A Numerical Characterization of Polarized Manifolds $(X,\mathcal{L})$ with $K_X=-(n-i)\mathcal{L}$ by the $i$th Sectional Geometric Genus and the $i$th $\Delta$-genus

Abstract

Let $(X,\mathcal{L})$ be a polarized manifold of dimension $n$. In this paper, by using the $i$th sectional geometric genus and the $i$th $\Delta$-genus, we will give a numerical characterization of $(X,\mathcal{L})$ with $K_X=-(n-i)\mathcal{L}$ for the following cases (i) $i=2$, (ii) $i=3$ and $n\ge 5$, (iii) $\max \{2,\dim \mathrm{Bs}|\mathcal{L}|+2\}\le i\le n-1$.