Stability conditions on free abelian quotients
Pith reviewed 2026-05-24 07:47 UTC · model grok-4.3
The pith
There is an analytic isomorphism between G-invariant geometric stability conditions on a smooth projective variety and hat G-invariant geometric stability conditions on its free abelian quotient.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show there is an analytic isomorphism between G-invariant geometric stability conditions on the cover and geometric stability conditions on the quotient that are invariant under the residual action of the group hat G of irreducible representations of G. We apply our results to describe a connected component inside the stability manifolds of free abelian quotients when the cover has finite Albanese morphism. This applies to varieties with non-finite Albanese morphism which are free abelian quotients of varieties with finite Albanese morphism, such as Beauville-type and bielliptic surfaces. We also give counterexamples to a conjecture of Fu--Li--Zhao concerning the Le Potier function, which
What carries the argument
The analytic isomorphism between spaces of G-invariant geometric stability conditions on the cover and hat G-invariant geometric stability conditions on the quotient.
If this is right
- A connected component of the stability manifold can be described explicitly for free abelian quotients of varieties with finite Albanese morphism.
- Varieties with non-finite Albanese morphism that arise as such quotients admit non-geometric stability conditions.
- The Le Potier function conjecture fails in some cases, with explicit counterexamples provided.
- Geometric stability conditions on any surface are classified using a refined version of the Le Potier function.
Where Pith is reading between the lines
- The correspondence may allow explicit computation of stability manifolds for additional classes of quotients by reducing to the cover.
- Similar invariance arguments could be tested on quotients by non-abelian groups or on varieties with non-free actions to see where the isomorphism breaks.
- The refined Le Potier description on surfaces suggests a way to check slope-semistability via Chern class inequalities in higher Picard rank cases.
Load-bearing premise
The finite abelian group G acts freely on the smooth projective variety.
What would settle it
A direct computation on an explicit free quotient such as a bielliptic surface showing that the spaces of invariant geometric stability conditions are not analytically isomorphic.
read the original abstract
We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is an analytic isomorphism between $G$-invariant geometric stability conditions on the cover and geometric stability conditions on the quotient that are invariant under the residual action of the group $\widehat{G}$ of irreducible representations of $G$. We apply our results to describe a connected component inside the stability manifolds of free abelian quotients when the cover has finite Albanese morphism. This applies to varieties with non-finite Albanese morphism which are free abelian quotients of varieties with finite Albanese morphism, such as Beauville-type and bielliptic surfaces. This gives a partial answer to a question raised by Lie Fu, Chunyi Li, and Xiaolei Zhao: if a variety $X$ has non-finite Albanese morphism, does there always exist a non-geometric stability condition on $X$? We also give counterexamples to a conjecture of Fu--Li--Zhao concerning the Le Potier function, which characterises Chern classes of slope-semistable sheaves. As a result of independent interest, we give a description of the set of geometric stability conditions on an arbitrary surface in terms of a refinement of the Le Potier function. This generalises a result of Fu--Li--Zhao from Picard rank $1$ to arbitrary Picard rank.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies slope-stable vector bundles and Bridgeland stability conditions on smooth projective varieties that are free quotients Y = X/G by a finite abelian group G. It claims an analytic isomorphism between the space of G-invariant geometric stability conditions on X and the space of Ĝ-invariant geometric stability conditions on Y. It further claims a description of a connected component of the stability manifold for such quotients when the Albanese morphism of X is finite (with applications to Beauville-type and bielliptic surfaces), a partial answer to a question of Fu–Li–Zhao on the existence of non-geometric stability conditions, counterexamples to a Fu–Li–Zhao conjecture on the Le Potier function, and a generalization to arbitrary Picard rank of the description of geometric stability conditions on surfaces in terms of a refined Le Potier function.
Significance. If the central claims hold, the work supplies a concrete correspondence that transfers stability data between covers and quotients, furnishes explicit counterexamples to an existing conjecture, and extends a Picard-rank-1 result of Fu–Li–Zhao to arbitrary rank. These contributions would be useful for the study of Bridgeland stability on surfaces and their quotients, particularly for varieties with non-finite Albanese morphisms.
minor comments (2)
- [Abstract] The abstract states that the results give 'a partial answer' to the Fu–Li–Zhao question on non-geometric stability conditions; a sentence clarifying the precise scope of this partial answer would improve readability.
- The manuscript invokes the standard equivalence D^b(Y) ≅ D^b_G(X) for free actions; a brief reminder of the precise hypotheses under which this equivalence preserves geometric stability conditions would help readers who are not specialists in equivariant derived categories.
Simulated Author's Rebuttal
We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments appear in the report provided.
Circularity Check
No significant circularity identified
full rationale
The paper's central claims—an analytic isomorphism between G-invariant geometric stability conditions on the cover and Ĝ-invariant ones on the quotient, plus a connected-component description and counterexamples to the Fu–Li–Zhao conjecture—rest on the standard derived equivalence D^b(Y) ≅ D^b_G(X) that holds precisely when G acts freely. This equivalence is invoked as an external fact under the explicitly stated setup assumption rather than derived internally. The generalization of the Le Potier function description to arbitrary Picard rank is presented as an independent extension of prior work by different authors. No load-bearing step reduces by the paper's own equations or self-citations to a fitted parameter, self-definition, or unverified uniqueness theorem; all new statements are independent of the inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Properties of geometric stability conditions and slope stability on smooth projective varieties
- domain assumption Existence and finiteness properties of the Albanese morphism
discussion (0)
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