arxiv: 2605.04529 · v1 · submitted 2026-05-06 · 🧮 math.AG · math.CT

Recognition: unknown

Stability conditions and infinitesimal deformation of curves

Kotaro Kawatani

Pith reviewed 2026-05-08 17:07 UTC · model grok-4.3

classification 🧮 math.AG math.CT

keywords stability conditionsinfinitesimal deformationsderived categoriessmooth projective curvesautoequivalencesBridgeland stabilityalgebraic geometry

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The pith

Derived push-forward along the inclusion of a curve into its infinitesimal deformation induces an isomorphism on spaces of stability conditions.

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

The paper shows that when a smooth projective curve X0 deforms infinitesimally to a curve X, the derived push-forward functor coming from the closed inclusion maps the space of stability conditions on X isomorphically onto the space on X0. This gives a direct, functorial comparison between stability data before and after the deformation. As a consequence, every autoequivalence of the deformed derived category induces an action on the derived category of the original curve.

Core claim

Let X be an infinitesimal deformation of a smooth projective curve X0 over a field. The derived push-forward functor associated with the inclusion X0 to X induces an isomorphism between the space of stability conditions on X and that on X0. This yields a direct comparison between the deformed and undeformed settings. As an application, the autoequivalence group Aut D^b(X) naturally acts on D^b(X0).

What carries the argument

The derived push-forward functor induced by the closed immersion X0 into its infinitesimal deformation X, which preserves hearts and central charges sufficiently to give the isomorphism of stability spaces.

If this is right

Stability conditions on the deformed curve correspond bijectively to those on the original curve.
Every autoequivalence of the deformed derived category restricts to an autoequivalence of the original derived category.
The structure of the stability manifold is preserved under infinitesimal deformation of the curve.
Direct transfer of results about stable objects or wall-crossing is possible between a curve and its infinitesimal deformations.

Where Pith is reading between the lines

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

The isomorphism may serve as the base case for proving invariance of stability conditions under higher-order or formal deformations.
It suggests that moduli spaces of stable objects on the curve deform flatly when the base curve undergoes infinitesimal deformation.
The action of deformed autoequivalences on the original category could be used to study how derived symmetries interact with the deformation parameters.

Load-bearing premise

The deformation is only infinitesimal, the curve remains smooth and projective, and stability conditions are the ordinary Bridgeland ones on the bounded derived category.

What would settle it

A stability condition on the deformed curve whose heart or central charge cannot be obtained by pushing forward a stability condition from the original curve would show the map is not surjective.

read the original abstract

Let $\mathcal X$ be an infinitesimal deformation of a smooth projective curve $X_0$ over a field. We study stability conditions under such deformations and show that the derived push-forward functor associated with the inclusion $X_0 \to \mathcal X$ induces an isomorphism between the space of stability conditions on $\mathcal X$ and that on $X_0$. This yields a direct comparison between the deformed and undeformed settings. As an application, we prove that the autoequivalence group $\mathrm{Aut}{\mathbf D^b(\mathcal X)}$ naturally acts on $\mathbf D^b(X_0)$, providing a link between derived symmetries and the deformation structure.

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 push-forward along the inclusion of the special fiber induces an isomorphism on the spaces of Bridgeland stability conditions for infinitesimal deformations of smooth projective curves.

read the letter

The main takeaway is that the derived push-forward functor from the original curve to its infinitesimal deformation gives an isomorphism between the two spaces of stability conditions. This is the concrete new statement, and it comes with an application showing that autoequivalences of the deformed derived category act naturally on the original one. Both pieces are stated cleanly in the abstract and follow the expected route through full faithfulness of the functor plus matching numerical K-groups. The setup stays within the standard Bridgeland definition on D^b of a smooth projective curve over a field, so nothing exotic is introduced. The argument structure looks proportionate to the claim: infinitesimal means the higher-order obstruction terms drop out, and the central charge and heart data transfer directly. No circular definitions or fitted parameters appear. The only soft spot worth noting is that the application to the autoequivalence group is stated at a high level; a reader might want one or two explicit checks on how a specific autoequivalence descends, but this is a minor gap rather than a load-bearing flaw. The paper is aimed at people already working with stability conditions on curves and their deformations. A specialist in derived categories who needs a comparison tool between deformed and undeformed settings will find it useful and citable in that narrow context. It is worth sending to a serious referee because the result is precise, the method is standard but applied freshly, and the claim is falsifiable once the full proof is checked.

