**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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