{"id":"d691f8db-7b79-4d30-8ef3-d8d15e30cf23","arxiv_id":"1908.11772","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Corrected proofs for the global Torelli theorem for hyperkähler manifolds are supplied by switching from the Teichmüller space to the marked moduli space and by adding an ergodic lemma on lattice isometry groups.","lead":"This erratum corrects a false claim in the author's earlier proof of the global Torelli theorem for hyperkähler manifolds. It shows the main results survive after replacing the Teichmüller space by the marked moduli space and the mapping class group by a finite-index quotient.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6's proof fixes a very ample L on (M,I), so it only covers projective deformations; the promised adjustment for non-algebraic I is never written, leaving Theorem 3.1(i)'s finite-orbit claim unproved for non-projective hyperkähler manifolds.","rationale":"The erratum's central claim is that the original global Torelli results survive after replacing the Teichmüller space by the marked moduli space and the mapping class group by a quotient of the Torelli group. The proof chain is Theorem 1.1, Theorem 2.6, and then Theorem 3.1, with 'Theorem 3.1 (i) is Theorem 2.6 (ii)'. The weakest link in that chain is the proof of Theorem 2.6, which chooses a very ample line bundle on (M,I). This is not available for non-projective hyperkähler manifolds, and the erratum's promise that the Bakker-Lehn proof 'needs some adjustments' is never fulfilled. That is load-bearing because Theorem 3.1(i) is exactly the finite-orbit statement needed for the corrected Torelli picture. I do not see the concern as fatal: the missing step is likely supplied by noting that every Teichmüller component contains projective points and that the monodromy group is constant on a component, so the projective proof can be run at a projective point in the same component. But the manuscript does not say this, and a reader cannot verify the non-projective case from the text as written. I also considered whether Theorem 1.1 is compromised by citing the erroneous [V1, Theorem 3.5] and whether Lemma 2.7's ergodicity step is too terse; these are secondary, since the erratum explicitly isolates the false part of [V1, Theorem 3.5] as (iv) and Lemma 2.7 is a standard Moore-theorem argument. The unresolved projective assumption is the single most decisive gap, and it matches the reader's weakest_assumption; therefore the CONDITIONAL verdict should stand.","tokens_in":8131,"tokens_out":20417,"duration_ms":190969,"concrete_test":"Analytical check: for a non-algebraic I with Picard rank 0, determine whether the connected component Teich_I contains a projective point J and whether Mon_I = Mon_J for J in Teich_I; if so, rewrite the proof of Theorem 2.6 with L chosen on (M,J) instead of (M,I) and verify every step (density of Teich_L, boundedness of the Hilbert scheme, Borel extension, Lemma 2.7) depends only on Teich_I. If the rewrite goes through, the concern is an expositorial gap; if not, Theorem 2.6(ii) and Theorem 3.1(i) are unsupported for non-projective hyperkähler manifolds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2, proof of Theorem 2.6, states a result for an arbitrary hyperkähler manifold (M,I), but the proof begins 'Fix a very ample line bundle L on (M,I)' and then works entirely with the Hilbert scheme, Teich_L, and Per_eta. A very ample line bundle exists only if (M,I) is projective; a very general non-algebraic hyperkähler deformation can have Picard rank 0. The text says the Bakker-Lehn proof 'needs some adjustments' for non-algebraic deformations, but no adjustment is exhibited. Since Theorem 3.1(i) is proved as an immediate consequence of Theorem 2.6(ii), the finite-orbit/finite-stabilizer action of the Torelli group on Teichmüller components—the substantive correction on which the erratum's central claim rests—is not established for non-projective I. A plausible repair exists (replace I by a projective J in the same Teichmüller component, using constancy of the monodromy group on a component), but the erratum does not state it; hence the argument is conditional, not complete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This erratum addresses an error in Verbitsky's earlier paper on the mapping class group and global Torelli theorem for hyperkähler manifolds. The error stems from a misquotation of Sullivan's theorem, exposed by Kreck and Su, which invalidated the claimed finiteness of the kernel of the mapping class group action on cohomology. The erratum states corrected versions: Theorem 1.1 gives finite-index image of the mapping class group in the isometry group of the Bogomolov–Beauville–Fujiki form; Theorem 2.6 claims finite index of the monodromy group and finiteness of components of the marked moduli space; Theorem 3.1 claims that the Torelli group acts on the components of the Teichmüller space with finitely many orbits and finite stabilizers, and that any element fixing a point acts trivially on the component. Section 4 lists the statements of the original paper that are false and indicates where corrected versions are given.","tokens_in":8447,"tokens_out":13023,"duration_ms":115259,"significance":"If the corrected statements hold, the global Torelli theorem survives in a modified but substantive form, with the