Mixed Hodge Modules and Canonical Perverse Extensions for Multi-Node Conifold Degenerations

Abstract

We study one-parameter conifold degenerations whose central fiber \(X_0\) has finitely many ordinary double points and construct a mixed-Hodge-module refinement of the corrected perverse variation-cone object associated with the degeneration. At each node \(p_k\), the local \(\can^H/\Var^H/N\) geometry determines a rank-one point-supported interface block \(W_k^H\simeq i_{k*}\Q^H_{\{p_k\}}(-1)\). These blocks assemble canonically into \(Q_Σ^H=\bigoplus_{k=1}^r W_k^H\), through which the node-supported monodromy factors. We prove that the perverse-normalized cone \(\Cone(\Var^H)\) is concentrated in the mixed-Hodge-module heart, thereby defining \(\mathcal P^H\in MHM(X_0)\), and that \(\rat(\mathcal P^H)\simeq\mathcal P\). In the forward Yoneda orientation, the point-supported quotient is instead \(R_Σ^H\simeq\bigoplus_{k=1}^r i_{k*}\Q^H_{\{p_k\}}(-2)\), exhibiting a Hodge-theoretic distinction invisible after rational realization. We further prove a canonical nodewise decomposition of the rational perverse extension space and construct a morphism from the formal local Hodge sector to the independently defined limiting mixed Hodge structure. Its image records the node contributions that survive in global limiting cohomology, allowing nontrivial global relations among vanishing cycles.

arXiv commentStrengthened the local and global mixed-Hodge-module constructions, clarified Tate normalization and Yoneda orientations, expanded the nodewise Ext and LMHS comparisons, added technical appendices, and revised the exposition, citations, notation, and figures throughout