TanakaThomas conjecture on JoyceSong pair invariants

Let $(S,H)$ be a smooth polarized surface with $H_1(S,\mathbb Z)=0$, $p_g(S)>0$, and $r\in\mathbb Z_{>0}$. Fix $\operatorname{ch}=(r,c_1,\frac12c_1^2-n)$. For $\nu\gg0$, let $P=P_{S,\nu}^H(r,c_1,n)$ be the moduli space of stable JoyceSong Higgs pairs, with its induced symmetric perfect obstruction theory and $\mathbb C^*$-fixed virtual localization invariant. Assume that $H$ is generic in the stated sense, and define $\mathrm{VW}_S^H(r,c_1,n)$ by

$$\int_{[P^{\mathbb C^*}]^{\mathrm{vir}}}\frac{1}{e(N^{\mathrm{vir}})}=(-1)^{\chi(\operatorname{ch}(\nu H))-1}\chi(\operatorname{ch}(\nu H))\,\mathrm{VW}_S^H(r,c_1,n).$$

Tanaka--Thomas conjecture. The invariant $\mathrm{VW}_S^H(r,c_1,n)$ is independent of the choice of $\nu\gg0$.

This removes the apparent dependence of the JoyceSong pair construction on the auxiliary twisting parameter $\nu$. The statement was conjectured by Tanaka and Thomas and proved on the vertical component by T. Laarakker; the general assertion remains open in the source.