Abstract
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.