{"id":"eced14e5-5fe2-4375-8ebd-495a4cc85fa9","arxiv_id":"2502.01149","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For parabolic automorphisms of hyperkähler manifolds with a Lagrangian fibration, the translation vector has maximal rank, making fibers with finite-order or dense orbits dense in the base.","lead":"The paper proves that parabolic automorphisms of hyperkähler manifolds with a Lagrangian fibration have translation vectors of maximal variation. This implies that fibers where the automorphism acts with finite order, as well as fibers with dense orbits, form dense subsets of the base.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Relative polarization construction in §4.1.2 has a root-degree mismatch; Proposition 4.1's use of Gauthier–Vigny depends on it.","rationale":"The reader's CONDITIONAL verdict is appropriate. I agree that the weakest point is §4.1.2. The proof strategy is otherwise coherent: Lemma 3.3 gives the local volume estimate in Betti coordinates; the new ingredient is to propagate it across singular fibers via Gauthier–Vigny. That propagation is not a black box: it requires a relative polarization with the exact equivariance (4.3). My computation shows the proposed (D^2-1)-th root correction does not achieve this equivariance; the correct root is D(D-1)-th, so there is a genuine mismatch in the manuscript. The phrase 'take the sum of them all' obscures rather than proves the global monodromy invariance. If the construction can be corrected, the volume propagation and Theorem A(2) likely go through; if not, Proposition 4.1 fails. I do not see a reason to reject the theorem, since the statement is plausible and the projective case is known by other methods; the issue is with the proof's new mechanism. The SYZ assumption is explicitly part of Theorem A and acknowledged, so it is not an internal flaw. Theorem C's sketch is a lesser concern because assertion (1) is already established in the literature and the cohomological growth argument follows from Verbitsky's embedding plus convexity. Therefore UNCHANGED: keep CONDITIONAL, with the polarization construction as the condition to be supplied.","tokens_in":21482,"tokens_out":9419,"duration_ms":95619,"concrete_test":"Take D=2 and a Lagrangian fiber X_b, with L_b a degree-one polarization. Compute M_b from m_2^*L_b = L_b^4 * M_b. Test every R_b in Pic^0(X_b) with R_b^3 = M_b and separately with R_b^2 = M_b: check whether m_2^*(L_b * R_b) = (L_b * R_b)^4; determine whether the tensor product of all roots is monodromy invariant and satisfies the same equation. If no choice works, the relative polarization hypothesis needed by [15] fails already on a single fiber; if some choice works, redo the global averaging over B^o and state the corrected root degree in §4.1.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem A(2) hinges on Proposition 4.1, which applies Gauthier–Vigny's local-to-global volume estimate to the rational fiberwise multiplication m_D. That application requires a relatively ample line bundle A on X with m_D^*A_b = A_b^{D^2} for b in a dense open subset. §4.1.2 attempts to construct A_b as follows: if L_b represents the ample generator H_b of R_b, then m_D^*L_b = L_b^{D^2} * M_b with M_b in Pic^0(X_b); one 'adds' a (D^2-1)-th root R_b of M_b. The computation is not consistent: for L'_b = L_b * R_b, m_D^*L'_b = L_b^{D^2} * M_b * R_b^D, while (L'_b)^{D^2} = L_b^{D^2} * R_b^{D^2}; these agree only if R_b^{D(D-1)} = M_b, not R_b^{D^2-1} = M_b. Also, 'take the sum of them all' is not a defined operation on line bundles; if it means the tensor product of all roots, the resulting class is not H_b and its m_D-equivariance is not verified. The global monodromy-invariant A_b over B^o is therefore not established. Without A_b satisfying (4.3), Proposition 3.3 of [15] cannot be invoked; the implication from local volume o(D^{2gk}) to global O(D^{(2g-2)k}) in (4.5), and hence estimate (4.11), is unsupported. Proposition 4.3 and Theorem A(2) are the next steps that collapse. The theorem may still be true, and the error may be repairable by taking D(D-1)-th roots and tracing monodromy, but as written the central new proof has a gap at this exact point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies parabolic automorphisms of irreducible hyperkähler manifolds of complex dimension 2g that admit an invariant Lagrangian fibration p_f: X -> B. The main result (Theorem A) states that for any p ≤ g the operator norm of (f^n)^* on H^{p,p}(X;R) grows as c_p n^{2p} + O(n^{2p-1}); that the translation vector of the iterate f^k has maximal variation; and that for each s ≤ g the set of base points for which orbit closures in the fiber have