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Derived isogenies between abelian varieties

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Pith's one-line read In characteristic zero, two abelian varieties of dimension at least two are derived isogenous if and only if they are connected by an isogeny whose degree is a perfect square.

desk verdict Substantial, likely-correct resolution of the twisted Torelli question, with a clean derived-isogeny criterion; send to referees but ask for full proofs of the sketched technical steps. read the letter →

arxiv 2510.22612 v3 pith:ENTGU3JK submitted 2025-10-26 math.AG

Zhiyuan Li , Ziwei Lu , Zhichao Tang This is my paper · ORCID
classification math.AG MSC 14F0814K05
keywords derivedcategoriesabelianvarietiestwistedsheavesequivariantprincipalisogenysymplecticisomorphismTorelli
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that for abelian varieties of dimension at least two over an algebraically closed field of characteristic zero, the category-theoretic notion of derived isogeny—being linked by a chain of derived equivalences between twisted abelian varieties—coincides exactly with the classical notion of principal isogeny, i.e., the existence of an isogeny whose degree is a perfect square. The route passes through a twisted derived Torelli theorem: a derived equivalence between twisted abelian varieties produces a symplectic isomorphism of canonical symplectic abelian varieties, and in characteristic zero the converse holds. The proof introduces a twisted Orlov functor that transports categorical equivalences into geometric isomorphisms, and decomposes every principal isogeny into spectrally paired isogenies, each step induced by a twisted derived equivalence. The result closes the question raised for abelian surfaces and extends the characterization to arbitrary dimension.

What carries the argument

The twisted Orlov functor Ξ_{X,α}: D^b(A(X,α), q^*α) → D^b(X×X, α^{-1}⊠α), built through equivariant derived categories for the action of X[n] on X×X twisted by a 2-cocycle representing α. It is the bridge that converts a Fourier–Mukai equivalence between twisted derived categories into an isomorphism of symplectic abelian varieties, by showing that each point z∈A(X,α) acts as an autoequivalence and that the adjoint action is tensor product with a line bundle. The other half is the decomposition theorem: any principal isogeny of degree a perfect square factors, after multiplication by [N], into spectrally paired isogenies—isogenies whose kernel is a product of groups (Z/aZ)^2—and each such i

What would settle it

Compute the two sides of Proposition 5.2 explicitly for an elliptic curve X with a nontrivial 2-torsion Brauer class and check whether the diagram (5.7) commutes for all n-torsion translations; a single failing step would break the construction. Alternatively, try to find two abelian varieties of dimension ≥2 over C admitting a principal isogeny of square degree that cannot be factored into spectrally paired isogenies—Theorem 7.6 asserts no such pair exists.

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Extended reading notes

Core claim

The central claim is that derived isogeny and principal isogeny are the same relation for abelian varieties of dimension g ≥ 2 over algebraically closed fields of characteristic zero: X and Y are derived isogenous if and only if there exists an isogeny X→Y whose degree is a perfect square (and for elliptic curves, derived isogeny forces isomorphism). The engine is a derived Torelli theorem for twisted abelian varieties: any derived equivalence D^b(X1,α1) ≃ D^b(X2,α2) gives a symplectic isomorphism ψ: A(X1,α1) → A(X2,α2) between the canonical symplectic quotients of Xi×X̂i, and in characteristic zero this condition is also sufficient. This makes the categorical invariant of derived isogeny co

Load-bearing premise

The load-bearing premise is the canonical identification, in Proposition 5.2, of the equivariant category D^b(X×X)^{G,η,a} with the twisted category D^b(X×X, μ^*(α^{-1}⊠α)); this bridge is checked by diagram chasing, and if the identification fails the twisted Orlov functor cannot produce the required symplectic isomorphism.

