Assessment of special k-stability shows that CM line bundles are big and nef, indicating projectivity in a specific family of varieties.
We show that the CM line bundle on a proper family parametrizing specially K-stable varieties with maximal variation is big and nef. As an application, we show projectivity of any proper subspace of the coarse moduli space of uniformly adiabatically K-stable klt-trivial fibrations over curves constructed in Hashizume and Hattori (Geom. Topol. 29 (2025), 1619–1691).
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Masafumi Hattori (2025) studied this question.
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