Vertical Vafa-Witten invariants

TL;DR

This paper establishes the well-defined nature of vertical contributions to Vafa-Witten invariants for surfaces with positive geometric genus, linking them to tautological integrals and analyzing stability via cosections of obstruction sheaves.

Contribution

It proves the well-definedness of vertical contributions to Vafa-Witten invariants for certain surfaces and relates these to tautological integrals, advancing the understanding of their structure.

Findings

Vertical contributions are well defined for surfaces with p_g > 0.

Vertical contributions are computed by the same tautological integrals as in the stable case.

Stability of universal families is controlled by cosections of obstruction sheaves.

Abstract

We show that \emph{vertical} contributions to (possibly semistable) Tanaka-Thomas-Vafa-Witten invariants are well defined for surfaces with $p_g(S)>0$, partially proving conjectures of \cite{TT2} and \cite{T}. Moreover, we show that such contributions are computed by the same tautological integrals as in the stable case, which we studied in \cite{L}. 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.