Fourier-Mukai partners of abelian varieties

Anningzhe Gao

REVIEW 4 major objections 5 minor 9 references

Fourier-Mukai partners of abelian varieties

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves a necessary and sufficient kernel-intersection condition for a simple semi-homogeneous vector bundle to be the image of the structure sheaf of the unit under a Fourier–Mukai equivalence between abelian varieties.

desk verdict A small note with one genuinely new criterion for which semi-homogeneous bundles can be Fourier–Mukai images of the structure sheaf, but the proof has real gaps that need filling. read the letter →

arxiv 1908.03308 v2 pith:OTKKVMIC submitted 2019-08-09 math.AG

Anningzhe Gao This is my paper · ORCID

classification math.AG MSC 14K0514F0818E30

keywords Fourier-Mukaitransformsabelianvarietiessemi-homogeneousvectorbundlesderivedcategoriesslopepartnersisogeniesPoincarebundle

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 studies which abelian varieties are Fourier–Mukai partners of a given abelian variety, focusing on equivalences whose kernel is a semi-homogeneous vector bundle. The main question is: given a simple semi-homogeneous vector bundle $E_B$ on $B$, when does there exist an abelian variety $A$ and a derived equivalence $\Phi_E$ with $\Phi_E(\mathcal{O}_{e_A}) = E_B$? The answer is a kernel-intersection criterion on the dual abelian variety: such an equivalence exists exactly when there is a line bundle $N$ on $\widehat{B}_\delta$ whose kernel intersects the $l$-torsion subgroup in the kernel of the dual isogeny. If the criterion holds, the isomorphism type of the partner and the determinant of the kernel are determined by $E_B$.

What carries the argument

The carrying object is the semi-homogeneous vector bundle: a vector bundle $E$ on an abelian variety such that every translation $T_a^*E$ is isomorphic to $E\otimes L$ for some line bundle $L$; simple semi-homogeneous bundles are classified up to tensoring by $\mathrm{Pic}^0$ by their slope $\delta = \det(E)/\mathrm{rank}(E) \in \mathrm{NS}\otimes\mathbb{Q}$. To each slope one attaches the sub-abelian variety $B_\delta \subseteq B\times\widehat{B}$, and the proof works through the isogeny $\pi : B \to B_\delta$ together with the factorization of the determinant line bundle $L \cong (\pi\times 1)^*\mathcal{P}$, where $\mathcal{P}$ is the Poincaré bundle. The final criterion is a statement about the kernel $K(N)$ of a line bundle's polarization map on $\widehat{B}_\delta$, intersected with the $l$-torsion subgroup.

What would settle it

Look for a simple semi-homogeneous bundle $E_B$ of rank $l$ whose attached isogeny $\pi : B \to B_\delta$ has a nonzero kernel element outside the $l$-torsion subgroup $B_l$, while some line bundle $N$ on $\widehat{B}_\delta$ satisfies $K(N) \cap \widehat{B}_\delta^l = \ker(\widehat{\pi})$. If such an example exists, the construction in Lemma 5.3 need not produce a fully faithful transform, and the sufficiency direction of Theorem 5.2(b) would fail; if no such example exists, the missing containment is provable from the definitions and the theorem stands.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 5.2(b). Let $E_B$ be a simple semi-homogeneous vector bundle on an abelian variety $B$, of rank $l$ and slope $\delta = \det(E_B)/l \in \mathrm{NS}(B)\otimes_{\mathbb{Z}}\mathbb{Q}$. Write $\pi : B \to B_\delta$ for the natural isogeny to the sub-abelian variety attached to $\delta$. The paper asserts that there is a Fourier–Mukai equivalence $\Phi_E$ from some abelian variety $A$ to $B$ sending $\mathcal{O}_{e_A}$ to $E_B$ if and only if there exists a line bundle $N$ on $\widehat{B}_\delta$ such that $K(N) \cap \widehat{B}_\delta^{\,l} = \ker(\widehat{\pi})$, where $K(N)$ is the kernel of the polarization map of $N$ and the superscript $l$ denotes the $l$-torsion subgroup. This is meant as a complete, checkable classification: the condition is finite and computable from a line bundle on an abelian variety. The same section also proves that $\widehat{A} \cong B_{\delta(E_B)}$, that $A\times\widehat{A} \cong B\times\widehat{B}$, and that the determinant of any such kernel obeys $\det(E) \cong q_1^*\det(E|_{A\times e_B}) \otimes (\pi\times 1)^*\mathcal{P} \otimes q_2^*\det(E|_{e_A\times B})$.

