Analysis reveals that the H-locus exhibits discrete or complete behavior over compact complex manifolds, suggesting implications for Riemann surfaces.
We study the deformation behavior of compact hyperbolic complex manifolds. Let π:X→ Δ be a smooth family of compact complex manifolds over the unit disk in C, and H a compact hyperbolic complex manifold. Then the H-locus ∈Δ: Xₜ H\ is either at most a discrete subset of $Δ$ or the whole $Δ$. For a smooth family over a compact Riemann surface Y, its H-locus is either at most finite or the whole Y. Furthermore, if Y is isomorphic to P¹ or an elliptic curve, then we conjecture that the H-locus is empty or the whole Y.
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Li et al. (2025) studied this question.
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