Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds

Abstract

We prove a projection-triangle statement for projective Calabi--Yau threefold conifold degenerations and use it to organize an intersection-space Hodge atom shadow package. For a nodal central fiber $X_0$, assume the relevant Banagl--Budur--Maxim, multi-node gluing, mixed-Hodge-module, and specialization-splitting hypotheses, so that $\psi_\pi(F)\simeq \mathcal{IS}^{H}_{X_0}\oplus\mathcal C^H_\Sigma$. Projection of the variation morphism to the intersection-space summand defines $\operatorname{var}_I:\phi_\pi(F)\to\mathcal{IS}^{H}_{X_0}$, and the octahedral axiom gives $P^H\to P^H_I\to\mathcal C^H_\Sigma\xrightarrow{+1}$, where $P^H=\operatorname{Cone}(\operatorname{var})[-1]$ and $P^H_I=\operatorname{Cone}(\operatorname{var}_I)[-1]$. This realizes the intersection-space atom shadow package $\mathsf{HA}^{I}(X_0)$ and compares it with the intersection-homology package $\mathsf{HA}^{IH}(X_0)$. Under the self-dual specialization-splitting hypothesis, the projected variation object satisfies $\mathbb D P^H_I\simeq Q^H_I(3)$, where $Q^H_I=\operatorname{Cone}(\operatorname{can}_I)[-1]$. Under the mixed-Hodge realization of Banagl's middle exact sequence, we isolate a rigid--vanishing filtration and identify the IIB vanishing atom with the realized kernel. For the classical $125$-node quintic, the middle-degree IC--intersection-space defect has rank $202$. The construction remains at Hodge-realization level and identifies $\mathcal C^H_\Sigma$ and $\Delta_{I/IC}(X_0)$ as geometry-side handoff objects for future DT/BPS comparisons.