  A notion of vector field cobordism for oriented manifolds was defined by
Bokstedt and Svane. We extend this notion to define complex section cobordism
for almost complex manifolds. We then determine the complex section cobordism
groups and a relevant cobordism category. We describe an obstruction which
tells us when a cobordism class contains a manifold, which can be equipped with
$r$ linearly independent complex sections, in terms of the Chern classes.
Finally, we show that this obstruction vanishes for certain multiplicative
generators in the complex cobordism ring.
