arxiv: 2604.17754 · v1 · submitted 2026-04-20 · 🧮 math.AG · hep-th

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Hodge Atoms at Conifold Degenerations: F-Bundles, Limiting Mixed Hodge Modules, and the Rigid-Flexible Decomposition

Abdul Rahman

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Pith reviewed 2026-05-10 04:26 UTC · model grok-4.3

classification 🧮 math.AG hep-th

keywords Hodge atomsconifold degenerationsCalabi-Yau threefoldsmixed Hodge modulesDubrovin connectionvanishing cyclesintersection matrixrigid-flexible decomposition

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The pith

Hodge atoms of nearby Calabi-Yau fibers decompose into rigid and flexible parts at conifold degenerations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Hodge atoms framework to one-parameter conifold degenerations of Calabi-Yau threefolds whose central fiber has ordinary double points. It constructs a canonical decomposition of the Hodge atoms of the nearby smooth fiber into a rigid atom preserved from the intersection cohomology of the central fiber and flexible rank-one atoms, one for each vanishing cycle. This decomposition attaches to the atom of a corrected mixed Hodge module P^H on the central fiber, which sits in an exact sequence whose non-split extensions are controlled by the intersection matrix of the vanishing cycles. The technical link is the Stokes-Extension Identification, which equates the Stokes matrix of the Dubrovin connection with the variation morphism realized in mixed Hodge modules. A reader would care because the result organizes how Hodge structures change in a controlled way across these degenerations.

Core claim

For a degeneration of a Calabi-Yau threefold with central fiber having r ordinary double points, the Hodge atoms of the nearby smooth fiber admit a rigid-flexible decomposition attached to the corrected mixed Hodge module P^H in MHM(X_0). The rigid atom A(IC^H_{X_0}) is preserved across the degeneration. The flexible atoms A(i_{k*} QQ^H_{p_k}(-1)) supply one rank-one contribution per vanishing cycle. The total degeneration atom A(P^H) fits into an exact sequence of atoms whose non-split structure is governed by the intersection matrix of the vanishing cycles.

What carries the argument

The corrected mixed Hodge module P^H on the central fiber X_0, whose Hodge atom A(P^H) carries the rigid-flexible decomposition, together with the Stokes-Extension Identification that equates the Stokes matrix of the Dubrovin connection to the variation morphism varF under mixed Hodge module realization.

Load-bearing premise

The corrected mixed Hodge module P^H exists in MHM(X_0) with the stated properties, and the Stokes-Extension Identification holds equating the Stokes matrix of the Dubrovin connection to the variation morphism.

What would settle it

A direct computation for any explicit conifold degeneration in which the matrix of the variation morphism varF differs from the Stokes matrix of the Dubrovin connection at the conifold locus would falsify the identification.

read the original abstract

We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration $\pi\colon X \to \Delta$ whose central fiber $X_0$ has $r$ ordinary double points, we construct a canonical rigid-flexible decomposition of the Hodge atoms of the nearby smooth fiber attached to the corrected degeneration object. The rigid atom $A(\IC^H_{X_0})$ is preserved across the degeneration, while the flexible atoms $A(i_{k*}\QQ^H_{\{p_k\}}(-1))$ are rank-one contributions, one for each vanishing cycle. The total degeneration atom $A(P^H)$ is the atom of the corrected mixed Hodge module $P^H\in\MHM(X_0)$ and fits into an exact sequence of atoms whose non-split structure is controlled by the intersection matrix $(\langle\delta_i,\delta_j\rangle)$. The technical core is the Stokes--Extension Identification, which identifies the Stokes matrix of the Dubrovin connection at the conifold locus with the matrix of the variation morphism $\varF\colon\varphi_\pi(F) \to \psi_\pi(F)$ under mixed Hodge module realization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Hodge atoms to conifold degenerations with a rigid-flexible split and a claimed Stokes identification, but the core step looks potentially circular without independent checks.

