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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.
