Cycle Relations and Global Gluing in Multi-Node Conifold Degenerations

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 $$R_{\mathrm{res}}=R_{\mathrm{sm}}=R_{\mathrm{ext}}=R_{\mathrm{blk}}$$.

• The paper demonstrates that local rank‐one contributions at ODPs are globally constrained by cycle-node incidence data, reducing admissible extension classes.
• It introduces a combinatorial cycle-node incidence datum to encode and formalize the global constraints on nodewise perverse sheaf and mixed Hodge module extensions.
• Under geometric admissibility and block-adaptedness, the work rigorously bridges perverse sheaf theory, mixed Hodge modules, and classical conifold transition techniques.

Cycle Relations and Global Gluing in Multi-Node Conifold Degenerations

Introduction and Problem Formulation

This work analyzes the extension problem for perverse sheaves, mixed Hodge modules, and associated categorical data in the setting of projective one-parameter conifold degenerations with finitely many ordinary double points (ODPs) as singularities of the central fiber. The conventional local theory ascribes a rank-one local contribution per ODP, reflecting the vanishing-cycle structure. However, the central focus of the paper is the nontrivial global constraint on these local sectors: when nodes are interconnected by homological or geometric cycles, the set of global extension classes is strictly smaller than the naive direct sum of local extension data.

To articulate and formalize this phenomenon, the author introduces a cycle-node incidence datum—a combinatorial–geometric object encoding which nodes are related via global cycles—and then determines a geometrically realized subspace of the ambient extension group. Under prescribed admissibility and block-adaptedness conditions, the distinguished extension class arising from the geometry factors through this subspace. The central claim is that global geometry, not just local singularity content, rigidifies or identifies extension directions.

Local and Global Structures in Finite-Node Conifold Degenerations

Degenerations of complex threefolds with isolated ODPs are prototypical for topological transitions (notably in Calabi–Yau geometry), with each ODP contributing a localized vanishing cycle, typically viewed via perverse sheaf theory as rank-one skyscraper sheaves. At the perverse-sheaf level, the correction object is realized as

$$\mathcal{P} := \mathrm{Cone}\left( \mathrm{var}_F \right)[-1], \qquad F = \mathbb{Q}_X[3],$$

living in a short exact sequence: $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$ where $IC_{X_0}$ is the intersection complex on the central fiber and $i_{k*}$ is the inclusion of $p_k$. An analogous refinement exists in the category of mixed Hodge modules, with each singular summand being a Tate-twisted, point-supported piece.

A categorical schober (perverse schober) structure further enhances the local-to-global picture, with one local sector per node and a "quiver shadow" encoding formal decategorification. However, the global extension class is naively an element of an $r$-dimensional vector space (one per node), but is not realized freely in the geometric category.

Cycle-Node Incidence and Relation Law

When nodes are linked via distinguished global cycles (holomorphic or topological), their extension classes become correlated. The author formalizes this via a cycle-node incidence datum, a map encoding which nodes are collectively constrained. The key construction is an incidence morphism

$$\iota_{\mathcal{C}} \colon \mathbb{Q}^A \to \mathbb{Q}^r,$$

where $A$ is the set of cycles and $r$ is the number of nodes. This map prescribes which nodewise extension classes are geometrically admissible, i.e., which combinations arise from the geometry. The image of this map, $$V_{\mathrm{geom}} = \mathrm{Im}(\iota_{\mathcal{C}})$$, is the subspace through which the corrected extension class factors.

Block-adaptedness refers to cases where the cycles partition the nodes into blocks such that the image consists of coefficient vectors constant within each block. For example, if two nodes are incident to the same cycle and another node is incident to a distinct one, the global extension space is, up to isomorphism, $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$0.

Main Theorems and Dimensionality Statements

Under geometric admissibility and block-adaptedness assumptions on the incidence datum, the following hold:

• The corrected perverse extension class and its mixed Hodge module counterpart factor through $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$1, a dimensionally reduced subspace of the ambient extension space.
• The dimension of the realized geometric extension space is $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$2, possibly much less than $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$3.

Specifically, in the block-separated cycle family—a setting where nodes within a block are constrained to move together—the relations match precisely those seen in homology among exceptional curves in resolutions and vanishing cycles in smoothings. The main comparison theorems assert this equality of relation spaces: $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$4 where these are the respective kernels of maps from the formal nodewise space to homology (resolution, smoothing) or the perverse/mixed Hodge extension quotient.

Lifting to Mixed Hodge Modules and Schober/Quiver Shadows

The incidence relation law is maintained upon passage to the mixed Hodge module category, compatible with Saito's realization functor. All such relation constraints on the perverse extension space have compatible counterparts in the realized and motivic settings.

On the categorified side, the quiver shadow (as the decategorification of a perverse schober) inherits a block structure, not only reflecting the node content but enforcing that global coupling data (walls, transport, or BPS invariants) must live in the restricted space $$0 \to IC_{X_0} \to \mathcal{P} \to \bigoplus_{k=1}^r i_{k*} \mathbb{Q}_{\{p_k\}} \to 0,$$5.

Examples and Model Configurations

Model configurations, such as two nodes constrained by one cycle or three nodes grouped into two blocks, explicitly illustrate the reduction in admissible extension directions. In these settings, the admissibility criteria are realized geometrically by families analytically locally trivial along cycle components off the nodes, ensuring that the actual extension space is a one- or two-dimensional subspace of the formal nodewise cube.

A projective example with symmetry demonstrates the block-separated theory in an explicit global (projective, equivariant) context, with invariant cycles and nodes exchanged under the symmetry.

Implications and Future Directions

The main theoretical implication is a nontrivial reduction in the space of admissible gluing data—correlating the local extension structure with global cycles—thus establishing the first truly global combinatorial and homological invariant of the corrected extension framework. This structure provides a precise bridge between the classical geometric conifold transition theory (resolution/smoothing) and perverse/mixed Hodge theoretic data.

Practically, this has implications for any further analysis involving local-to-global geometry in degenerations, including wall-crossing, BPS state counts, and the construction of transport (scattering) functors, which must now be formulated on the restricted (relation-controlled) space rather than the naive free nodewise arena.

Looking forward, the author points toward two directions: (1) extension of the theory beyond ODP singularities and (2) investigation of how monodromy/transport and wall-crossing operations interact with, or preserve, these cycle-induced block structures, possibly leading to new phenomena in the global study of degenerations and their categorical/physical counterparts.

Conclusion

This paper identifies and characterizes the first genuinely global geometric constraint in the theory of perverse sheaves and their mixed Hodge/categorical avatars for multi-node conifold degenerations. Through the introduction of cycle-node incidence data and rigorous dimensional analysis, it demonstrates that nodewise extension classes are not in general freely determined, but rather are universally constrained by the global cycle geometry of the degeneration. This result sharpens the link between Hodge-theoretic, perverse-sheaf, and classical geometric perspectives, and lays a necessary structural foundation for future advances in degeneration theory, wall-crossing, and categorical transport.