Riv. Mat. Univ. Parma, Vol. 6, No. 2, 2015

Yoshiaki Fukuma[1]

Sectional class of ample line bundles on smooth projective varieties

Pages: 215-240
Received: 11 November 2013   
Accepted: 2 February 2015
Mathematics Subject Classification (2010): Primary 14C20; Secondary 14C17, 14J30, 14J35, 14J40, 14J60, 14M99, 14N15.
Keywords: Ample vector bundle, (multi-)polarized manifold, class, sectional Euler number, sectional Betti number.
Author address:
[1] : Department of Mathematics, Faculty of Science, Kochi University, Akebono-cho, Kochi 780-8520, Japan

Abstract: Let \(X\) be an \(n\)-dimensional smooth projective variety defined over the field of complex numbers, let \(L_{1}, \dots , L_{n-i}, A_{1}\) and \(A_{2}\) be ample line bundles on \(X\). In this paper, we will define the sectional class \(\mbox{cl}_{i}(X,L_{1}, \dots , L_{n-i}; A_{1},A_{2})\) for every integer \(i\) with \(0\leq i\leq n\), and we will investigate this invariant. In particular, for every integer \(i\) with \(0\leq i\leq n\), by setting \(L_{1}=\cdots=L_{n-i}=L\) and \(A_{1}=A_{2}=L\), we give a classification of polarized manifolds \((X,L)\) by the value of \(\mbox{cl}_{i}(X,L):=\mbox{cl}_{i}(X,\underbrace{L, \cdots , L}_{n-i};L,L)\).

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