Teichmüller space replaced by the marked moduli space and with explicit finite-orbit/finite-stabilizer control on the Torelli action. The erratum is honest and useful: it publicly identifies the misquotation and the affected results, and it provides a clear target statement in Theorem 3.1. The paper also presents a self-contained Lemma 2.7 with an ergodicity argument, which is a useful ingredient. However, as submitted, the proof of the central replacement theorem is incomplete for non-projective hyperkähler manifolds, and some citations to the partially retracted original paper are not sufficiently precise. The erratum is therefore a valuable but not yet fully rigorous correction.","major_comments":[{"comment":"Theorem 2.6 is stated for an arbitrary hyperkähler manifold (M,I), but the proof begins by fixing a very ample line bundle L on (M,I), which exists only when (M,I) is projective. The sentence immediately before this step says that the Bakker–Lehn proof 'needs some adjustments' for non-algebraic deformations, yet no such adjustment is written. Because Theorem 3.1(i) is deduced directly from Theorem 2.6(ii), the finite-orbit and finite-stabilizer statement for the Torelli group action is not proved for non-projective hyperkähler manifolds. A repair using a projective deformation in the same Teichmüller component and the constancy of monodromy on a component is plausible, but it needs to be stated and checked.","section":"Section 2, proof of Theorem 2.6"},{"comment":"The proof of Theorem 1.1 asserts that the image of φ is described in [V1, Theorem 3.5], while Section 4 declares Theorem 3.5 to be false as stated and only the image statement is retained in a corrected form. Since Theorem 2.6 later relies on Theorem 1.1, the erratum should isolate exactly which assertions in [V1, Theorem 3.5] remain valid and indicate why the retracted statements do not affect the image computation; as written, this is an unsupported reliance on a partially retracted theorem.","section":"Section 1, Theorem 1.1"},{"comment":"The finiteness of the group of complex automorphisms acting trivially on H^2(M) is attributed to [V1, Theorem 4.26], but Section 4 lists parts (ii) and (iii) of that theorem as false. If the finiteness assertion is part (i) and remains valid, the manuscript should say so explicitly. The sketch given in the proof, namely that the group of isometries of a compact metric space is compact, only yields compactness; finiteness requires the additional fact that Aut(M,I) is discrete, which is not stated.","section":"Section 2, Claim 2.1"}],"minor_comments":[{"comment":"The assertion that 'Teich_L is dense in Teich_η and has the same number of connected components' needs a proof, since a dense open subset of a connected space can be disconnected; the authors should specify why the complement has codimension at least two or otherwise justify the connectedness statement.","section":"Section 2, proof of Theorem 2.6"},{"comment":"The introductory paragraph quotes Sullivan's theorem for manifolds with nilpotent fundamental group and dimension at least 5, while Theorem 1.3 states the theorem for compact simply-connected manifolds; the hypotheses should be stated consistently.","section":"Section 1, introductory paragraph"},{"comment":"The final paragraph says that Γ is replaced by Γ/K0, 'where K0 is a subgroup of all elements acting trivially on M'; this should presumably be 'acting trivially on H^2(M)', since the Torelli group is defined via the cohomology action.","section":"Section 4, final summary"},{"comment":"The reference [Kn] is listed in the bibliography but is never cited in the text.","section":"References"},{"comment":"The statement that each connected component of Teich/K is diffeomorphic to a corresponding component of Teich depends on the pointwise triviality of the stabilizer of a component; this use of Claim 2.1 should be made explicit.","section":"Section 2, Definition 2.2"}],"recommendation":"major_revision","confidential_remarks":"The erratum is a genuine attempt to correct a serious error and the intended replacement theorem is plausible. However, the proof of Theorem 2.6 is incomplete in the non-projective case, and the reliance on [V1, Theorem 3.5] and [V1, Theorem 4.26] is not made precise after the partial retraction. These gaps are fillable, so I do not recommend rejection, but the current text is not yet a complete correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: this erratum is honest and mostly convincing, but the key correction is not fully proved as written. If you work on hyperkähler geometry, you should read it carefully and probably cite the corrected statements.\n\nWhat is actually new: the paper retracts the false Sullivan quotation and gives corrected versions of the main claims. Theorem 1.1 fixes the mapping class group image, Theorem 2.6 fixes the monodromy group and marked moduli, Theorem 3.1 fixes the Torelli group action on Teichmüller space, and there is a new Lemma 2.7 about generating finite-index subgroups of lattice isometry groups from stabilizers of positive vectors. The Lemma 2.7 proof via Moore ergodicity is a nice piece of work and seems plausible. The erratum also openly lists which statements in [V1] are false and which sections contain the corrections.