Cycle Relations and Global Gluing in Multi-Node Conifold Degenerations

TL;DR

This paper investigates how global geometric relations influence the extension classes in conifold degenerations, revealing that these classes are constrained by cycle and homological relations, with implications across various mathematical frameworks.

Contribution

It introduces a cycle-node incidence framework and proves that global relations restrict extension classes, connecting local variation data with global geometric structures.

Findings

Corrected perverse extension factors through a relation-controlled subspace.

Relation law lifts compatibly to the mixed-Hodge-module setting.

In block-separated cycle families, various invariants coincide.

Abstract

We study projective one-parameter conifold degenerations whose central fiber has finitely many ordinary double points. Existing finite-node theory isolates one rank-one local sector per node on the perverse-sheaf, mixed-Hodge-module, and categorical sides, but does not determine which global extension classes are actually realized by geometry. We show that when the nodes are linked by common cycle geometry or homological relations, the corrected extension is not free nodewise data, but is forced into a smaller relation-controlled subspace. To formalize this, we introduce a cycle-node incidence datum and the associated geometrically realized subspace of the ambient nodewise extension space. Under geometrically admissible and block-adapted incidence hypotheses, we prove that the corrected perverse extension factors through this subspace, with incidence compatibility derived from propagation of local variation data along admissible cycle components, and we show that the same relation law lifts compatibly to the mixed-Hodge-module setting. We then compare this relation law with the resolution and smoothing sides and, in the block-separated cycle family, obtain $R_{\mathrm{res}}=R_{\mathrm{sm}}=R_{\mathrm{ext}}=R_{\mathrm{blk}}$.