Editorial extensions

If this is right

  • Derived isogeny classes of abelian varieties of dimension ≥2 over C are exactly principal isogeny classes; the categorical relation adds no extra structure beyond square degree.
  • Specialization: if the generic fibers of two families of abelian varieties over a DVR are derived isogenous, then the special fibers are derived isogenous (when the residue field has characteristic 0).
  • Kuga–Satake varieties: two complex projective K3 surfaces that are derived isogenous have derived-isogenous Kuga–Satake abelian varieties, provided those have dimension at least 2.
  • In positive characteristic p>2, a prime-to-p principal isogeny between abelian varieties of dimension ≥2 implies derived isogeneity; the remaining inseparable case is left open.
  • For elliptic curves, derived isogeny is the same as isomorphism, since Brauer groups are trivial and derived equivalence implies isomorphism for curves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The equivalence suggests that the group of derived autoequivalences up to 'isogeny' is controlled purely by the symplectic and Hodge-theoretic data of A(X,α), so finer categorical invariants may be redundant for detecting isogeny.
  • The decomposition into spectrally paired isogenies is an arithmetic statement about matrices in GL_{2g}(Z); it may be reusable in other contexts where isogeny factoring by square kernels is needed, such as constructing Kuga–Satake varieties or moduli of abelian varieties.
  • A natural test is whether Theorem 1.6 extends to positive characteristic: the paper proves only the prime-to-p direction, leaving open whether inseparable principal isogenies also induce derived isogenies; a counterexample there would sharpen the boundary.
  • The twisted derived Torelli theorem may lead to a modular description of twisted Fourier–Mukai partners for abelian varieties of arbitrary dimension, paralleling known surface results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proves a twisted derived Torelli theorem for abelian varieties and uses it to characterize derived isogenies. For twisted abelian varieties (X,α) with ord(α) invertible in k and char(k)≠2, a derived equivalence D^b(X1,α1)≃D^b(X2,α2) is shown to induce a symplectic isomorphism A(X1,α1)≅A(X2,α2); the converse is obtained in characteristic zero via Polishchuk's machinery (Theorem 1.1). The main application is Theorem 1.6: over an algebraically closed field of characteristic zero, two abelian varieties of dimension g≥2 are derived isogenous if and only if they are principally isogenous, i.e. connected by an isogeny whose degree is a perfect square; for elliptic curves the condition is isomorphism. The proof strategy follows Orlov's method: a twisted Orlov functor Ξ_{X,α} identifies D^b(A(X,α), q^*α) with D^b(X×X, α^{-1}⊠α), and a derived equivalence Φ induces an isomorphism of the associated symplectic abelian varieties. Principal isogenies are then decomposed into 'spectrally paired' isogenies, each giving a twisted derived equivalence. The paper also gives a Hodge-theoretic criterion over C and applications to Kuga–Satake varieties. The overall architecture is coherent and the main theorems are plausible, but several load-bearing technical identifications are only sketched.

Significance. If the central results are correct, the paper resolves a natural question posed in [LZ25] and provides a complete categorical characterization of derived isogeny classes of abelian varieties in characteristic zero, extending the surface case to all dimensions. The twisted Orlov functor and the 'spectrally paired isogeny' decomposition are new structural tools that are likely to be useful beyond this paper. The authors use external results (Orlov, Polishchuk, Beckmann–Oberdieck, Elagin) rather than assuming the target theorem, and I find no circularity in the main argument. However, the correctness of Theorem 1.1 currently rests on unverified cocycle identifications in Proposition 5.2 and on a proof sketched in Proposition 6.1; conversely, the decomposition theorem for principal isogenies (Theorem 7.6) has a concrete algebraic error in the displayed factorization. These are load-bearing gaps, so the paper needs substantial revision before the claims can be accepted.

