Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866918478099775488 |
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| author | Rahman, Abdul |
| author_facet | Rahman, Abdul |
| contents | 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 $ψ_π(F)\simeq \mathcal{IS}^{H}_{X_0}\oplus\mathcal C^H_Σ$. Projection of the variation morphism to the intersection-space summand defines $\operatorname{var}_I:ϕ_π(F)\to\mathcal{IS}^{H}_{X_0}$, and the octahedral axiom gives $P^H\to P^H_I\to\mathcal C^H_Σ\xrightarrow{+1}$, 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 $\mathcal C^H_Σ$ and $Δ_{I/IC}(X_0)$ as geometry-side handoff objects for future DT/BPS comparisons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01455 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds Rahman, Abdul Algebraic Geometry High Energy Physics - Theory Category Theory 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 $ψ_π(F)\simeq \mathcal{IS}^{H}_{X_0}\oplus\mathcal C^H_Σ$. Projection of the variation morphism to the intersection-space summand defines $\operatorname{var}_I:ϕ_π(F)\to\mathcal{IS}^{H}_{X_0}$, and the octahedral axiom gives $P^H\to P^H_I\to\mathcal C^H_Σ\xrightarrow{+1}$, 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 $\mathcal C^H_Σ$ and $Δ_{I/IC}(X_0)$ as geometry-side handoff objects for future DT/BPS comparisons. |
| title | Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds |
| topic | Algebraic Geometry High Energy Physics - Theory Category Theory |
| url | https://arxiv.org/abs/2605.01455 |