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arxiv: 2605.03166 · v1 · submitted 2026-05-04 · 🧮 math.AG · math-ph· math.DG· math.MP

Recognition: unknown

Formal moduli and the splitting theory of supermanifolds

Mauricio Corr\^ea, Simone Noja

Pith reviewed 2026-05-08 17:05 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.DGmath.MP
keywords complex supermanifoldssplitting problemAtiyah classGreen obstructionsformal modulidg Lie algebrasL-infinity modelsKuranishi theory
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The pith

The affine Atiyah class encodes the complete Green obstruction tower for splitting complex supermanifolds.

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

The paper develops a formal moduli theory for the splitting problem of complex supermanifolds. It starts from Green's obstruction tower and builds a finite-step filtered dg Lie algebra that controls splittings through filtered Maurer-Cartan theory. The authors prove that the affine Atiyah class already contains this entire tower in a filtered sense, with the Donagi-Witten component supplying the primary obstruction and higher obstructions appearing as successive projected defects and residual classes from the same cocycle. They transfer the setup to a minimal filtered L∞-model whose higher brackets capture the Kuranishi relations among obstruction coordinates and extend the theory to families, yielding relative tangent-obstruction and Kuranishi presentations under standard hypotheses. Explicit examples illustrate non-split cases with residual obstructions and nonlinear relations.

Core claim

In a precise filtered sense, the affine Atiyah class contains the entire Green obstruction tower: the Donagi-Witten component gives the primary obstruction, while the higher obstructions arise as successive projected defects and residual classes of the same Atiyah cocycle. The authors construct a finite-step filtered dg Lie algebra that controls splittings by filtered Maurer-Cartan theory, recover the classical obstruction classes as leading terms of adapted Maurer-Cartan representatives, and transfer the theory to a minimal filtered L∞-model whose higher brackets give the intrinsic Kuranishi relations. They identify the formal neighbourhood of the split section with the fibrewise deformati

What carries the argument

The finite-step filtered dg Lie algebra built from Green's obstruction tower, which controls splittings via filtered Maurer-Cartan theory and whose leading terms recover the classical obstruction classes; equivalently, the affine Atiyah class viewed in this filtered sense.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework suggests that splitting obstructions could be computed by determining the Atiyah class once instead of constructing the full tower step by step.
  • The filtered L∞-model may connect splitting theory to deformation problems in derived algebraic geometry.
  • The formalism could be tested on real supermanifolds or other variants of the splitting problem to see if the same encoding holds.
  • Examples with nonlinear Kuranishi relations point toward possible new higher-order phenomena in supergeometric moduli.

Load-bearing premise

The relative tangent-obstruction and Kuranishi presentations for families hold under standard finiteness, base-change, and descent hypotheses.

What would settle it

A concrete complex supermanifold in which the higher Green obstruction classes fail to match the projected defects and residual classes extracted from the affine Atiyah cocycle.

read the original abstract

We develop a formal moduli theory for the splitting problem of complex supermanifolds. Starting from Green's obstruction tower, we construct a finite-step filtered dg Lie algebra which controls splittings by filtered Maurer-Cartan theory. We prove that the classical successive obstruction classes are recovered as the leading terms of adapted Maurer-Cartan representatives, and we transfer the theory to a minimal filtered $L_\infty$-model whose higher brackets give the intrinsic Kuranishi relations among the obstruction coordinates. We also prove that, in a precise filtered sense, the affine Atiyah class contains the entire Green obstruction tower: the Donagi-Witten component gives the primary obstruction, while the higher obstructions arise as successive projected defects and residual classes of the same Atiyah cocycle. We then pass to families, by constructing the fixed-retract formal moduli problem with prescribed split model and by identifying the formal neighbourhood of the split section with the fibrewise deformation theory of the split model; under standard finiteness, base-change, and descent hypotheses this yields relative tangent-obstruction and Kuranishi presentations. Finally, we work out explicit examples of non-split supergeometries, including cases with residual higher obstructions and a first nonlinear Kuranishi relation. These examples illustrate the interaction between Green obstructions, Atiyah classes, and higher $L_\infty$-operations.

