  In the previous paper, we studied the existence of complex sections on almost
complex manifolds up to cobordism. We proved a necessary and sufficient
condition for their existence in $\Omega_{*}^U\otimes \mathbb{Q}$ in terms of
the Chern classes. In this paper we study the existence question in
$\Omega^{U}_*$. We call corresponding obstructions torsion and we determine
them for small values of $r$. Calculations with the Adams-Novikov spectral
sequence for the Thom spectra defined in the previous paper allow us to show
the vanishing of the torsion obstruction for low $r$. For prime $p$, we show
the vanishing of $p$ torsion obstructions for finding $r$ complex sections for
$r<p^2-p$.
