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

Abstract: We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration $π\colon X \to Δ$ 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 $A(\IC<sup>H_{X_0})$ is preserved across the degeneration, while the flexible atoms $A(i_{k*}\QQ<sup>H_{{p_k}}(-1))$ are rank-one contributions, one for each vanishing cycle. The total degeneration atom $A(P<sup>H)$ is the atom of the corrected mixed Hodge module $P<sup>H\in\MHM(X_0)$ and fits into an exact sequence of atoms whose non-split structure is controlled by the intersection matrix $(\langleδ<em>i,δ_j\rangle)$. 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 $\varF\colon\varphi</em>π(F) \to ψ_π(F)$ under mixed Hodge module realization.