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A Residual-Excess Framework for Defect Resolutions: Finite-Node Saito Gluing and Integral Hodge Obstructions

Authors: Rahman, Abdul;

A Residual-Excess Framework for Defect Resolutions: Finite-Node Saito Gluing and Integral Hodge Obstructions

Abstract

Defect data attached to a morphism admit two orthogonal refinements. The first is factorization sensitive: a factorization \( A\xrightarrow{u}B\xrightarrow{v}A^\dagger \) with vu = R, gives a residual triple (Cone(u), Cone(v), Cone(R)) constrained by the octahedral axiom. The second is coefficient-sensitive: an excess object is the cone of a comparison morphism between integral or finite-coefficient residual data and its rational or mixed-Hodge realization. We organize these two directions into a residual–excess matrix. The main residual witness is a finite ordinary-double-point conifold degeneration. Saito divisor gluing produces a node-supported Hodge interface \( W^H_\Sigma=\bigoplus_{p_k\in\Sigma}i_{k*}\mathbb Q^H_{\{p_k\}}(-1) \) factoring finite-node monodromy through \( \psi^H_{\pi,1}\to W^H_\Sigma\to\psi^H_{\pi,1}(-1) \). The variation cone yields the corrected extension \( 0\to IC^H_{X_0}\to P^H_{\mathrm{var},\Sigma}\to W^H_\Sigma\to0 \), whose class decomposes into nodewise Ext residual classes. Ordinary double points calibrate the zero-excess regime because their Milnor fibers are integrally torsion-free. Diaz’s Enriques-product Bockstein mechanism calibrates the nonzero-excess regime through integral torsion killed by rationalization. The resulting framework compares finite-node Saito gluing with integral Hodge obstruction channels without conflating rational residual data with coefficient-change defects.

Keywords

mixed Hodge modules, Saito gluing, nearby cycles, vanishing cycles, conifold degenerations, ordinary double points, monodromy, perverse sheaves, residual triples, excess defects, octahedral axiom, integral Hodge obstructions, Computer Science and Mathematics, Geometry and Topology, 14D06, 14F10, 14F08, 14F45, 32S35, 18G80

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
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