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

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 $t\ge 2$. In this paper, we consider the case where $(X,L)$ is a quasi-polarized manifold. First we will prove $g_2(X,L)\ge h^1({\mathcal{O}}_X)$ for the following cases: (a) $n=3$, $\kappa (X)=-\infty$ and $\kappa (K_X+L)\ge 0$. (b) $n\ge 3$ and $\kappa (X)\ge 0$. 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.

20 pages. We improved some results and corrected some errata in the second version of this paper

Cited by in corpus (1)

• Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds