Cobordism Obstructions to Complex Sections II: Torsion Obstructions

In the previous paper, we studied obstructions to the existence of complex sections on almost complex manifolds up to cobordism. We determined the obstruction rationally, in terms of the Chern classes. In this paper, we study the torsion obstructions, that is, the obstructions which vanish after tensoring with $\mathbb{Q}$ or multiplication by an integer. Calculations with the Adams-Novikov spectral sequence for the Thom spectra $\mathbf{MTU}(d)$ allow us to show the torsion obstructions for low $r$. For prime $p\geq 3$, we show that torsion obstructions for finding $r$ complex sections of order $p$ vanish for $r<p^2-p$.