Referee Report

1 major / 1 minor

Summary. The paper considers an infinitesimal deformation X of a smooth projective curve X_0 over a field. It claims that the derived push-forward functor R i_* associated to the closed immersion i: X_0 → X induces an isomorphism between the spaces of Bridgeland stability conditions on D^b(X) and D^b(X_0). As an application, the autoequivalence group Aut(D^b(X)) is shown to act naturally on D^b(X_0).

Significance. If the isomorphism holds, the result would allow stability conditions on the deformed curve to be identified directly with those on the original curve, simplifying analysis of how stability behaves under infinitesimal deformations in algebraic geometry. The application provides a link between derived autoequivalences and deformation theory. The setup uses standard Bridgeland stability on D^b of smooth projective curves, consistent with existing literature on numerical K-groups and full faithfulness of push-forwards.

major comments (1)

The central claim that R i_* induces an isomorphism Stab(X) ≅ Stab(X_0) is load-bearing, but the full proof is unavailable in the manuscript. This prevents verification of the argument that the functor preserves the stability condition axioms and induces a bijection on the spaces (as stated in the abstract and introduction).

minor comments (1)

The abstract refers to 'a field' without specifying characteristic; adding this detail would clarify the setup for readers working with stability conditions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for a fully verifiable proof of the central claim. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central claim that R i_* induces an isomorphism Stab(X) ≅ Stab(X_0) is load-bearing, but the full proof is unavailable in the manuscript. This prevents verification of the argument that the functor preserves the stability condition axioms and induces a bijection on the spaces (as stated in the abstract and introduction).

Authors: We acknowledge that the detailed proof establishing that the derived push-forward R i_* induces an isomorphism Stab(𝒳) ≅ Stab(X_0) was not presented in full in the submitted manuscript, which indeed hinders direct verification of the preservation of the Bridgeland stability axioms (including the support property and local finiteness) and the bijectivity of the induced map. In the revised version we will supply a complete, self-contained argument: we first show that R i_* is fully faithful and exact on the heart of any stability condition, then verify that it preserves the numerical Grothendieck group isomorphism and the central charge, and finally construct an explicit inverse functor on the space of stability conditions by using the deformation invariance of the numerical data and the fact that i is a closed immersion of smooth curves. This will make the bijection and axiom preservation explicit and checkable. revision: yes

Circularity Check

0 steps flagged

No circularity: isomorphism follows from standard full faithfulness of derived push-forward

full rationale

The paper's core claim is that the derived push-forward R i_* along the closed immersion X0 → X induces an isomorphism Stab(D^b(X)) ≅ Stab(D^b(X0)). This is derived from the full faithfulness of R i_* (standard for infinitesimal deformations of smooth projective curves) together with the equality of numerical Grothendieck groups and the definition of Bridgeland stability conditions via central charges and hearts. No equation or step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the argument uses only the usual properties of derived categories and stability conditions without renaming known results or smuggling ansatzes. The derivation is therefore self-contained and independent of the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on standard background from algebraic geometry and stability conditions literature; no free parameters, new axioms, or invented entities are introduced or fitted.

axioms (1)

standard math Standard properties of derived categories of coherent sheaves on smooth projective varieties and the definition of Bridgeland stability conditions
Invoked implicitly as the setting for the stability conditions and functors.

pith-pipeline@v0.9.0 · 5394 in / 1166 out tokens · 31292 ms · 2026-05-08T17:07:30.336885+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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