dimension s is dense in B. The proof is new: it avoids the Ax-Schanuel machinery used by Gao and others, relying instead on Betti coordinates, a volume-growth criterion for non-maximal variation, and a local-to-global propagation theorem of Gauthier-Vigny for relatively polarized fiberwise endomorphisms. The projective case is established first; a degenerate twistor deformation is used to pass to the non-projective case.","tokens_in":21848,"tokens_out":21909,"duration_ms":216676,"significance":"If the proof is completed, this gives a uniform and conceptually simpler derivation of maximal variation of Betti maps in the hyperkähler setting, covering both projective and non-projective manifolds and isotrivial/non-isotrivial fibrations. The use of the Gauthier-Vigny local-to-global principle is a promising new ingredient. However, the central technical construction on which the proof relies currently has a gap, so the paper is not yet ready for publication.","major_comments":[{"comment":"The construction of the relatively ample line bundle A_b satisfying m_D^*A_b = A_b^{D^2} is not valid as written. For a chosen L_b, m_D^*L_b = L_b^{D^2} ⊗ M_b, and setting L'_b = L_b ⊗ R_b gives m_D^*L'_b = L_b^{D^2} ⊗ M_b ⊗ R_b^D, whereas (L'_b)^{D^2} = L_b^{D^2} ⊗ R_b^{D^2}; equality would require R_b^{D(D-1)} = M_b, not R_b^{D^2-1} = M_b. Moreover, \"take the sum of them all\" is undefined for line bundles, and a product of all roots would not keep the class H_b. Consequently, the existence of a monodromy-invariant A_b satisfying (4.3) is not established, and the application of [15, Proposition 3.3] that yields the global estimate (4.5) and then (4.11) is unsupported.","section":"§4.1.2"},{"comment":"The identity f^{Dk} = m_D^k ∘ f is not correct under the Betti-coordinate conventions of the paper. Since f_Φ(u,x)=(u,x+t_f(u)) and m_D,Φ(u,x)=(u,Dx), one has f^{Dk}(u,0)=(u,Dk·t_f(u)) while m_D^k(f(u,0))=(u,D^k·t_f(u)); these are equal only if D^k = Dk. If the intended statement is f^{D^k} = m_D^k ∘ f, then the notation in Proposition 4.1 and Proposition 4.3 should be changed to the subsequence n = D^k, and the argument that this subsequence suffices for the full statement of Proposition 4.3 needs to be supplied. As written, the proof of Proposition 4.1 is invalid at this point.","section":"§4.3.1, Eqs. (4.8)–(4.10)"}],"minor_comments":[{"comment":"With T_k as defined in (4.14), A(x,y,z)=x+y-z gives f^{-Dk}(S), not f^{Dk}(S); either redefine A as x+z-y or take z=f^{-Dk}(y). The error is harmless for the growth bound but should be corrected.","section":"§4.3.1, Step 2"},{"comment":"The inequality \"0 < r q(a,σ) < q(a,h)\" is not meaningful because q(a,σ) is a complex number; please state the intended condition with real/imaginary parts or absolute values.","section":"§7.2, Proposition 7.2"},{"comment":"Theorem C is stated with only a sketch, and since Assertion (1) of Theorem A depends on it, the authors should give a complete proof or a precise reference for the Khovanskii-Teyssier concavity step and the Verbitsky embedding used for the lower bound.","section":"§5, Theorem C"},{"comment":"The extension of Lo Bianco's theorem to the non-projective case is only a short paragraph; a fuller explanation of why the projective conclusion on X_t transfers back to X_0 would be useful.","section":"§7.4"}],"recommendation":"major_revision","confidential_remarks":"The main theorem may well be true and the gaps appear repairable, but the missing polarization construction and the incorrect commutation identity are too central to be fixed by local copyediting. I recommend major revision, with a request to rework §4.1.2 and §4.3.1 and to re-check the notation for the sequence n = D^k."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is the volume-propagation proof of maximal variation for the translation vector, replacing the Ax–Schanuel machinery from Bakker–Gao–Voisin with a Gauthier–Vigny type estimate on the fiberwise multiplication. That is a real improvement, and the extension to non-projective Kähler hyperkähler manifolds via twistor deformations is a nice touch. The paper is also honest about the projective case being already known, which I respect.\n\nThe good parts hold up: the statement of Theorem A is clean, the derivation of (3) from (2) is purely topological, and Theorem C, though sketched, is a useful synthesis. The volume computation in Lemma 3.3 is elementary and convincing.