\n\nWhere it lands soft: the proof of Theorem 2.6 starts with \"Fix a very ample line bundle L on (M,I)\". That only exists when (M,I) is projective, yet the theorem is stated for all hyperkähler I. The author says the Bakker–Lehn proof \"needs some adjustments\" for non-algebraic deformations but never exhibits them. Since Theorem 3.1(i) is proved as an immediate consequence of Theorem 2.6(ii), the finite-orbit claim for the Torelli group action is not established for non-projective manifolds. This is fixable — a standard density argument should let you replace I by a projective J in the same Teichmüller component — but it is not in the paper. The second soft spot is smaller: Claim 2.1 cites [V1, Theorem 4.26] for finiteness of certain automorphisms, even though parts (ii)-(iii) of that theorem are retracted in Section 4. The specific part used appears to survive, but the citation should have been made precise.\n\nOverall, the central argument for the corrected global Torelli theorem holds up, modulo the projective gap. The paper deserves a serious referee, who should ask the author to supply the missing non-projective adjustment and to clarify the V1 citation. It is useful reading for anyone working on hyperkähler moduli or Torelli, and I would probably bring it to the reading group.","headline":"An honest erratum with a real but fixable gap: Theorem 2.6 is proved only for projective deformations, and the promised adjustment for non-algebraic ones is missing.","tokens_in":8898,"tokens_out":7341,"would_cite":true,"duration_ms":62926,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C26","32G13"],"pacs":[],"model":"deepseek-v4-flash","headline":"The global Torelli theorem for hyperkähler manifolds still holds after a corrected mapping-class-group argument.","keywords":["hyperkähler manifold","irreducible holomorphically symplectic manifold","global Torelli theorem","mapping class group","Torelli group","Teichmüller space","monodromy group","period map"],"falsifier":"Produce a compact hyperkähler manifold whose complex structure is not projective and for which the monodromy group maps to a subgroup of $O(H^2(M,\\mathbb{Z}))$ of infinite index; equivalently, a marked moduli space with infinitely many connected components. A concrete calculation for a known non-projective deformation of a generalized Kummer variety or of a Hilbert scheme of points on a K3 surface would settle the point.","tokens_in":7931,"feed_emoji":"","tokens_out":7687,"duration_ms":67287,"temperature":0.7,"pith_summary":"An earlier proof of the global Torelli theorem for compact hyperkähler manifolds relied on a quotation of a classical theorem claiming that the mapping class group maps to cohomology automorphisms with finite kernel. That quotation is wrong: the kernel can be infinite, as shown by counterexamples. This erratum retracts the false statements and replaces them with a corrected version in which the global Torelli theorem still holds after changing terminology. The key corrected statement is Theorem 3.1: the Torelli group acts on the connected components of the Teichmüller space with finitely many orbits and finite stabilizers, and any element fixing a point acts trivially on its component. A reader should care because the global Torelli theorem is the standard structural result for moduli of hyperkähler manifolds; the paper shows it stands, but in a weaker form than originally claimed.","feed_headline":"Torelli theorem survives a false quotation","feed_subtitle":"Corrected proof: Torelli group has finitely many orbits on Teichmüller components, finite stabilizers.","key_machinery":"The carrying object is the period map from each component of the Teichmüller space to the Grassmannian of positive 2-planes in $H^2(M,\\mathbb{R})$, together with the monodromy group $\\mathrm{Mon}_I$. The proof that Torelli elements act trivially on fixed components uses the identification of a period fiber with the set of Kähler chambers in $H^{1,1}$; the proof of finiteness of orbits uses polarization by a very ample line bundle (a line bundle whose sections embed the manifold into projective space), boundedness of Hilbert schemes, the compactification of arithmetic quotients of bounded symmetric domains, an extension theorem for period maps, and an ergodicity lemma (Lemma 2.7) showing that finite-index fixators of all positive-square vectors generate a finite-index subgroup of the orthogonal group. Each element fixing a point is a complex automorphism, and automorphisms acting trivially on $H^2$ are finite because they preserve the unique Calabi–Yau metric.","core_discovery":"The central claim is that the global Torelli theorem for compact hyperkähler manifolds remains true after replacing the Teichmüller space by the marked moduli space and after replacing the mapping class group by a finite-index subgroup of the Torelli group. Concretely, the Torelli group $K$, the kernel of the action on $H^2(M,\\mathbb{R})$, acts on the set of connected components of the Teichmüller space with finitely many orbits and finite stabilizers, and every element of $K$ that fixes a point fixes its entire connected component. This makes the quotient $\\mathrm{Teich}/K$ a finite union of components, each diffeomorphic to a component of $\\mathrm{Teich}$, and it implies the monodromy group has finite-index image in the integral orthogonal group of the Bogomolov–Beauville–Fujiki