{"id":"d2bfc007-6608-43ac-8a13-f2e58afc389a","arxiv_id":"1908.03308","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"This note characterizes when a simple semi-homogeneous vector bundle on an abelian variety B is the image of the structure sheaf under a Fourier-Mukai equivalence from another abelian variety; the condition is that a kernel-intersection equation K(N)∩A_l=Ker(π) admits a line bundle N.","lead":"This paper gives a criterion for when a special kind of vector bundle can be the image of a structure sheaf under a derived equivalence of abelian varieties. It also collects structural facts about such equivalences first proved by Orlov and Mukai.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4's asserted equality of finite subgroups is unproved; Theorem 5.2(b) sufficiency rests on it.","rationale":"The reader identified a plausible weak point in Lemma 5.3, namely the implication l(a−a′)≠0 ⇒ π(a−a′)≠0 or, equivalently, Ker(π)⊂A_l. That is a real but secondary issue: it is likely true for the canonical isogeny because the finite group scheme killed by π is Cartier dual to a subgroup of B_l, so the paper's 'by definition' is terse but repairable. The more load-bearing problem is in Lemma 5.4, where the proof of sufficiency for Theorem 5.2(b) depends on an equality of two order-l^2 subgroups that is asserted without derivation. The assumption K(N)∩A_l = Ker(π) constrains the kernel of h_N on A_l but not its image; the image could in principle be a different order-l^2 subgroup of \\hat A, in which case the constructed bundle would not have the required rank and Prop 5.1 would not yield an equivalence. The necessity half is similarly only asserted in Remark 5.1; the forward computation gives only containment, not the equality required by the theorem. Since the central classification claim is plausible and likely correct, but the written proof has a genuine gap in the sufficiency argument, the fair verdict remains CONDITIONAL, as the reader already concluded. My recommendation is therefore UNCHANGED: the paper needs a completed proof of Lemma 5.4 (and a written necessity argument) before the main theorem can be accepted.","tokens_in":9419,"tokens_out":40715,"duration_ms":417776,"concrete_test":"Work out the smallest nontrivial case g=2, l=2, with B a product of two elliptic curves and M_B a polarization such that K(M_B)⊂B_2 has order 4. Compute A = \\widehat{B_δ}, the groups A_2, Ker(π), and \\hatπ(B_2) explicitly as matrices on 2-torsion. Enumerate line bundles N on A for which K(N)∩A_2 = Ker(π), compute h_N(A_2), and test whether h_N(A_2) = \\hatπ(B_2) and whether |{(h_N(a)+\\hatπ(b), h_M_B(b)+π(a)) | a∈A_2,b∈B_2}| = 4. If a valid N gives a different image or a different cardinality, Lemma 5.4 is false; if the equality always holds, the missing argument should be identified and written out.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.2(b) claims a necessary and sufficient condition for a simple semi-homogeneous bundle to be the image of the structure sheaf under a Fourier–Mukai equivalence. In the converse direction, Lemma 5.4 is supposed to show that a line bundle N satisfying K(N)∩A_l = Ker(π) meets the rank condition of Lemma 5.3. The proof reduces this to the equality {(h_N(a), π(a)) | a∈A_l} = {(\\hatπ(b), h_M_B(b)) | b∈B_l}, and then says 'we can check easily'. This equality is not a formal consequence of K(N)∩A_l = Ker(π): the assumption fixes only the kernel of h_N on A_l; it does not determine the image h_N(A_l), which is an order-l^2 subgroup of \\hat A. The right-hand side is the image of a specific canonical section determined by the dual isogeny. Without a proof that h_N(A_l) equals that section, the rank of the bundle constructed in Lemma 5.3 is not shown to be l, and Φ_E need not be an equivalence. Necessity, asserted in Remark 5.1 by taking N = det(E_A), is also not proved: the forward calculation in §5 yields only Ker(π) ⊂ K(N)∩A_l, while the theorem needs equality. A related terse step in Lemma 5.3 says 'if l(a−a′)≠0 then π(a−a′)≠0', but the actual requirement is π(a−a′)∉Σ(E_B); the proof does not explain why the canonical isogeny guarantees this. These are fixable gaps if the result is correct, but the current text does not supply the missing arguments.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9757,"tokens_out":6622,"duration_ms":61943,"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":[{"comment":"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":"Section 5, Lemma 5.3"},{"comment":"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":"Section 5, Lemma 5.4"},{"comment":"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":"Section 5, Remark 5.1"},{"comment":"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.","section":"Section 5, proof of Lemma 5.3"}],"minor_comments":[{"comment":"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":"Throughout"},{"comment":"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":"Section 2, Theorem 2.2"},{"comment":"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":"Section 4, proof of Lemma 4.1"},{"comment":"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":"Section 5, Lemma 5.4"},{"comment":"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.","section":"Section 5, Example 5.