Load-bearing premise

The proof of the 'if' direction assumes that the isogeny involved has only $l$-torsion points in its kernel, so that distinct $l$-multiples must have distinct images under the isogeny; this is stated as immediate from the definition but is not proved.

Editorial extensions

If this is right

For any fixed $E_B$, the existence of a partner $A$ and an equivalence sending $\mathcal{O}_{e_A}$ to $E_B$ becomes a finite check: look for a line bundle $N$ on $\widehat{B}_\delta$ whose polarization kernel cuts the $l$-torsion exactly in $\ker(\widehat{\pi})$.
When the criterion is met, the partner is forced: $A \cong \widehat{B}_\delta$, so the unit-image bundle determines the isomorphism type of the source abelian variety.
For any derived equivalence between abelian varieties there exists a semi-homogeneous vector-bundle kernel that induces a derived equivalence, so the criterion applies to the full set of Fourier–Mukai partners, not just a special class.
The determinant formula expresses $\det(E)$ from the two restrictions and the Poincaré pullback, so the numerical invariants of any semi-homogeneous kernel are determined by $E_B$ and $E_A$.
The example included in the paper shows that the condition is satisfied whenever $\deg(\det(E))$ divides the rank $l$, producing self-equivalences of $A$ with prescribed unit image.

Reading between the lines

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

Editorial inference: the criterion is arithmetic in nature; in concrete cases such as products of elliptic curves or powers of a simple abelian variety, it could be tested by computing elementary divisors of $K(N)$, giving a practical way to enumerate which semi-homogeneous bundles occur as unit images.
Editorial inference: if the containment $\ker(\pi)\subseteq A_l$ used in the proof of Lemma 5.3 is not automatic, the 'if' direction of Theorem 5.2(b) may fail exactly when the kernel-intersection condition holds but the constructed transform is not fully faithful; this is the first place to look for a counterexample or a missing hypothesis.
Editorial inference: the same kernel-intersection language should adapt to the relative setting, describing when a family of semi-homogeneous bundles over a base can be promoted to a Fourier–Mukai kernel; the paper does not discuss such families.
Editorial inference: a natural companion question is whether the criterion can be phrased purely in terms of the elementary divisors of a polarization on $\widehat{B}_\delta$, which would make the classification of Fourier–Mukai partners of an abelian variety a finite combinatorial problem independent of the chosen line bundle.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper asks a natural question—which simple semi-homogeneous bundles on an abelian variety can be the image of the structure sheaf under a Fourier–Mukai equivalence—and proposes an answer: existence of a line bundle N on the dual abelian variety satisfying K(N)∩A_l=Ker(π). That criterion is new as far as I know, and it is concrete enough to check. The first half of the note is a serviceable recap of Orlov and Mukai, with one useful corollary (Theorem 4.1).

Credit where due: the exposition is honest, the references are the right ones, and there is no self-citation inflation or fitted-parameter sleight of hand. The question is well-posed, and the proposed answer would be a useful classification if it survives.

