  We conjecture a formula for the virtual elliptic genera of moduli spaces of
rank 2 sheaves on minimal surfaces $S$ of general type. We express our
conjecture in terms of the Igusa cusp form $\chi_{10}$ and Borcherds type lifts
of three quasi-Jacobi forms which are all related to the Weierstrass elliptic
function. We also conjecture that the generating function of virtual cobordism
classes of these moduli spaces depends only on $\chi(\mathcal{O}_S)$ and
$K_S^2$ via two universal functions, one of which is determined by the
cobordism classes of Hilbert schemes of points on $K3$. We present
generalizations of these conjectures, e.g. to arbitrary surfaces with $p_g>0$
and $b_1=0$.
  We use a result of J. Shen to express the virtual cobordism class in terms of
descendent Donaldson invariants. In a prequel we used T. Mochizuki's formula,
universality, and toric calculations to compute such Donaldson invariants in
the setting of virtual $\chi_y$-genera. Similar techniques allow us to verify
our new conjectures in many cases.