\n\nNow the soft spot, and it is load-bearing. The stress-test note is right: in §4.1.2 the construction of the relative polarization A_b for m_D has a root-degree error. If L_b is the ample generator and m_D^*L_b = L_b^{D^2} ⊗ M_b with M_b in Pic^0, then to make m_D^*L'_b = (L'_b)^{D^2} you need M_b = R_b^{D(D-1)}, not M_b = R_b^{D^2-1} as the text says. And “take the sum of them all” is not an operation on line bundles. Since Gauthier–Vigny’s Proposition 3.3 is invoked precisely to get the local-to-global volume estimate (4.5), the current proof of Proposition 4.1, and hence of Theorem A(2), does not go through as written. This is not a minor typo; it is the central mechanism.\n\nThat said, the gap looks repairable. Choosing D(D-1)-th roots and tracking monodromy should produce the needed A_b, and the rest of the argument is coherent. The non-projective extension in §7.4 is also admittedly thin, just a paragraph, but the projective case is the core and contains the actual issue.\n\nVerdict: the theorem is probably true and the paper deserves peer review, but a referee needs to check §4.1.2 carefully. If the polarization construction can be fixed, this is a solid contribution. Send it out.","headline":"A genuinely new proof route with a repairable but load-bearing gap in the relative polarization construction of §4.1.2 — definitely deserves a serious referee.","tokens_in":22418,"tokens_out":4008,"would_cite":true,"duration_ms":38095,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J50","53C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"For parabolic automorphisms of irreducible hyperkähler manifolds with an invariant Lagrangian fibration, the paper proves the translation vector has maximal variation, so the fibers where the induced translation has finite order and the…","keywords":["parabolic automorphisms","hyperkähler manifolds","Lagrangian fibrations","Betti maps","translation vector","orbit closures","twistor deformations","volume estimates"],"falsifier":"Compute, for a concrete parabolic automorphism with invariant Lagrangian fibration (for instance on a K3 surface with an elliptic fibration or on the Hilbert scheme of two points on a K3), the generic rank of the translation vector in Betti coordinates; if any example yields rank strictly less than $2g$, Assertion (2) of Theorem A is false. Alternatively, test the relative-polarization construction by checking whether $m_D^* A_b$ equals $A_b^{\\otimes D^2}$ for a single line bundle $A_b$ rather than a formal sum of roots of a Pic$^0$ twist; a failure there would break Proposition 4.1 and the proof of maximal variation.","tokens_in":21244,"feed_emoji":"🌀","tokens_out":8513,"duration_ms":92441,"temperature":0.7,"pith_summary":"Parabolic automorphisms of hyperkähler manifolds are automorphisms whose cohomological action is parabolic, meaning a unipotent isometry with quadratic growth of norms; up to taking a power, they act as translations on the smooth fibers of an invariant Lagrangian fibration. The paper proves that this translation never degenerates: in Betti coordinates its translation vector has maximal rank $2g$, so the associated Betti map is an open mapping. From that, it follows that for every dimension $s$ the set of fibers in which orbit closures have dimension $s$ is dense in the base, and in particular the fibers on which the translation has finite order are dense. The proof is new: it propagates local volume estimates via a fiberwise multiplication-by-$D$ map, and it extends from projective to arbitrary Kähler hyperkähler manifolds by deforming the complex structure. This settles a natural density question that previously required deep transcendence results in the projective case and extends the answer to non-projective manifolds.","feed_headline":"Parabolic hyperkähler maps force dense orbit fibers","feed_subtitle":"New volume-propagation proof covers projective and non-projective cases, settling density of finite-order and dense-orbit fibers.","key_machinery":"The central object is the translation vector $t_{f^k}$ of a parabolic automorphism in local Betti coordinates: above a small simply connected open set $U$ in the regular locus of the fibration, the automorphism acts as $(u,x) \\mapsto (u, x + t_{f^k}(u))$, and maximal variation means that $t_{f^k}$ is an open mapping, equivalently that its generic rank is $2g$. The main mechanism is the fiberwise multiplication-by-$D$ map $m_D$, which acts by $z \\mapsto Dz$ on each smooth abelian fiber and satisfies the key identity $f^{Dk}(S_0) = m_D^k(f(S_0))$ over $U$ once a section $S_0$ is fixed. Using the local-to-global volume propagation developed for relatively polarized fibered endomorphisms, the paper shows that if $t_{f^k}$ were not of maximal rank, then the volume of the images of multisections would grow at most like $n^{2g-2}$, contradicting the cohomological theorem that embeds symmetric powers into $H^{2p}(X;\\mathbb R)$ and forces growth of