Abstract
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.
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Acknowledgements
I thank Amin Gholampour, my Ph.D. advisor Martijn Kool, and Richard Thomas for useful discussions. I thank Richard Thomas for hosting me at Imperial College, where part of the work presented here was done.
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