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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)

14M17 Homogeneous spaces and generalizations
32M10 Homogeneous complex manifolds
14C20 Divisors, linear systems, invertible sheaves
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