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

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

2010-03-30

arxiv.org/abs/1003.5736v3

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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