Rigidity Criterion for Certain Calabi-Yau families

TL;DR

This paper introduces a new criterion for the rigidity of certain Calabi-Yau families, linking geometric singularities with deformation properties, and verifies it for specific types of singularities.

Contribution

It proposes a conjecture relating singular fiber singularities to rigidity and proves it for cases with isolated singularities like double points and cusps.

Findings

Rigidity criterion based on singular fiber properties.

Verification of the conjecture for specific singularities.

Analysis combining Hodge theory and vanishing cycles.

Abstract

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 it admits a smooth compactification whose singular fiber has only isolated singularities. We verify this conjecture for singularities with a concentrated mixed Hodge spectrum class including ordinary double points and cusps. The proof combines an analysis of the vanishing cycle exact sequence and limiting mixed Hodge structure with a tensor-product decomposition of the associated variation of Hodge structures.