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

TL;DR

This paper constructs a refined mixed-Hodge-module framework for conifold degenerations with multiple nodes, providing a Hodge-theoretic enhancement of the canonical perverse extension in such degenerations.

Contribution

It introduces a new mixed-Hodge-module refinement of the corrected perverse object for multi-node conifold degenerations, linking local and global Hodge structures.

Findings

Constructed a global mixed-Hodge-module object $ ext{P}^H$ for the degeneration.

Proved $ ext{P}^H$ fits into an exact sequence with the intersection complex and local contributions.

Established the relation between the extension, perverse extension, and limiting mixed Hodge structure.

Abstract

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 $\bigoplus_{k=1}^r i_{k*}\Q^H_{\{p_k\}}(-1)$, 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 $0 \to IC^H_{X_0} \to \mathcal P^H \to \bigoplus_{k=1}^r i_{k*}\Q^H_{\{p_k\}}(-1) \to 0$, 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.