  We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2
dimensional support on a smooth projective surface in an ambient non-compact
Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface.
We prove that in certain cases, when the rank 2 bundle is chosen appropriately,
the universal truncated Atiyah class of these codimension 2 sheaves reduces to
one, defined over the moduli space of such sheaves realized as torsion
codimension 1 sheaves in a noncompact divisor (threefold) embedded in the
ambient fourfold. Such reduction property of universal Atiyah class enables us
to relate our fourfold DT theory to a reduced DT theory of a threefold and
subsequently then to the moduli spaces of sheaves on the base surface using
results in arXiv:1701.08899 and arXiv:1701.08902 and . We finally make
predictions about modularity of such fourfold invariants when the base surface
is an elliptic K3.
