Balanced hyperbolicity along balanced deformations

Determine whether every sufficiently small balanced deformation of a balanced hyperbolic compact complex manifold remains balanced hyperbolic, assuming that all sufficiently small fibres remain balanced.

Background

The paper constructs families whose central fibre is balanced hyperbolic while nearby fibres are not even balanced, proving that balanced hyperbolicity is not deformation-open without additional hypotheses. The authors therefore isolate the sharper unresolved case in which balancedness of all nearby fibres is assumed in advance.

Several later criteria establish balanced-hyperbolicity persistence under additional cohomological, transversality, or topological hypotheses, but none resolves the unrestricted question for families whose nearby fibres are merely known to remain balanced.

References

The proof is given in \cref{sec:counterexample}. It leaves open the sharper question of whether balanced hyperbolicity persists when the nearby fibres are assumed to remain balanced.

\begin{question}\label{q:balanced-locus} Suppose $X_0$ is balanced hyperbolic and every sufficiently small $X_t$ is known to be balanced. Must every sufficiently small $X_t$ be balanced hyperbolic? \end{question}

— Deformations of Kähler and Balanced Hyperbolicity  (2609.04816 - Fu et al., 4 Sep 2026) in Question 1 (immediately after Theorem 1)