15 papers
From Finite-Node Conifold Geometry to BPS Structures III: Mediated Triangle Transport and Graded Interaction Data
Abdul Rahman
In previous work, we extracted from a finite-node conifold degeneration the state-data package and then constructed the support-level interaction package encode…
From Diaz's Enriques Product to an -Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples
Abdul Rahman
We reinterpret Diaz's construction of Chow-trivial smooth projective varieties violating the integral Hodge conjecture as the level-two case of an \(n\)-fold cup-product Bockstein…
Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds
Abdul Rahman
We prove a projection-triangle statement for projective Calabi--Yau threefold conifold degenerations and use it to organize an intersection-space Hodge atom shadow package. For a n…
Torsion Trajectories from Local Discriminants to Global Obstructions
Abdul Rahman
For a normal surface singularity, the discrepancy between the ordinary and dual middle-perversity intersection complexes over \(\mathbb Z\) is measured by a finite group \(E\). In…
Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy
Abdul Rahman
For a germ of a normal complex analytic surface, let , where and denote the ordinary and dual middl…
Interacting Multi-Node Conifold Light Sectors
Abdul Rahman
We study finite-node conifold degenerations of Calabi--Yau threefolds from the point of view of interacting light sectors. Although each ordinary double point contributes a rank-on…
From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly
Abdul Rahman
In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form , where is the finite node-in…
From Finite-Node Conifold Geometry to BPS Structures I: Algebraic State Data
Abdul Rahman
Let be a one-parameter degeneration whose central fiber is a complex threefold with finitely many ordinary double points . Associat…
Hodge Atoms at Conifold Degenerations: F-Bundles, Limiting Mixed Hodge Modules, and the Rigid-Flexible Decomposition
Abdul Rahman
We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration who…
Cycle Relations and Global Gluing in Multi-Node Conifold Degenerations
Abdul Rahman
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 secto…
GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural Kähler Potentials
Abdul Rahman
We present \emph{GlobalCY}, a JAX-based framework for globally defined and symmetry-aware neural Kähler-potential models on projective hypersurface Calabi--Yau geometries. The cent…
Finite-Node Perverse Schobers and Corrected Extensions for Conifold Degenerations
Abdul Rahman
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…
Mixed Hodge Modules and Canonical Perverse Extensions for Multi-Node Conifold Degenerations
Abdul Rahman
We study one-parameter conifold degenerations whose central fiber \(X_0\) has finitely many ordinary double points and construct a mixed-Hodge-module refinement of the corrected pe…
Perverse Extensions and Limiting Mixed Hodge Structures for Conifold Degenerations
Abdul Rahman
Let be a one-parameter degeneration whose central fiber has a single ordinary double point. The nearby- and vanishing-cycle formalism determines a canonical perver…
Perverse Schober Structures for Conifold Degenerations
Abdul Rahman
We study a one parameter degeneration of Calabi Yau threefolds whose central fiber contains a single ordinary double point. Using the nearby and vanishing cycle formalism, we const…