read the letter

The new piece is a decomposition of the limiting Hodge atoms for a one-parameter conifold degeneration of a Calabi-Yau threefold. The rigid atom comes from the intersection cohomology of the central fiber and stays fixed, while each ordinary double point contributes a rank-one flexible atom tied to its vanishing cycle. These fit into an exact sequence for the atom of a corrected mixed Hodge module P^H, with the extension class controlled by the intersection matrix on the vanishing cycles. The technical move is the Stokes-Extension Identification that equates the Stokes matrix of the Dubrovin connection with the variation morphism under mixed Hodge module realization. This organizes the data in a way that could help track limiting behavior in mirror symmetry calculations. The construction is formally presented and builds directly on the Katzarkov-Kontsevich-Pantev-Yu framework, which is a reasonable starting point. The main concern is that P^H appears tailored so the identification holds, rather than constructed first from the geometry and then shown to satisfy the Stokes property. The abstract and outline do not include an explicit example or a computation that would confirm the functors commute and the extension is non-split in the expected way. Without that, it is difficult to rule out that the non-split structure is imposed rather than derived. The nearby-cycle and weight-filtration compatibilities are asserted but not walked through in enough detail to check independence from the target decomposition. This is the sort of technical paper that algebraic geometers and string theorists working on degenerations might want to see, provided the proofs close the gaps. It is coherent enough on its own terms to go to referees who can verify the mixed Hodge module constructions and the connection to the Dubrovin flat structure.

Referee Report

3 major / 2 minor

Summary. The manuscript extends the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi-Yau threefolds. For a degeneration π: X → Δ with central fiber X_0 having r ordinary double points, it constructs a canonical rigid-flexible decomposition of the Hodge atoms of the nearby smooth fiber attached to the corrected degeneration object P^H ∈ MHM(X_0). The rigid atom A(IC^H_{X_0}) is preserved across the degeneration, while the flexible atoms A(i_{k*} QQ^H_{{p_k}}(-1)) are rank-one contributions, one for each vanishing cycle. The total degeneration atom A(P^H) fits into an exact sequence of atoms whose non-split structure is controlled by the intersection matrix (⟨δ_i, δ_j⟩). The technical core is the Stokes--Extension Identification, which equates the Stokes matrix of the Dubrovin connection at the conifold locus with the matrix of the variation morphism varF : φ_π(F) → ψ_π(F) under mixed Hodge module realization.

Significance. If the central claims hold, the paper advances the understanding of Hodge structures under conifold degenerations by providing a canonical decomposition into rigid and flexible atoms, directly tied to the geometry of vanishing cycles via the intersection matrix. Linking this to limiting mixed Hodge modules and the Dubrovin connection offers a new categorical perspective that may clarify phenomena in mirror symmetry and moduli spaces of Calabi-Yau threefolds. The explicit control of the extension class by geometric data is a notable strength, potentially enabling generalizations to other singularities.

major comments (3)

[Abstract and §3] Abstract and §3 (Construction of P^H): The corrected mixed Hodge module P^H is introduced as existing in MHM(X_0) with atom A(P^H) fitting the claimed exact sequence, but the construction appears defined so that the Stokes--Extension Identification holds by fiat. No independent a priori definition (e.g., via explicit gluing of nearby-cycle data without reference to Dubrovin flat sections) is given, creating a risk that the non-split extension class controlled by ⟨δ_i, δ_j⟩ is not canonically determined.
[§4] §4 (Stokes--Extension Identification): This is the load-bearing technical core equating the Stokes matrix of the Dubrovin connection with the matrix of varF : φ_π(F) → ψ_π(F) under MHM realization. The argument does not verify that the MHM realization commutes with the flat connection at the conifold point or that the weight filtration on the atoms is compatible, which is required to establish invariance of the rigid atom A(IC^H_{X_0}).
[§5] §5 (Exact sequence for A(P^H)): The claim that A(P^H) fits into a non-split exact sequence of atoms with extension class given by the intersection matrix is central to the rigid-flexible decomposition. The manuscript asserts this without an explicit computation or check against known vanishing-cycle data for conifolds (e.g., for r=1 or r=2), leaving open whether the sequence is indeed non-split as stated.

minor comments (2)

[Notation] Notation throughout: The functors φ_π, ψ_π, varF, and their relation to F-bundles and limiting mixed Hodge modules would benefit from a short preliminary table or diagram summarizing the relevant exact sequences and weight filtrations.
[References] References: Update citations to the Hodge atoms framework to the most recent version of Katzarkov--Kontsevich--Pantev--Yu and add references to existing literature on mixed Hodge modules at ordinary double points if not already present.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and have revised the manuscript to strengthen the exposition and add explicit verifications where needed.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (Construction of P^H): The corrected mixed Hodge module P^H is introduced as existing in MHM(X_0) with atom A(P^H) fitting the claimed exact sequence, but the construction appears defined so that the Stokes--Extension Identification holds by fiat. No independent a priori definition (e.g., via explicit gluing of nearby-cycle data without reference to Dubrovin flat sections) is given, creating a risk that the non-split extension class controlled by ⟨δ_i, δ_j⟩ is not canonically determined.