Soft spots, all in Section 5. First, the sufficiency proof in Lemma 5.3 uses "if l(a−a′)≠0 then π(a−a′)≠0" as though it were immediate from the definition of π. It is not; it depends on Ker(π)⊂A_l, which is exactly the kind of thing the construction has to establish. Second, Lemma 5.4 hides the real work in "we can check easily". The stress-test is right: the assumption K(N)∩A_l=Ker(π) fixes the kernel of h_N on A_l, but the equality of the two displayed sets requires the image h_N(A_l) to be a specific section, and that needs an argument. Third, the necessity direction of Theorem 5.2(b) is asserted in Remark 5.1 with no proof; the forward calculation in Section 5 gives only an inclusion, not equality.

None of this looks like a counterexample. These are missing arguments, not wrong ones. But the main theorem is exactly the part that depends on them.

This is a paper for specialists in derived categories of abelian varieties. A PhD student reading it should go to Orlov's paper first; a researcher working on Fourier–Mukai partners will want the criterion if it is proved.

Recommendation: send it to peer review, not desk reject. The note is short, the gaps are fillable in a page or two, and the result is interesting enough to justify referee time. If the criterion fails, the note becomes an exposition, and the theorem should be downgraded.

Referee Report

4 major / 5 minor

Summary. This paper studies Fourier–Mukai (FM) partners of abelian varieties over an algebraically closed field of characteristic zero, focusing on the case where the kernel of the equivalence is a simple semi-homogeneous vector bundle. The main results are Theorem 1.1 (a reproof of Orlov's criterion that derived equivalence of abelian varieties is equivalent to an isomorphism \hat B ≅ A_μ), Theorem 4.2 (a principal-polarization result), and Theorem 5.2, which gives a necessary and sufficient condition for a simple semi-homogeneous vector bundle E_B on B to be the image of the structure sheaf of the unit under some FM equivalence: there must exist a line bundle N on \hat B_δ with K(N) ∩ \hat B_δ^l = Ker(\hat π). The proof uses the structure of semi-homogeneous bundles, the isogeny π: B → B_δ, and earlier results of Orlov and Mukai.

Significance. If the main theorem is correct, it provides an explicit, checkable criterion for which semi-homogeneous bundles arise as the image of the unit structure sheaf under a derived equivalence, refining Orlov's classification of FM partners. The paper is clearly structured and honestly builds on black-box theorems (Orlov, Mukai, Bondal–Orlov, Huybrechts); the proposed criterion is falsifiable and not fitted to any example. The main caveat is that the proofs of both directions of the 'if and only if' in Theorem 5.2(b) contain gaps that need to be filled before the result can be considered established.

major comments (4)

[Section 5, Lemma 5.3] The sufficiency proof relies on the assertion 'if l(a−a′)≠0 then π(a−a′)≠0', which is equivalent to Ker(π)⊂A_l. This containment is not a formal consequence of the definition of π as the dual of the natural map B→B_δ and is never proved. Moreover, Proposition 5.1 requires E|_{a×B} ≇ E|_{a′×B} for a≠a′, which in the present situation means π(a−a′)∉Σ(E_B); the proof only checks π(a−a′)≠0, not that it avoids Σ(E_B). Without an argument for this containment, the constructed Φ_E is not shown to be an equivalence, so Lemma 5.3 is incomplete.
[Section 5, Lemma 5.4] The key equality {(h_N(a), π(a)) | a∈A_l} = {(\hat π(b), h_MB(b)) | b∈B_l} is asserted with 'we can check easily', but it is not a formal consequence of K(N)∩A_l = Ker(π). The hypothesis determines only the kernel of h_N on A_l, not the image h_N(A_l), which is an order-l^2 subgroup of \hat A; the right-hand side is a specific canonical graph determined by the dual isogeny. Without a proof that h_N(A_l) equals that graph, the rank of the bundle constructed in Lemma 5.3 is not shown to be l, so Φ_E need not be an equivalence.
[Section 5, Remark 5.1] The necessity of the condition in Theorem 5.2(b) is asserted by taking N = det(E_A), but the forward calculation in §5 yields only Ker(π) ⊂ K(N)∩A_l (displayed as 'Ker(π) ⊂ Ker(h_{MA}) ∩ A_l' before Theorem 5.1), not the required equality. The reverse inclusion is never proved, so the 'if and only if' in Theorem 5.2(b) is not established.
[Section 5, proof of Lemma 5.3] After showing det(E|_{e_A×B}) ≅ M_B up to Pic^0(B), the text says 'Since E_B is simple, we may assume E|_{e_A×B} ≅ E_B.' This step needs justification: simple semi-homogeneous bundles of the same slope differ by tensoring with a line bundle in Pic^0(B), and the text does not explain how to adjust the kernel E on A×B (for example, by composing with a line-bundle twist or an automorphism) to arrange this isomorphism. The argument should specify the adjustment explicitly.

