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On second exterior power of tangent bundles of threefolds. (English) Zbl 0824.14037
In an earlier paper [Math. Ann. 289, No. 1, 169-187 (1991; Zbl 0729.14032)], the authors have classified – among other things – all projective 3-folds $$X$$ whose tangent bundles $$T_ X$$ are nef. The purpose of this paper is to classify 3-folds such that $$\bigwedge^ 2 T_ X$$ is nef. The main result is that, if $$T_ X$$ is not nef, then $$X$$ is either the blow-up of $$\mathbb{P}_ 3$$ in one point or $$X$$ is a Fano 3- fold of index 2 with $$b_ 2 = 1$$ except for those arising as double covers of the Veronese cone in $$\mathbb{P}_ 6$$. Also a new invariant for a Fano manifold is introduced, namely the unique number $$\lambda$$ such that $$T_ X \otimes {\mathcal O} (- \lambda K_ X)$$ is nef but not ample. $$\lambda$$ is computed in several cases. As a conjecture we ask whether $$\lambda$$ is always rational.

##### MSC:
 14J30 $$3$$-folds 14J45 Fano varieties
##### Keywords:
nef tangent bundles; 3-folds; Fano manifold
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##### References:
 [1] Barth, W. , Peters, C. , and van de Ven, A. , Compact complex surfaces . Erd. d. Math. Bd. 3, Springer 1984. · Zbl 0718.14023 [2] Campana, F. , and Peternell, Th. , Manifolds whose tangent bundles are numerically effective . Math. Ann. 289, 169-187 (1991). · Zbl 0729.14032 [3] Fulton, W. , Intersection Theory . Erg. d. Math. 2. Springer 1984. · Zbl 0541.14005 [4] Hartshorne, R. , Ample vector bundles . Publ. Math. IHES 29, 319-350 (1966). · Zbl 0173.49003 [5] Iskovskih, V.A. , Fano 3-folds I, II . Math. USSR Isv. 11, 485-527 (1977),ibid. 12, 469-506 (1978). · Zbl 0424.14012 [6] Iskovskih, V.A. and Shokurov, V.V. , Biregular theory of Fano 3-folds . Lecture Notes in Math. 732, 171-182 Springer 1978. · Zbl 0417.14025 [7] Kobayashi , Ochiai, T. , Characterizations of complex projective spaces and hyperquadrics . J. Math. Kyoto Univ. 13, 31-47 (1973). · Zbl 0261.32013 [8] Mori, S. and Mukai, S. , On Fano 3-folds with b2 \succcurleq 2 . Adv. Studies in Math. 1, 101-129 (1983). · Zbl 0537.14026 [9] Mori, S. , Threefolds whose canonical bundles are not numerically effective . Ann. Math. 116, 133-176 (1982). · Zbl 0557.14021 [10] Peternell, Th. , Calabi-Yau manifolds and a conjecture of Kobayashi . Math. Z. 207, 305-318 (1991). · Zbl 0735.14028 [11] Shokurov, V.V. , The existence of a straight line on Fano 3-folds . Math. USSR Isv. 15, 173-209 (1980). · Zbl 0444.14027 [12] Demazure, M. , Surface de del Pezzo I-V , Lecture Notes in Math. 777, 23-70. Springer 1980. · Zbl 0444.14024 [13] Yau, S.T. , Calabi’s conjecture and some new results in algebraic geometry . Proc. Math. Acad. Sci. USA 74, 1789 (1977). · Zbl 0355.32028
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