Rigidity Criterion for Certain Calabi-Yau Families

We prove a new rigidity criterion for families of polarized Calabi--Yau manifolds. Motivated by known non-rigid examples, we conjecture that a family over a quasi-projective curve is rigid if, near a boundary point, the total space is smooth, the relative canonical bundle is trivial, and the boundary fiber contains an isolated singular point. We verify this conjecture when one such isolated singularity has a concentrated mixed Hodge spectrum, a class including ordinary double points and cusps. The proof combines a local vanishing-cycle analysis with a global tensor-product decomposition of the associated variation of Hodge structures.