order $n^{2g}$. The passage from projective to Kähler manifolds uses degenerate twistor deformations: the automorphism remains holomorphic on all deformed complex structures, and some of the deformed manifolds are projective, so the projective result transfers back.","core_discovery":"Theorem A states that if $X$ is an irreducible hyperkähler manifold of dimension $2g$, $f$ is a parabolic automorphism with an invariant Lagrangian fibration $p_f: X \\to B$, and $k \\geq 1$ satisfies $p_f \\circ f^k = p_f$, then three conclusions hold. First, for every $p \\in \\{1,\\dots,g\\}$ the operator norm of $(f^n)^*$ on $H^{p,p}(X;\\mathbb R)$ equals $c_p(f)\\,n^{2p} + O(n^{2p-1})$ for a positive constant $c_p(f)$. Second, the translation vector of $f^k$ has maximal variation, equivalently the Betti map is generically of maximal rank $2g$ and its image is open. Third, for every $s \\in \\{1,\\dots,g\\}$ the set of base points $b$ for which the orbit closures of $f^k$ in the fiber $X_b$ have dimension $s$ is dense in $B$ for the Euclidean topology; in particular, fibers where every orbit is dense and fibers where the translation has finite order are both dense. The paper obtains these conclusions through a new route that avoids functional-transcendence theorems, using cohomological growth estimates, volume propagation along fiberwise multiplication maps, and a degenerate-twistor deformation argument to pass from projective to non-projective hyperkähler manifolds.","pith_inferences":["If the open Lagrangian conjecture that every nef isotropic class is semi-ample is proved, the invariant-fibration assumption in Theorem A would become automatic, so the theorem would apply to every parabolic automorphism.","The volume-propagation method is not tied to the hyperkähler cohomology theorem except for the lower bound, so the same strategy may prove maximal variation for other fibered automorphisms once an analogous cohomological growth estimate is available.","The density of finite-order fibers suggests that torsion values of sections of Lagrangian torus fibrations should be dense even in non-projective families, connecting the result to unlikely-intersection problems in a setting where transcendence methods do not apply.","A concrete computational check of the relative-polarization construction on a known hyperkähler example such as the Hilbert scheme of two points on a K3 surface would provide a direct verification of the polarization step used in the proof."],"forward_implications":["For every invariant Lagrangian fibration of a parabolic automorphism, the fibers on which the induced translation has finite order form a dense subset of the base, as do the fibers on which every orbit is dense.","The operator norm of $(f^n)^*$ on each $H^{p,p}(X;\\mathbb R)$ grows exactly like $c_p(f)\\,n^{2p}$ with no faster growth, for every $p$ up to $g$.","The maximal-variation conclusion holds without any projectivity assumption on the hyperkähler manifold, so the density statements apply to all Kähler examples.","For two parabolic automorphisms with distinct Lagrangian fibrations, generic orbits of the group they generate are dense, and for large powers the set of finite orbits is $\\varepsilon$-dense in $X$.","The proof provides a route to maximal rank of Betti maps in the hyperkähler setting that bypasses functional-transcendence theorems and instead relies on volume estimates and cohomological growth."],"supporting_citations":[{"why":"Establishes that a positive iterate of a parabolic automorphism acts as a fiberwise translation on smooth fibers and that orbits are dense on almost all smooth fibers, providing the starting point and the notion of translation vector.","marker":"[1]"},{"why":"Supplies the local-to-global volume estimate for relatively polarized fibered endomorphisms that is the core mechanism for propagating volume bounds from open subsets of the base to all of $X$.","marker":"[15]"},{"why":"Gives the quasi-finite modular-map result that underlies the earlier projective derivation of maximal variation and clarifies the role of the modular map in the hyperkähler setting.","marker":"[4]"},{"why":"Provides the previous maximal-rank theorem for Betti maps of abelian schemes under simplicity and dimension assumptions, the projective-case baseline that the new proof bypasses.","marker":"[14]"},{"why":"Gives the cohomological embedding theorem for hyperkähler manifolds that yields the $n^{2p}$ lower bound and hence the exact growth of the operator norm on $H^{p,p}$.","marker":"[32]"},{"why":"Shows that the induced automorphism of the base has finite order and that an invariant Kähler form exists on the base, which is needed for the fibration