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.

Referee Report

0 major / 3 minor

Summary. The paper develops a formal moduli theory for the splitting problem of complex supermanifolds. Starting from Green's obstruction tower, it constructs a finite-step filtered dg Lie algebra controlling splittings via filtered Maurer-Cartan theory. It proves that the affine Atiyah class encodes the entire Green obstruction tower (Donagi-Witten component as primary obstruction, higher terms as projected defects and residuals), transfers the theory to a minimal filtered L_∞-model whose brackets yield Kuranishi relations, extends the construction to families to obtain relative tangent-obstruction and Kuranishi presentations under standard finiteness/base-change/descent hypotheses, and illustrates the results with explicit examples of non-split supergeometries including residual higher obstructions and nonlinear Kuranishi relations.

Significance. If the central claims hold, the work provides a unified filtered algebraic framework that links Green's obstruction tower, Atiyah classes, and L_∞ structures for the splitting problem in supergeometry. This yields new Kuranishi-type presentations and relative deformation theories for families, which could advance formal moduli problems in algebraic geometry and supergeometry. The explicit examples and the transfer to minimal models are concrete strengths that enhance applicability.

minor comments (3)
  1. The abstract and introduction state several theorems (e.g., the filtered Maurer-Cartan correspondence and the encoding of the Green tower by the Atiyah class) without indicating the precise section or equation where the key filtered correspondence is established; adding forward references would improve readability.
  2. In the discussion of the minimal filtered L_∞-model, the notation for the higher brackets and their relation to Kuranishi relations could be made more explicit by including a small diagram or table summarizing the correspondence between obstruction coordinates and bracket operations.
  3. The examples section would benefit from a brief statement of the base field and dimension assumptions used in the computations to ensure reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary, and positive assessment of the manuscript, along with the recommendation for minor revision. No specific major comments appear in the report, so there are no points requiring point-by-point rebuttal or defense.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper starts from Green's obstruction tower and the affine Atiyah class as given external inputs, then constructs a finite-step filtered dg Lie algebra controlling splittings via filtered Maurer-Cartan theory and transfers it to a minimal filtered L_∞ model whose brackets yield Kuranishi relations. The central claim that the Atiyah class encodes the entire tower (Donagi-Witten as primary, higher terms as projected defects) is proved as a new filtered identification rather than by redefining or fitting the inputs to match outputs. Passage to families and relative presentations relies on standard finiteness/base-change/descent hypotheses without self-definitional loops or fitted parameters renamed as predictions. No load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work appear; the explicit examples are illustrations of the independent constructions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Review performed on abstract only; full definitions of the filtered dg Lie algebra, the precise notion of adapted Maurer-Cartan representatives, and the explicit higher brackets are not supplied, so the ledger cannot be completed exhaustively.

axioms (2)
  • domain assumption Existence and properties of Green's obstruction tower for supermanifold splittings
    The paper starts from this tower and builds the new algebra on top of it.
  • domain assumption Standard finiteness, base-change, and descent hypotheses for families
    Invoked to obtain relative tangent-obstruction and Kuranishi presentations.
invented entities (2)
  • Finite-step filtered dg Lie algebra controlling splittings no independent evidence
    purpose: Encodes all splittings via filtered Maurer-Cartan theory
    New algebraic object constructed from Green's tower
  • Minimal filtered L_infty model no independent evidence
    purpose: Captures intrinsic Kuranishi relations among obstruction coordinates via higher brackets
    Transferred from the dg Lie algebra

pith-pipeline@v0.9.0 · 5542 in / 1310 out tokens · 17870 ms · 2026-05-08T17:05:28.507480+00:00 · methodology

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Reference graph

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