Rigidity Criterion for Certain Calabi-Yau families

TL;DR

The paper develops a rigidity criterion for families of polarized Calabi–Yau manifolds over quasi-projective curves, showing rigidity when the central fiber has only isolated singularities with concentrated mixed Hodge spectrum (including ODPs and cusps). The approach combines vanishing cycle analysis, limiting mixed Hodge structures, and a Deligne-type tensor-product decomposition obstruction for VHS to rule out nontrivial deformations. It establishes nontrivial monodromy on the holomorphic volume form via both Hodge-theoretic and differential-geometric period arguments, and leverages the concentration property to exclude any spreading of Hodge filtrations across tensor factors. The work also discusses connections to the Weil–Petersson metric via Wang’s incompleteness result, highlighting how rigidity constrains the period geometry in CY families and clarifies when metric considerations yield contradictions. Overall, it provides a robust, spectrum-based obstruction to nonrigidity in CY degenerations with isolated boundary singularities.

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.

Theorems & Definitions (6)

• proof
• proof
• proof : Proof of Proposition \ref{['non-triv-T']}
• proof : Proof of Proposition \ref{['prop: ordinary double points']}
• proof : Proof of Lemma \ref{['lemma: vanishing cycles in ODP']}
• proof : Proof of Theorem \ref{['main-thm']}