Thomas's virtual localisation conjecture for sheaf-counting invariants

Let $S$ be a polarized smooth projective surface, let $X=K_S$, and let $P^\perp_\alpha(n)$ denote the virtual localisation pair invariant associated with a sheaf class $\alpha$. For decompositions $\alpha_i=\delta_i\alpha$ with $\sum_{i=1}^\ell\delta_i=1$, write $\chi(\alpha_i(n))$ for the associated Euler characteristics. If $H^{0,1}(S)=0=H^{0,2}(S)$, then there exist invariants $\mathsf{VW}_{\alpha_i}(S)\in\mathbb Q$ such that

$$P^\perp_{\alpha}(n)=\mathop{\sum_{\ell\ge 1,\,(\alpha_i=\delta_i\alpha)_{i=1}^\ell:}}_{\sum_{i=1}^\ell\delta_i=1}\frac{(-1)^\ell}{\ell!}\prod_{i=1}^\ell(-1)^{\chi(\alpha_i(n))}\chi(\alpha_i(n))\,\mathsf{VW}_{\alpha_i}(S)$$

for $n\gg0$. When either $H^{0,1}(S)$ or $H^{0,2}(S)$ is nonzero, only the first term in the sum is taken, namely

$$P^\perp_{r,L,c_2}(n)=(-1)^{\chi(\alpha(n))-1}\chi(\alpha(n))\mathsf{VW}_{r,L,c_2}(S).$$

Thomas's virtual localisation conjecture. The invariants should furthermore satisfy $\mathsf{VW}_\alpha=\mathsf{vw}_\alpha$ whenever $\deg K_S\leq0$. The preceding formulas conjecturally extend the virtual-localisation invariant to the semistable case and relate it to the Behrend-localised Vafa–Witten invariant; the source gives no resolution status for this conjecture.