Deformations of Kähler and Balanced Hyperbolicity

We study the deformation stability of Kähler and balanced hyperbolicity. Balanced hyperbolicity is not open in general: in every complex dimension $N\geq5$ we construct a one-parameter family with balanced hyperbolic central fibre and non-balanced nearby fibres. For positive results, we develop three complementary mechanisms. A finite-dimensional moving-intersection framework tracks $\widetilde d$-bounded de Rham classes through moving pure-type loci; its Aeppli and Dolbeault realizations yield continuation, transversality, and positivity criteria for balanced and Kähler hyperbolicity. A topological mechanism combines the graded-ideal property of hyperbolic cohomology with the hard Lefschetz theorem to obtain saturation and higher-power propagation results. Finally, on the universal cover, we reduce the passage from a bounded $(\partial+\bar\partial)$-potential to a bounded $d$-primitive to a single bounded top-row $\partial$-equation, and package the dependence on the potential into a canonical quotient obstruction. Together, these viewpoints yield a range of deformation stability results.