On Quasi-Polarized Manifolds Whose Sectional Genus is Equal to the Irregularity

Abstract

Let $(X,L)$ be a quasi-polarized manifold of dimension $n$. In our previous paper, we proved that if $\dim X=3$ and $h^0(L)\ge 2$, then $g(X,L)\ge h^1({\mathcal{O}}_X)$ holds. Here $g(X,L)$ denotes the sectional genus of $(X,L)$. In this paper, we give the classification of quasi-polarized $3$-folds $(X,L)$ with $h^0(L)\ge 3$ and $g(X,L)=h^1({\mathcal{O}}_X)$. Moreover as an application of this result, we also give the classification of polarized manifolds $(X,L)$ with $\dim \text{Bs}|L|=1$, $h^0(L)\ge n$ and $g(X,L)=h^1({\mathcal{O}}_X)$.