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

We study one-parameter conifold degenerations whose central fiber has finitely many ordinary double points and construct a mixed-Hodge-module refinement of the canonical corrected perverse object associated with the degeneration. We build a rank-one point-supported mixed-Hodge-module block at each node, identify the global singular quotient as ${⨁}_{k=1}^r{i}_{k\ast }{\text{Q}}_{\{p_k\}}^H(-1)$, and assemble these local blocks via Saito’s divisor-case gluing formalism into a global object ${\mathcal{P}}^H\in MHM(X_0)$. We prove that ${\mathcal{P}}^H$ realizes the corrected perverse object, fits into an exact sequence $0\to I{C}_{X_0}^H\to {\mathcal{P}}^H\to {⨁}_{k=1}^r{i}_{k\ast }{\text{Q}}_{\{p_k\}}^H(-1)\to 0$, and that the same quotient realizes the finite local vanishing sector in the nearby-cycle formalism. We further relate the mixed-Hodge-module extension, its realized perverse extension, and the induced extension on hypercohomology carrying the limiting mixed Hodge structure. This gives a theorem-level Hodge-theoretic refinement of the corrected perverse extension in the finite multi-node ordinary double point setting.