Remarks on the second sectional geometric genus of quasi-polarized manifolds and their applications

TL;DR

This paper extends previous work on the second sectional geometric genus to quasi-polarized manifolds, establishing new bounds and exploring the dimension of global sections of certain line bundles, with applications to 3-folds.

Contribution

It proves a lower bound for the second sectional geometric genus of quasi-polarized manifolds in specific cases and applies this to study global sections of canonical bundles plus multiples of the polarization.

Findings

Proved $g_{2}(X,L) \,\geq\, h^{1}(\mathcal{O}_{X})$ for certain cases.

Established bounds for the dimension of sections of $K_{X}+tL$ on 3-folds.

Extended geometric genus inequalities to quasi-polarized manifolds.

Abstract

In our previous papers, we investigated a lower bound for the second sectional geometric genus $g_{2}(X,L)$ of $n$-dimensional polarized manifolds $(X,L)$ and by using these, we studied the dimension of global sections of $K_{X}+tL$ with $t\geq 2$. In this paper, we consider the case where $(X,L)$ is a quasi-polarized manifold. First we will prove $g_{2}(X,L)\geq h^{1}(\mathcal{O}_{X})$ for the following cases: (a) $n=3$, $\kappa(X)=-\infty$ and $\kappa(K_{X}+L)\geq 0$. (b) $n\geq 3$ and $\kappa(X)\geq 0$. Moreover, by using this inequality, we will study $h^{0}(K_{X}+tL)$ for the case where $(X,L)$ is a quasi-polarized 3-fold.