Hodge Atoms at Conifold Degenerations: F-Bundles, Limiting Mixed Hodge Modules, and the Rigid-Flexible Decomposition

TL;DR

This paper extends the Hodge atoms framework to conifold degenerations of Calabi--Yau threefolds, constructing a canonical decomposition of Hodge structures that captures the degeneration's geometric and Hodge-theoretic features.

Contribution

It introduces a rigid-flexible decomposition of Hodge atoms for conifold degenerations, linking the Stokes matrix with the variation morphism in mixed Hodge modules.

Findings

Constructed a canonical decomposition of Hodge atoms for conifold degenerations.

Identified the Stokes matrix with the variation morphism matrix in mixed Hodge modules.

Provided a detailed description of the intersection matrix's role in the non-split structure.

Abstract

We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration $\pi\colon X \to \Delta$ whose central fiber $X_0$ has $r$ ordinary double points, we construct a canonical rigid-flexible decomposition of the Hodge atoms of the nearby smooth fiber attached to the corrected degeneration object. The rigid atom $A(\IC^H_{X_0})$ is preserved across the degeneration, while the flexible atoms $A(i_{k*}\QQ^H_{\{p_k\}}(-1))$ are rank-one contributions, one for each vanishing cycle. The total degeneration atom $A(P^H)$ is the atom of the corrected mixed Hodge module $P^H\in\MHM(X_0)$ and fits into an exact sequence of atoms whose non-split structure is controlled by the intersection matrix $(\langle\delta_i,\delta_j\rangle)$. The technical core is the Stokes--Extension Identification, which identifies the Stokes matrix of the Dubrovin connection at the conifold locus with the matrix of the variation morphism $\varF\colon\varphi_\pi(F) \to \psi_\pi(F)$ under mixed Hodge module realization.