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a somewhat rough preprint, and the main result in §5 is promising but not yet proven in its current form. The gaps are localized but concern exactly the 'if and only if' claim in Theorem 5.2(b): the missing containment Ker(π)⊂A_l in Lemma 5.3, the unproved graph equality in Lemma 5.4, and the missing reverse inclusion in Remark 5.1. If the author can supply these arguments, the result is potentially publishable. The paper is within the journal's scope and does not rely on unacknowledged prior work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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).\n\nCredit 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.\n\nSoft 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.\n\nNone 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.\n\nThis 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":10306,"tokens_out":2873,"would_cite":false,"duration_ms":29312,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14K05","14F08","18E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Fourier-Mukai transforms","abelian varieties","semi-homogeneous vector bundles","derived categories","slope","Fourier-Mukai partners","isogenies","Poincare bundle"],"falsifier":"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.","tokens_in":9196,"feed_emoji":"🔁","tokens_out":10338,"duration_ms":91103,"temperature":0.7,"pith_summary":"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 $E_B$.","feed_headline":"Criterion decides which semihomogeneous bundles are equivalence images","feed_subtitle":"The check: a line bundle on the dual abelian variety must have kernel intersecting the l-torsion in the right way.","key_machinery":"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.","core_discovery":"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 $\\det(E) \\cong q_1^*\\det(E|_{A\\times e_B}) \\otimes (\\pi\\times 1)^*\\mathcal{P} \\otimes q_2^*\\det(E|_{e_A\\times B})$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the theorem that any derived equivalence between abelian varieties has a semi-homogeneous kernel and defines the sub-abelian variety $A_\\mu$; the paper's question and Theorem 5.2 build directly on it.","marker":"[8]"},{"why":"Supplies the structure theory of simple semi-homogeneous vector bundles, including slope, $\\Sigma(E)$, and the degree statement used in Lemma 5.1 throughout Section 5.","marker":"[5]"},{"why":"Supplies the fully faithfulness and equivalence criterion (Theorem 2.2) and background on Fourier–Mukai transforms used in Proposition 5.1.","marker":"[3]"},{"why":"Supplies the adjunction and kernel-uniqueness statements used to pass from equivalences to kernels and back.","marker":"[1]"},{"why":"Supplies the reconstruction and finiteness result invoked in Corollary 2.1 to reduce the case where the unit structure sheaf maps to a line bundle.","marker":"[2]"}],"fun_headline_variants":["Semihomogeneous bundle criterion for Fourier-Mukai partners","Kernel condition for Fourier-Mukai images of semihomogeneous bundles","Line bundle kernel check classifies Fourier-Mukai partners","l-torsion condition for semihomogeneous bundle equivalences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Semihomogeneous bundle criterion for Fourier-Mukai partners","Kernel condition for Fourier-Mukai images of semihomogeneous bundles","Line bundle kernel check classifies Fourier-Mukai partners","l-torsion condition for semihomogeneous bundle equivalences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000349,"raw_usage":{"total_tokens":1923,"prompt_tokens":976,"completion_tokens":947,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":873}},"tokens_in":592,"tokens_out":947,"duration_ms":9244,"temperature":1.0,"reasoning_tokens":873,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:18:11.680652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that any derived equivalence between abelian varieties has a semi-homogeneous kernel and defines the sub-abelian variety $A_\\mu$; the paper's question and Theorem 5.2 build directly on it."},{"cited_title":"Mukai et al","cited_arxiv_id":null,"evidence_quote":"Supplies the structure theory of simple semi-homogeneous vector bundles, including slope, $\\Sigma(E)$, and the degree statement used in Lemma 5.1 throughout Section 5."},{"cited_title":"Huybrechts","cited_arxiv_id":null,"evidence_quote":"Supplies the fully faithfulness and equivalence criterion (Theorem 2.2) and background on Fourier–Mukai transforms used in Proposition 5.1."},{"cited_title":"Bridgeland","cited_arxiv_id":null,"evidence_quote":"Supplies the adjunction and kernel-uniqueness statements used to pass from equivalences to kernels and back."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the reconstruction and finiteness result invoked in Corollary 2.1 to reduce the case where the unit structure sheaf maps to a line bundle."}],"review_version":1}