  We investigate the possible values for geometric genera of subvarieties in a
smooth projective variety. Values which are not attained are called gaps. For
curves on a very general surface in $\mathbb{P}^3$, the initial gap interval
was found by Xu (see [7] in References), and the next one in our previous paper
(see [4] in References), where also the finiteness of the set of gaps was
established and an asymptotic upper bound of this set was found. In the present
paper we extend some of these results to smooth projective varieties of
arbitrary dimension using a different approach.