major comments (3)
  1. [§5.2 (Prop. 5.2, eqs. (5.5)–(5.8))] The proof of the canonical equivalence (5.3) is the bridge between the equivariant action and the twisted Orlov functor, and it is not fully verified. After restricting to 0×G and passing to the quotient action (G×0, ρ̄, ā), the text asserts that Ψ identifies this action with the (G, η, a)-action because 'both are canonically twisted by the same 2-cocycle a'. This assertion contains two sign-sensitive data that must be checked explicitly: (i) the character by which ρ̄ acts on the 0×G-linearization must be exactly ⟨σ′, φ_α(σ)⟩, i.e. the descent character of P_{−φ_α(σ)}; (ii) the quotient cocycle must be a_{σ1,σ2}, not a_{σ2,σ1} or a_{σ1,σ2}^{-1} times a cross-term. The diagram (5.7) and the surrounding formulas give the ingredients, but the actual verification is omitted. Since Propositions 5.4, 5.7 and hence Theorem 1.1 all depend on this identification, a complete proof—or a precise ref
  2. [§6.1 (Prop. 6.1)] Proposition 6.1 is the step that turns the composed equivalence Ψ into an isomorphism of twisted abelian varieties, but its proof is only a sketch. It asserts that Ψ sends skyscrapers on an open neighborhood U of 0_{A1} to skyscrapers, and then claims that these observations suffice to imply that Ψ sends every skyscraper to a skyscraper and is induced by an isomorphism ψ plus a line bundle N. The reduction is not fully justified: the cited results [CS07, Cor. 5.3] and [Huy06, Cor. 5.23] require checking their hypotheses, and the passage from 'open neighborhood' to 'all skyscrapers' via the diagram with w1 is compressed. Additionally, Step 2 of the proof of Proposition 5.7, which identifies the induced morphism ψ with the symplectic isomorphism ψ_α, is asserted to follow from 'explicit construction' and a commutative diagram (5.14) whose derivation is not shown. Because Proposition 6.1 is
  3. [§7.2 (Thm. 7.6, eq. (7.5))] The decomposition theorem for principal isogenies has a concrete algebraic error. In (7.5), the first factor on the right is diag(a_1 d_{2g}, …, a_{2g-3} d_{2g}, a_{2g-2}, 1, 1). For g=3, take d_1=…=d_5=1 and d_6=4; then a=(1,1,1,1,1,4) and the first factor is diag(4,4,4,1,1,1). Its cokernel is (Z/4)^3, which is not spectrally paired since 3 is odd. Thus the displayed factorization does not satisfy the theorem's own requirement that each integral factor have spectrally paired cokernel. The subsequent induction 'by repeatedly applying this decomposition' also needs a more general formulation, since after one step the new diagonal matrix need not retain the divisibility chain a_i = ∏_{j≤i} d_j. The theorem may still be true, but the proof as written is invalid. This decomposition is essential for the converse direction of Theorems 1.6 and 1.7, so it must be repaired or replaced by a correc
minor comments (5)
  1. [§1.2 (Def. 1.4)] The diagram (1.1) is hard to parse as printed; labeling the equivalences and explicitly indicating that each arrow is an equivalence of twisted derived categories would improve readability.
  2. [§2.3 (Prop. 2.5)] The proof states 'The interpretation yields a canonical equivalence'; it would help to note that the equivalence D^b(X,α)≃D^b(Y)^{G,ϱ,σ} depends on the chosen lift [σ] of α, and to specify the map H^2(G,G_m)→Br(X) more explicitly in terms of the finite étale cover.
  3. [§3.1 (Def. 3.1)] In the definition of a Lagrangian subvariety, the expression 'X ≃ [A/X' is missing a closing bracket; it should read 'A/X'.
  4. [§3.3 (Prop. 3.6)] In the proof, the displayed set contains a typo: 'A (x,α)' should be 'A(X,α)'.
  5. [§7.4 (Cor. 1.9)] The sentence 'If two complex projective K3 surfaces over are derived isogenous' contains a stray 'over'. Also, the dimension condition for Kuga–Satake varieties is stated as 'at least 2' in Corollary 1.9 but as 'greater than 2' in Remark 7.7; the relationship should be clarified.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: central claims are proved from Orlov/Polishchuk and explicit linear-algebra decompositions; [LZ25] self-citation is terminological and motivational only.

