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Let 
$\mathcal{E}$ be an ample vector bundle of rank 
$r$ on a projective variety 
$X$ with only log-terminal singularities. We consider the nefness of adjoint divisors 
${{K}_{X}}\,+\,\left( t-r \right)\,\det \,\mathcal{E}$ when 
$t\,\ge \,\dim\,X$ and 
$t\,>\,r$ . As an application, we classify pairs 
$\left( X,\,\mathcal{E} \right)$ with 
${{c}_{r}}$ -sectional genus zero.
Let 
$X$ be a smooth projective surface over the complex number field and let 
$L$ be a nef-big divisor on $X$ . Here we consider the following conjecture; If the Kodaira dimension 
$\kappa (X)\ge 0$ , then 
${{K}_{X}}L\,\ge \,2q(X)\,-\,4$ , where 
$q\left( X \right)$ is the irregularity of $X$ . In this paper, we prove that this conjecture is true if (1) the case in which 
$\kappa (X)=0$ or 1, (2) the case in which 
$\kappa (X)=2$ and 
${{h}^{0}}(L)\,\ge \,2$ , or (3) the case in which 
$\kappa (X)=2$ , $X$ is minimal, 
${{h}^{0}}(L)\,=\,1$ , and $L$ satisfies some conditions.
