# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 40 (2011) pp.251-277

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### Abstract

Let $X$ be a smooth complex projective variety of dimension $n$ and let $L$ be an ample line bundle on $X$. In our previous paper, in order to investigate the dimension of $H^{0}(K_{X}+tL)$ more systematically, we introduced the invariant $A_{i}(X,L)$ for every integer $i$ with $0\leq i\leq n$. Main purposes of this paper are (1) to study a lower bound of $A_{i}(X,L)$ for the following two cases: (1.a) the case where $L$ is merely ample and $i\leq 3$, (1.b) the case of $h^{0}(L)>0$, and (2) to evaluate a lower bound for the dimension of $H^{0}(K_{X}+tL)$ by using $A_{i}(X,L)$.

MSC(Primary) 14C20 14C17, 14J30, 14J35, 14J40 Polarized manifold, adjoint bundles, the $i$-th sectional $H$-arithmetic genus, the $i$-th sectional geometric genus