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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)

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