×

zbMATH — the first resource for mathematics

The nef value and defect of homogeneous line bundles. (English) Zbl 0808.14042
Let \(X \subset \mathbb{P}^ N\) be a smooth projective variety, and let \(L\) be the very ample line bundle on \(X\) defined by the hyperplane section. Let the canonical bundle \(K_ X\) be not nef, and let \(\tau = \tau (X,L) = \min \{t : K_ X \otimes L^ t\) is nef} be the nef value of \((X,L)\). In particular, let \(X=G/P\) \((G\) a semisimple complex Lie group, \(P\) a parabolic subgroup of \(G)\) be a homogeneous variety, and let \(L\) be a homogeneous line bundle on \(X\), defined by its weight \(\mu\). In theorem 2.2 the author gives an explicit formula for the nef value of such \(L\) – only in terms of the weight \(\mu\) of \(L\), and of the weight of the (not nef) canonical bundle \(K_ X\). Let moreover \(P\) be maximal, and let \(L_ 0\) be the ample generator of \(\text{Pic} (X) = \mathbb{Z}\). In corollary 2.4 the exact values of \(\tau (X,L_ 0)\) are tabulated for all the cases of simple Lie groups \(G=A_ 1\), \(B_ 1\), \(C_ 1\), \(D_ 1\), \(E_ 6\), \(E_ 7\), \(E_ 8\), \(F_ 4\), and \(G_ 2\).
In §3 some known facts are collected discussing the connection between the nef value of \((X,L)\) and the defect of the dual variety \(X'\) of \(X\). These facts are used in §4 to find a simpler proof of the classification of F. Knop and G. Menzel [see Comment. Math. Helv. 62, 38-61 (1987; Zbl 0625.14028)] of homogeneous spaces with positive defect. Finally, a list of the self-dual homogeneous spaces is derived (corollary 4.3), and the connection of these spaces and the classification of real hypersurfaces in \(\mathbb{P}^ N\) is explained (corollary 4.4).
Reviewer: A.Iliev (Sofia)

MSC:
14M17 Homogeneous spaces and generalizations
32M10 Homogeneous complex manifolds
14C20 Divisors, linear systems, invertible sheaves
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Jürgen Berndt, Real hypersurfaces with constant principal curvatures in complex space forms, Geometry and topology of submanifolds, II (Avignon, 1988) World Sci. Publ., Teaneck, NJ, 1990, pp. 10 – 19. · Zbl 0729.53027
[2] Mauro C. Beltrametti, M. Lucia Fania, and Andrew J. Sommese, On the discriminant variety of a projective manifold, Forum Math. 4 (1992), no. 6, 529 – 547. · Zbl 0780.14023 · doi:10.1515/form.1992.4.529 · doi.org
[3] Mauro C. Beltrametti, Andrew J. Sommese, and Jarosław A. Wiśniewski, Results on varieties with many lines and their applications to adjunction theory, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 16 – 38. · Zbl 0777.14012 · doi:10.1007/BFb0094508 · doi.org
[4] Armand Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0186.33201
[5] Thomas E. Cecil and Patrick J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481 – 499. · Zbl 0492.53039
[6] Lawrence Ein, Varieties with small dual varieties. I, Invent. Math. 86 (1986), no. 1, 63 – 74. · Zbl 0603.14025 · doi:10.1007/BF01391495 · doi.org
[7] Lawrence Ein, Varieties with small dual varieties. II, Duke Math. J. 52 (1985), no. 4, 895 – 907. · Zbl 0603.14026 · doi:10.1215/S0012-7094-85-05247-0 · doi.org
[8] Phillip Griffiths and Joseph Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 355 – 452. · Zbl 0426.14019
[9] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004
[10] Steven L. Kleiman, Tangency and duality, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 163 – 225. · Zbl 0601.14046
[11] Friedrich Knop and Gisela Menzel, Duale Varietäten von Fahnenvarietäten, Comment. Math. Helv. 62 (1987), no. 1, 38 – 61 (German). · Zbl 0625.14028 · doi:10.1007/BF02564437 · doi.org
[12] Alain Lascoux, Degree of the dual of a Grassmann variety, Comm. Algebra 9 (1981), no. 11, 1215 – 1225. · Zbl 0462.14017 · doi:10.1080/00927878108822641 · doi.org
[13] R. Lazarsfeld and A. Van de Ven, Topics in the geometry of projective space, DMV Seminar, vol. 4, Birkhäuser Verlag, Basel, 1984. Recent work of F. L. Zak; With an addendum by Zak. · Zbl 0564.14007
[14] Dennis M. Snow, Vanishing theorems on compact Hermitian symmetric spaces, Math. Z. 198 (1988), no. 1, 1 – 20. · Zbl 0631.32025 · doi:10.1007/BF01183035 · doi.org
[15] Ryoichi Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495 – 506. · Zbl 0274.53062
[16] Jacques Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Springer-Verlag, Berlin-New York, 1967 (German). · Zbl 0166.29703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.