  A new genus $g=g(X,\ce)$ is defined for the pairs $(X,\ce)$ that consist of
$n$-dimensional compact complex manifolds $X$ and ample vector bundles $\ce$ of
rank $r$ less than $n$ on $X$. In case $r=n-1$, $g$ is equal to curve genus.
Above pairs $(X,\ce)$ with $g$ less than two are classified. For spanned $\ce$
it is shown that $g$ is greater than or equal to the irregularity of $X$, and
its equality condition is given.
