Finite-Node Perverse Schobers and Corrected Extensions for Conifold Degenerations

TL;DR

This paper develops a finite-node categorical formalization for conifold degenerations with multiple nodes, extending perverse schober theory and integrating mixed-Hodge modules, without claiming universality.

Contribution

It introduces a minimal finite-node formalism for conifold degenerations, formalizes local and global data, and connects perverse schober extensions with mixed-Hodge modules.

Findings

Established compatibility of shadows with corrected perverse extension

Isolated localized categorical sectors for each node

Created a combinatorial skeleton encoding nodewise patterns

Abstract

We study one-parameter conifold degenerations whose central fiber has finitely many ordinary double points. Working within a deliberately minimal finite-node bulk/localized-sector formalism, we identify the first categorical layer suggested by the corrected finite-node perverse extension and its mixed-Hodge-module package. Assuming that the local ordinary-double-point coupling pattern admits categorical realization in this finite-node setting, we formalize the corresponding local and finite-node data over a chosen bulk category, prove compatibility of their specified shadows with the corrected finite-node perverse extension established in earlier work, and isolate one localized categorical sector per node. We also extract a first finite combinatorial skeleton encoding the nodewise coupling pattern. The paper does not claim a universal perverse-schober theory for arbitrary singular Calabi--Yau degenerations, nor a categorical wall-crossing theory. Rather, it provides the foundational finite-node categorical formalization layer above the corrected perverse and mixed-Hodge-module packages in the conifold degeneration.