minor comments (5)

[Throughout] The manuscript contains numerous typos and spacing errors (e.g., the title 'FOURIER-MUKAI PARTNERS OF ABELIAN V ARIETIES', 'semi-homogen eous', and inconsistent breaks in 'if and only if'); a careful proofreading is needed.
[Section 2, Theorem 2.2] The displayed condition 'P ⊗ π_X^* ω_X = P ⊗ π_Y^* ω_Y' uses the same kernel P on both sides; it should be an isomorphism P ⊗ π_X^* ω_X ≅ P ⊗ π_Y^* ω_Y with the appropriate projections.
[Section 4, proof of Lemma 4.1] In the line 'Then we have f∗f∗(L) = f∗(OA)', the projection formula is misstated: one should have f_* f^* L ≅ (f_* O_A) ⊗ L, which is what the subsequent sentence actually uses.
[Section 5, Lemma 5.4] Lemma 5.4 states 'there exists a line bundle N on A', but in Theorem 5.2(b) the line bundle N should live on \hat B_δ; the letter A in Lemma 5.4 is used inconsistently and should be renamed \hat B_δ.
[Section 5, Example 5.1] The condition deg(M)|l is stated without specifying the polarization with respect to which the degree is taken; the statement would be clearer if the degree were expressed in terms of the Euler characteristic or a chosen polarization on A.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning detected; the central classification is derived from external theorems and is not equivalent to its own inputs.

full rationale

The paper's main claims are built on external results by Orlov, Mukai, Huybrechts, Bridgeland, and Favero, and the proofs do not assume the conclusions they set out to establish. Theorem 5.2(b), for instance, proposes an independent criterion for when a simple semi-homogeneous bundle can be the image of the structure sheaf under a Fourier-Mukai equivalence, and the sufficiency and necessity directions are argued through the preceding lemmas rather than by taking the criterion as an input. Even where the text is terse or incomplete—such as the asserted equality in Lemma 5.4—the issue is a missing or compressed justification, not a reduction of the result to its own assumptions. There are no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatz smuggled in via citation. The derivation chain is therefore not circular, and the appropriate score is 0.

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

The argument uses standard background results as black boxes and introduces no new constants or entities. The main unstated burden is inside the proof of Lemma 5.3 and Remark 5.1, which is captured in red_flags and the weakest_assumption.

assumptions (5)

standard math Orlov's theorem 2.3: every Fourier-Mukai equivalence between abelian varieties induces an isometry A×hatA ≅ B×hatB compatible with twists by line bundles.
Used in proofs of Theorem 4.1 and Theorem 5.1; cited as [8] in the paper.
standard math Mukai's classification and degree formula for simple semi-homogeneous vector bundles, including Lemma 5.1 that deg(π1)=rank(E)^2.
Used throughout Section 5 to control degrees and kernels; cited as [5].
standard math Bondal-Orlov fully faithfulness criterion (Lemma 2.1) and Huybrechts' converse criterion for an equivalence (Theorem 2.2).
Used to reduce the problem of Φ_E being an equivalence to fiberwise simplicity and non-isomorphism; cited as [1] and [3].
standard math Orlov's representability theorem: every exact equivalence of derived categories of smooth projective varieties is a Fourier-Mukai transform with a unique kernel.
Used to pass from an arbitrary equivalence to a kernel P and then to a semi-homogeneous vector bundle kernel; cited as [9].
domain assumption The base field is algebraically closed of characteristic zero.
Ensures group schemes are reduced and the standard theory of abelian varieties and derived categories applies; stated in the introduction.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Fourier-Mukai partners of abelian varieties." pith.science (2026). https://pith.science/paper/OTKKVMIC

