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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)].

##### MSC:
 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