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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
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References:
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