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 "ℤ2-(almost) complex structure" under certain assumptions. In the case of a $4$-manifold or a quaternion-Kähler manifold, we can recover a compatible ℤ2-complex structure on the whole manifold including the zero locus of a. In general, we can recover a ℤ2-(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 U(1)-invariant solution to the Vafa-Witten equation on a compact Kähler surface. This hints at a possibility to generalise the notion of 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 U(1)-invariant solution", must be trivial if the $4$-manifold admits no compatible ℤ2-complex structure.
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BibTeX
@misc{cao2026recovering,
title = {{Recovering complex structure from $(n,0)$-form with implications for Vafa-Witten equation}},
author = {Zhengxiong Cao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27114},
eprint = {2609.27114},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27114},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recovering complex structure from (n,0)-form with implications for Vafa-Witten equation
AU - Zhengxiong Cao
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.27114
UR - https://arxiv.org/abs/2609.27114
AB - 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 "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 ℤ_2-complex structure on the whole manifold including the zero locus of a. In general, we can recover a 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 U(1)-invariant solution to the Vafa-Witten equation on a compact Kähler surface. This hints at a possibility to generalise the notion of 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 U(1)-invariant solution", must be trivial if the $4$-manifold admits no compatible ℤ_2-complex structure.
ER -