AI 中文总结
该研究针对凯勒与平衡双曲性的形变稳定性，证明平衡双曲性非开性质，提出三类互补机制，得到相关形变稳定性成果。
AI 中文摘要
我们研究凯勒（Kähler）与平衡双曲性的形变稳定性。平衡双曲性一般不是开性质：在每个复维数 $N\geq5$ 中，我们构造了一个单参数族，其中心纤维具有平衡双曲性，而邻近纤维则非平衡双曲性。对于正面结果，我们提出三种互补机制：其一，有限维移动相交框架追踪 $\widetilde d$ 有界 de Rham 类穿过移动纯型轨迹，其Aeppli与Dolbeault实现给出平衡及凯勒双曲性的延拓、横截性与正性准则；其二，拓扑机制结合双曲上同调的分次理想性质与强莱夫谢茨定理，得到饱和性与高次传播结果；其三，在万有覆叠上，我们将从有界 $(\partial+\bar\partial)$-势到有界 $d$-本原元的转化，简化为单个有顶行 $\partial$-方程，并将对势的依赖打包为典范商阻碍。这些视角共同得到一系列形变稳定性结果。
英文摘要
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.
Comments36 pages, comments are welcome!