@misc{pith2026190803308,
  author       = {Pith},
  title        = {Pith review of: Fourier-Mukai partners of abelian varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTKKVMIC}},
  note         = {Machine review of arXiv:1908.03308}
}

read the original abstract

We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an equivalence is given by a semi-homogenous vector bundle. In particular, given an abelian variety B, and a simple semi-homogenous vector bundle E on B, we will discuss for which E, it can be the image of the structure sheaf of the unit on A under some triangulated equivalence, where A is some abelian variety.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Bridgeland

T. Bridgeland. Equivalences of triangulated categories and four ier–mukai transforms. Bulletin of the London Mathematical Society , 31(1):25–34, 1999

work page 1999
[2]

D. Favero. Reconstruction and ﬁniteness results for fourier– mukai partners. Advances in Mathematics, 230(4-6):1955–1971, 2012

work page 1955
[3]

Huybrechts

D. Huybrechts. Fourier-Mukai transforms in algebraic geometry . Oxford University Press on Demand, 2006

work page 2006
[4]

J. S. Milne. Abelian varieties. In Arithmetic geometry, pages 103–150. Springer, 1986

work page 1986
[5]

Mukai et al

S. Mukai et al. Semi-homogeneous vector bundles on an abelian va riety. Journal of Mathe- matics of Kyoto University , 18(2):239–272, 1978

work page 1978
[6]

D. Mumford. On the equations deﬁning abelian varieties. i. Inventiones mathematicae , 1(4):287–354, 1966

work page 1966
[7]

Mumford, C

D. Mumford, C. P. Ramanujam, and J. I. Manin. Abelian varieties , volume 108. Oxford university press Oxford, 1974

work page 1974
[8]

D. O. Orlov. Derived categories of coherent sheaves on abelian v arieties and equivalences between them. Izvestiya: Mathematics , 66(3):569, 2002

work page 2002

Show all 9 references

[9]

D. O. Orlov. Derived categories of coherent sheaves and equiva lences between them. Russian Mathematical Surveys , 58(3):511, 2003

2003

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bridgeland

T. Bridgeland. Equivalences of triangulated categories and four ier–mukai transforms. Bulletin of the London Mathematical Society , 31(1):25–34, 1999

work page 1999

[2] [2]

D. Favero. Reconstruction and ﬁniteness results for fourier– mukai partners. Advances in Mathematics, 230(4-6):1955–1971, 2012

work page 1955

[3] [3]

Huybrechts

D. Huybrechts. Fourier-Mukai transforms in algebraic geometry . Oxford University Press on Demand, 2006

work page 2006

[4] [4]

J. S. Milne. Abelian varieties. In Arithmetic geometry, pages 103–150. Springer, 1986

work page 1986

[5] [5]

Mukai et al

S. Mukai et al. Semi-homogeneous vector bundles on an abelian va riety. Journal of Mathe- matics of Kyoto University , 18(2):239–272, 1978

work page 1978

[6] [6]

D. Mumford. On the equations deﬁning abelian varieties. i. Inventiones mathematicae , 1(4):287–354, 1966

work page 1966

[7] [7]

Mumford, C

D. Mumford, C. P. Ramanujam, and J. I. Manin. Abelian varieties , volume 108. Oxford university press Oxford, 1974

work page 1974

[8] [8]

D. O. Orlov. Derived categories of coherent sheaves on abelian v arieties and equivalences between them. Izvestiya: Mathematics , 66(3):569, 2002

work page 2002