A generalization of curve genus for ample vector bundles. I

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