full rationale

The derived-isogeny characterization is not built from its conclusion. In the derived-to-principal direction, a twisted derived equivalence is fed into Theorem 1.1, whose proof constructs a symplectic isomorphism via the twisted Orlov functor (Prop. 5.2) and then invokes Polishchuk's Theorem 3.4 to obtain a principal isogeny; each step is a separate mathematical theorem, not a restatement of Definition 1.5. In the converse direction, Theorem 7.6 decomposes a principal isogeny into spectrally paired isogenies by an explicit matrix factorization of diag(a_i), and Corollary 7.5 realizes each spectral isogeny by a twisted derived equivalence through Theorem 7.3 plus Polishchuk's symplectic-to-derived criterion. The choices of α and β in the spectral construction are existential choices allowed by Definition 1.4, so this is construction, not a fitted parameter renamed as a prediction. The only self-citation, [LZ25], supplies the term 'principal isogeny', the abelian-surface antecedent, and motivation; it is not invoked in the proof of Theorem 1.6 for g≥2 or in the proof of Theorem 1.1. The 'one can check' identification in Prop. 5.2 is a computational proof obligation about cocycles and characters; even if it contained a sign error it would be a correctness defect, not a circular reduction forced by the definitions of the target claims. The main chain is anchored in independent external results (Orlov 2002, Polishchuk 1996, Elagin 2015, Ploog, Huybrechts), so the derivation is self-contained apart from normal reliance on the literature.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants or data-derived parameters appear; this is a pure mathematics proof. The paper introduces new definitions (derived isogeny, spectrally paired isogeny, twisted Orlov functor), but these are internal proof devices, not postulated entities with independent empirical handles. The main unproved inputs are standard derived-category theorems from the cited literature, plus the stated field/characteristic restrictions.

assumptions (5)
  • standard math Orlov–Polishchuk derived Torelli theorem for untwisted abelian varieties
    Invoked as external benchmark in §1.1 and used as background; not re-proved in the paper.
  • standard math Elagin's equivalence D^b(Y)^G ≃ D^b(X) for finite étale Galois covers when gcd(|G|,char(k))=1
    Used in Proposition 2.5 and Corollary 2.6 to identify twisted categories with equivariant categories.
  • standard math Polishchuk's symplectic-biextension correspondence ([Pol96, Theorem 4.2])
    Used in Theorem 3.4 and Theorem 7.3 to pass from symplectic isomorphisms to twisted derived equivalences; the finiteness/characteristic conditions are assumed as stated.
  • domain assumption Algebraically closed field k with char(k) ≠ 2 and ord(α_i) coprime to char(k)
    Theorems 1.1, 1.6, and 1.7 are stated under these field/characteristic conditions; characteristic zero is the main case.
  • standard math Homogeneous finite abelian group property cited from [CF91]/[Lam07]
    Lemma 7.2 depends on this classification property to realize a spectrally paired subgroup as the image of an anti-symmetric homomorphism.

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Pith. "Pith review of Derived isogenies between abelian varieties." pith.science (2026). https://pith.science/paper/ENTGU3JK

@misc{pith2026251022612,
  author       = {Pith},
  title        = {Pith review of: Derived isogenies between abelian varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENTGU3JK}},
  note         = {Machine review of arXiv:2510.22612}
}
abstract

In this paper, we establish a derived Torelli Theorem for twisted abelian varieties. Starting from this, we explore the relation between derived isogenies and classical isogenies. We show that two abelian varieties of dimension $\geq 2$ are derived isogenous if and only if they are principally isogenous over fields of characteristic zero. This generalized the result for abelian surfaces and completely solves the question raised in [arXiv:2108.08710].

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Works this paper leans on

5 extracted references · 1 canonical work pages

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