] Ein, L.: An analogue of Max Noether’s theorem. Duke Math. J.52, 689-706 (1985) · Zbl 0589.14034 · doi:10.1215/S0012-7094-85-05235-4
[5] Elencwajg, G.: Les fibrés uniformes de rang 3 sur ?2 sont homogènes. Math. Ann.231, 217-227 (1978) · Zbl 0378.14003 · doi:10.1007/BF01420242
[6] Faltings, G.: Formale Geometrie und homogene Räume. Invent. Math.64, 123-165 (1981) · Zbl 0459.14010 · doi:10.1007/BF01393937
[7] Flenner, H.: The infinitesimal Torelli problem for zero sets of sections of vector bundles. Math. Z.193, 307-322 (1986) · Zbl 0613.14010 · doi:10.1007/BF01174340
[8] Griffiths, P.: Periods of integrals on algebraic manifolds, II. Am. J. Math.90, 804-865 (1968) · Zbl 0183.25501
[9] Grothendieck, A.: Quelques problèmes de modules. Sém. H. Cartan, 1960/61, no 16 · Zbl 0234.14007
[10] Kurke, H., Theel, H.: Some examples of vector bundles on the flag variety \(\mathbb{F}(1,2)\) . Algebraic geometry. Proceedings. Bucharest 1982. Lect. notes in math. 1056. Berlin Heidelberg New York: Springer (1984) · Zbl 0559.14007
[11] Schuster, H.: Formale Deformationstheorien. Habilitationsschrift, München 1971
[12] Wavrik, J.: A theorem on completeness for families of compact analytic spaces. Trans. AMS163, 519-532 (1972) · Zbl 0205.38803 · doi:10.1090/S0002-9947-1972-0294702-5
[13] Wehler, J.: Deformation of varieties defined by sections in homogeneous vector bundles. Math. Ann.268, 519-532 (1984) · Zbl 0542.32008 · doi:10.1007/BF01451856
[14] Wehler, J.: Cyclic coverings: Deformation and Torelli theorem. Math. Ann.274, 443-472 (1986) · Zbl 0593.32017 · doi:10.1007/BF01457228
[15] Wehler, J.:K3-surfaces with Picard number 2. Arch. d. Math. (to appear) · Zbl 0602.14038
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