We show that vertical contributions to (possibly semistable) Tanaka–Thomas–Vafa–Witten invariants are well defined for surfaces with \(p_g(S){>}0\), partially proving conjectures of Tanaka and Thomas (Pure Appl Math Q 13:517–562, 2017) and Thomas (Commun Math Phys 378(2):1451–1500, 2020). Moreover, we show that such contributions are computed by the same tautological integrals as in the stable case, which we studied in Laarakker (Geom Topol 24(6):2781–2828, 2020). Using the work of Kiem and Li, we show that stability of universal families of vertical Joyce–Song pairs is controlled by cosections of the obstruction sheaves of such families.

我们表明，对于具有\(p_g(S){>}0\) 的曲面，对（可能是半稳定的）Tanaka–Thomas–Vafa–Witten 不变量的垂直贡献是明确定义的，部分证明了 Tanaka 和 Thomas 的猜想（Pure Appl Math Q 13 :517–562, 2017) 和 Thomas (Commun Math Phys 378(2):1451–1500, 2020)。此外，我们表明这些贡献是通过与稳定情况相同的重言式积分计算的，我们在 Laarakker (Geom Topol 24(6):2781–2828, 2020) 中研究了这种情况。使用 Kiem 和 Li 的工作，我们表明垂直 Joyce-Song 对的通用族的稳定性受此类族的障碍滑轮的切面控制。