setup and for Theorem B.","marker":"[23]"},{"why":"Provides the degenerate twistor deformation technique used to pass from projective to non-projective hyperkähler manifolds.","marker":"[33]"},{"why":"Supplies the construction of C-symplectic forms whose associated complex structures make the twistor deformation work while preserving the automorphism.","marker":"[31]"}],"fun_headline_variants":["Parabolic hyperkähler maps yield dense orbit fibers","Dense orbits and finite-order fibers for parabolic hyperkähler maps","New proof: parabolic hyperkähler maps have dense orbit fibers","Parabolic hyperkähler automorphisms: dense orbit fibers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the Section 4.1.2 claim that the fiberwise multiplication map $m_D$ is relatively polarized, with the line bundle constructed by 'taking the sum of all' the $(D^2-1)$-th roots of a Pic$^0$ twist even though line-bundle operations are multiplicative, so if that polarization is not actually a line bundle the volume estimate and the maximal-variation contradiction collapse; the theorem also simply assumes an invariant Lagrangian fibration exists, a condition verified in all known examples but not proved in general.","fun_headline_variants_meta":{"raw":{"variants":["Parabolic hyperkähler maps yield dense orbit fibers","Dense orbits and finite-order fibers for parabolic hyperkähler maps","New proof: parabolic hyperkähler maps have dense orbit fibers","Parabolic hyperkähler automorphisms: dense orbit fibers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000756,"raw_usage":{"total_tokens":3374,"prompt_tokens":972,"completion_tokens":2402,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":2328}},"tokens_in":588,"tokens_out":2402,"duration_ms":20731,"temperature":1.0,"reasoning_tokens":2328,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:26:16.115065+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete parabolic automorphism with invariant Lagrangian fibration (for instance on a K3 surface with an elliptic fibration or on the Hilbert scheme of two points on a K3), the generic rank of the translation vector in Betti coordinates; if any example yields rank strictly less than $2g$, Assertion (2) of Theorem A is false. Alternatively, test the relative-polarization construction by checking whether $m_D^* A_b$ equals $A_b^{\\otimes D^2}$ for a single line bundle $A_b$ rather than a formal sum of roots of a Pic$^0$ twist; a failure there would break Proposition 4.1 and the proof of maximal variation.","supporting_citations":[{"cited_title":"Parabolic automo rphisms of hyperkähler mani- folds","cited_arxiv_id":null,"evidence_quote":"Establishes that a positive iterate of a parabolic automorphism acts as a fiberwise translation on smooth fibers and that orbits are dense on almost all smooth fibers, providing the starting point and the notion of translation vector."},{"cited_title":"The geometric dynam ical Bogomolov and Northcott properties","cited_arxiv_id":null,"evidence_quote":"Supplies the local-to-global volume estimate for relatively polarized fibered endomorphisms that is the core mechanism for propagating volume bounds from open subsets of the base to all of $X$."},{"cited_title":"Mixed Ax-Schanuel for the universal abelia n varieties and some applica- tions","cited_arxiv_id":null,"evidence_quote":"Provides the previous maximal-rank theorem for Betti maps of abelian schemes under simplicity and dimension assumptions, the projective-case baseline that the new proof bypasses."},{"cited_title":"Cohomology of compact hyperkähler ma nifolds and its applications","cited_arxiv_id":null,"evidence_quote":"Gives the cohomological embedding theorem for hyperkähler manifolds that yields the $n^{2p}$ lower bound and hence the exact growth of the operator norm on $H^{p,p}$."},{"cited_title":"An application of p-adic integration to the dynamics of a birational transformation preserving a fibration","cited_arxiv_id":null,"evidence_quote":"Shows that the induced automorphism of the base has finite order and that an invariant Kähler form exists on the base, which is needed for the fibration setup and for Theorem B."},{"cited_title":"Degenerate twistor spaces for hyperk ähler manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the degenerate twistor deformation technique used to pass from projective to non-projective hyperkähler manifolds."},{"cited_title":"The Moser isot opy for holomorphic symplec- tic and C-symplectic structures","cited_arxiv_id":null,"evidence_quote":"Supplies the construction of C-symplectic forms whose associated complex structures make the twistor deformation work while preserving the automorphism."}],"review_version":1}