form. The earlier claim that the kernel of $\\Gamma \\to \\mathrm{Aut}(H^*(M))$ is finite is withdrawn; only the image is controlled. The correction preserves the applications, including the finite-component statements for marked moduli and the Torelli theorem in its amended form.","pith_inferences":["If the non-algebraic gap is genuine, the theorem as stated may be narrower than claimed: finite-index monodromy would be established only for projective deformations, and the non-projective case would remain open.","The correction points to the algebraic mapping class group built from the minimal model, not just cohomology, as the right invariant; one could test whether the Torelli group of other hyperkähler examples is infinite and hence whether their Teichmüller spaces have infinitely many components.","Lemma 2.7 is a purely lattice-theoretic statement that may extend to other arithmetic groups: any collection of finite-index fixators of positive vectors in a quadratic lattice generates a finite-index subgroup, so similar finite-orbit conclusions could hold for other moduli problems with period maps."],"forward_implications":["The global Torelli theorem holds in its amended form: the marked moduli space $\\mathrm{Teich}/K$ is a finite union of components, each diffeomorphic to a component of the Teichmüller space.","The image of the mapping class group in $O(H^2(M,\\mathbb{Z}))$ has finite index, even though its kernel may be infinite; the orthogonal group remains the right target for the monodromy.","The Teichmüller space has finitely many connected components exactly when the Torelli group is finite; for manifolds with infinite Torelli group, such as the generalized Kummer fourfold, it has infinitely many components.","Any Torelli element that fixes a Teichmüller point fixes the whole component, so the original rigidity phenomenon is preserved.","The previously cited erroneous uses are repairable: Theorem 3.1 supplies enough to prove the affected results, including the universal-fibration application, with a missing construction supplied by a later source."],"supporting_citations":[{"why":"Supplies the counterexamples showing the kernel of the map to cohomology automorphisms can be infinite, motivating the correction.","marker":"[KS]"},{"why":"Provides the correct theorem on commensurability of the mapping class group with the integer points of the algebraic mapping class group, replacing the misquoted finite-kernel claim.","marker":"[Su]"},{"why":"The original paper whose statements are corrected; its hyperkähler techniques (local Torelli, cohomology computation, automorphism finiteness) are reused.","marker":"[V1]"},{"why":"Supplies the method for proving finite index of the monodromy image, adapted here for projective deformations.","marker":"[BL]"},{"why":"Gives compactification of arithmetic quotients of bounded symmetric domains, used to make the period quotient quasiprojective.","marker":"[BB]"},{"why":"Extension theorem for maps to arithmetic quotients of symmetric spaces, used to show algebraicity of the period map.","marker":"[B]"},{"why":"Ergodicity theorem for flows on homogeneous spaces, used in Lemma 2.7 to show the generated group acts ergodically on the quadric of positive vectors.","marker":"[Mo]"},{"why":"Provides the arithmetic-lattice property used in Lemma 2.7 to control covolume and conclude finite index.","marker":"[WM]"},{"why":"Identifies period fibers with the set of Kähler chambers, used to show a Torelli element fixing a point fixes its component.","marker":"[Ma1]"}],"fun_headline_variants":["Torelli theorem survives error correction","False quotation fixed: Torelli still holds","Hyperkähler Torelli: corrected version stands","Mapping class group fix keeps Torelli theorem","Torelli theorem for hyperkähler: erratum preserves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the central finiteness statement assumes the hyperkähler manifold is projective, choosing a very ample line bundle whose sections embed the manifold into projective space, while the theorem is declared for all hyperkähler complex structures; the text says the non-projective case needs adjustments and does not supply them.","fun_headline_variants_meta":{"raw":{"variants":["Torelli theorem survives error correction","False quotation fixed: Torelli still holds","Hyperkähler Torelli: corrected version stands","Mapping class group fix keeps Torelli theorem","Torelli theorem for hyperkähler: erratum preserves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000645,"raw_usage":{"total_tokens":2957,"prompt_tokens":928,"completion_tokens":2029,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":1953}},"tokens_in":544,"tokens_out":2029,"duration_ms":12452,"temperature":1.0,"reasoning_tokens":1953,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:07:01.140195+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a compact hyperkähler manifold whose complex structure is not projective and for which the monodromy group maps to a subgroup of $O(H^2(M,\\mathbb{Z}))$ of infinite index; equivalently, a marked moduli space with infinitely many connected components. A concrete calculation for a known non-projective deformation of a generalized Kummer variety or of a Hilbert scheme of points on a K3 surface would settle the point.","supporting_citations":[],"review_version":1}