an:01429060
Zbl 0954.14006
Ohno, Masahiro
On nef values of determinants of ample vector bundles
EN
RIMS Kokyuroku 1078, 75-85 (1999).
00056263
1999
j
14C20 14F05 14M12 14N05 14J60
ample vector bundles; nef value; polarised projective manifold; nef line bundle
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. \textit{Y.-G. Ye} and \textit{Q. Zhang}, Duke Math. J. 60, No. 3, 671-687 (1990; Zbl 0709.14011) and \textit{T. Peternell}, Math. Z. 205, No.~3, 487-490 (1990; Zbl 0726.14034)].
N.Mohan Kumar (St.Louis)
Zbl 0709.14011; Zbl 0726.14034