Authors: We thank the referee for this observation. The original presentation of P^H in §3 was indeed phrased in a way that could suggest the construction presupposes the identification. In the revised manuscript we have added an independent definition of P^H as the unique extension in MHM(X_0) obtained by gluing the intersection cohomology sheaf IC^H_{X_0} with the rank-one skyscraper modules i_{k*} Q^H_{p_k}(-1) via the variation morphism varF, using only the local geometry of the ordinary double points and the intersection pairing on vanishing cycles. The Stokes--Extension Identification is then proved as a theorem rather than imposed by definition, so that the extension class is canonically fixed by the geometric data. revision: yes
Referee: [§4] §4 (Stokes--Extension Identification): This is the load-bearing technical core equating the Stokes matrix of the Dubrovin connection with the matrix of varF : φ_π(F) → ψ_π(F) under MHM realization. The argument does not verify that the MHM realization commutes with the flat connection at the conifold point or that the weight filtration on the atoms is compatible, which is required to establish invariance of the rigid atom A(IC^H_{X_0}).

Authors: We agree that these compatibilities require explicit verification. The revised §4 now contains a new subsection establishing that the mixed Hodge module realization functor commutes with the flat sections of the Dubrovin connection at the conifold locus (using regularity of the singularity and the fact that nearby-cycle functors preserve the weight filtration for the relevant mixed Hodge modules). This directly implies the invariance of the rigid atom A(IC^H_{X_0}) across the degeneration. revision: yes
Referee: [§5] §5 (Exact sequence for A(P^H)): The claim that A(P^H) fits into a non-split exact sequence of atoms with extension class given by the intersection matrix is central to the rigid-flexible decomposition. The manuscript asserts this without an explicit computation or check against known vanishing-cycle data for conifolds (e.g., for r=1 or r=2), leaving open whether the sequence is indeed non-split as stated.

Authors: We accept that low-rank checks strengthen the exposition. While the general proof in §5 already relies on the non-degeneracy of the intersection matrix of vanishing cycles (which is standard for conifolds), we have added a remark with explicit computations: for r=1 the extension class is the self-intersection -2, yielding a non-split sequence; for r=2 the 2×2 intersection matrix produces a non-split extension class. These checks confirm the claim in small cases, and the general argument follows identically from the same non-degeneracy. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent MHM constructions

full rationale

The paper constructs the rigid-flexible decomposition of Hodge atoms for the corrected mixed Hodge module P^H at conifold degenerations, with the Stokes--Extension Identification presented as the technical core equating Stokes matrices to variation morphisms under MHM realization. No quoted equations or steps in the abstract reduce the central claims (exact sequence for A(P^H), invariance of rigid atom A(IC^H_{X_0}), or non-split extension controlled by intersection matrix) to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The framework extends prior Hodge atoms work but does not exhibit the forbidden patterns; the identification is asserted as a proven equivalence rather than imposed by construction. This is the expected honest non-finding for a construction paper whose core steps remain externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Ledger populated from abstract only; the paper relies on the prior Hodge atoms framework and introduces the corrected degeneration object and the Stokes-Extension Identification without independent evidence supplied here.

axioms (2)

domain assumption The Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu applies to the smooth fibers of the degeneration.
The paper states it extends this framework to conifold degenerations.
ad hoc to paper There exists a corrected mixed Hodge module P^H on the central fiber whose atom is the total degeneration atom.
Introduced in the abstract as the source of the total atom A(P^H).

invented entities (2)

Rigid atom A(IC^H_{X_0}) no independent evidence
purpose: Preserved Hodge atom across the degeneration
Defined as the rigid part of the decomposition.
Flexible atoms A(i_{k*} QQ^H_{{p_k}}(-1)) no independent evidence
purpose: Rank-one contributions from each vanishing cycle
One per ordinary double point in the central fiber.

pith-pipeline@v0.9.0 · 5529 in / 1537 out tokens · 49863 ms · 2026-05-10T04:26:44.687293+00:00 · methodology

discussion (0)

Reference graph

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