Recovering complex structure from $(n,0)$-form with implications for Vafa-Witten equation

Abstract

Given an even-dimensional smooth manifold and a (co)(closed) bundle-valued $n$-form $a$ that "resembles" a scalar $(n,0)$-form, we may (or may not) recover a "$\mathbb{Z}_2$-(almost) complex structure" under certain assumptions. In the case of a $4$-manifold or a quaternion-Kähler manifold, we can recover a compatible $\mathbb{Z}_2$-complex structure on the whole manifold including the zero locus of $a$. In general, we can recover a $\mathbb{Z}_2$-(almost) complex structure wherever $a$ satisfies a certain nondegeneracy condition, but not necessarily on the whole manifold, with a counterexample identified. The above condition of "resembling" a scalar $(2,0)$-form is satisfied by any $\mathrm{U}(1)$-invariant solution to the Vafa-Witten equation on a compact Kähler surface. This hints at a possibility to generalise the notion of $\mathrm{U}(1)$-invariant solutions to oriented Riemannian $4$-manifolds. However, our results above imply any solution satisfying the condition, and thus seemingly any potential "generalised $\mathrm{U}(1)$-invariant solution", must be trivial if the $4$-manifold admits no compatible $\mathbb{Z}_2$-complex structure.