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On nef values of determinants of ample vector bundles. (English) Zbl 0954.14006
Let \((M,L)\) be a polarised projective manifold of dimension \(n\). Then the nef value \(\tau(M,L)\) is defined as the infimum of \(t\), such that \(K_M+tL\) is nef. The author proves that under certain conditions on \(\tau(M,\det E)\) for a rank \(r\) ample vector bundle on \(M\), the manifold is very special. Here is a sample result:
\(\tau(M,\det E)\leq (n+1)/r\) and equality holds if and only if \((M,E)\cong ({\mathbb{P}}^n, {\mathcal O}(1)^{\oplus r}).\)
Similar results have been obtained in lesser generality by various authors [e.g. Y.-G. Ye and Q. Zhang, Duke Math. J. 60, No. 3, 671-687 (1990; Zbl 0709.14011) and T. Peternell, Math. Z. 205, No. 3, 487-490 (1990; Zbl 0726.14034)].

14C20 Divisors, linear systems, invertible sheaves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M12 Determinantal varieties
14N05 Projective techniques in algebraic geometry
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli