The universal-invariant conjecture for Gieseker semistable sheaves on surfaces

Let $X$ be a projective complex surface with geometric genus $p_g=0$, and let $\mathcal A=\operatorname{Coh}(X)$. For an ample line bundle on $X$, let $\tau$ denote the resulting Gieseker stability condition, and let the remaining data be as in the universal enumerative-invariant setup. Universal-invariant conjecture for surfaces. The universal-invariant conjecture holds for $\mathcal A=\operatorname{Coh}(X)$ with these choices. This predicts rational-homology invariants for all Gieseker semistable sheaf classes, agreeing with virtual classes in the stable case and satisfying the associated wall-crossing and structural properties. The conjecture is motivated by existing virtual-class constructions for stable sheaves and by Mochizuki’s invariants, but the full package is not established in the stated generality.