Vanishing of the universal-cover correction obstruction

Prove that the canonical obstruction class $\operatorname{Ob}_{\infty}(\widetilde\Omega)$ vanishes for every relevant lifted balanced form whose $(n-1)$-st power admits a bounded $(\partial+\bar\partial)$-potential, thereby establishing that Gauduchon hyperbolicity implies balanced hyperbolicity in the general setting considered.

Background

On the universal cover, the paper reduces the existence of a bounded real dd-primitive to solving a bounded top-row ∂\partial-equation for at least one admissible bounded potential. The authors package the resulting potential-dependence into the choice-independent quotient obstruction Ob⁡∞\operatorname{Ob}_{\infty}.

The paper gives conditional vanishing criteria and shows that solving the equation for every bounded potential is false in general. What remains unresolved is whether the canonical obstruction always vanishes in the general Gauduchon-hyperbolic situation; this is equivalent to proving the implication from Gauduchon hyperbolicity to balanced hyperbolicity.

References

The general vanishing of $\operatorname{Ob}_{\infty}$ remains open.

— Deformations of Kähler and Balanced Hyperbolicity  (2609.04816 - Fu et al., 4 Sep 2026) in Introduction, paragraph “The universal-cover correction” and Section 6, immediately after Corollary 6.8

Khelifati formulates

\text{balanced and Gauduchon hyperbolic}\quad\Longrightarrow\quad\text{balanced hyperbolic}

as Conjecture~2.20 and explains that the required complete-manifold equations with $L{\infty}$-control are not presently available Conjecture~2.20.

— Deformations of Kähler and Balanced Hyperbolicity  (2609.04816 - Fu et al., 4 Sep 2026) in Section 6.1, “Motivation”