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Correspondences for hyperk\"ahler varieties with large Picard numbers

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Pith's one-line read Hyperkähler manifolds of K3^[n] type with large Picard number are related by algebraic correspondence to abelian varieties, generalizing Morrison's theorem for K3 surfaces.

desk verdict Clean, short packaging of Morrison's Oda solution for projective K3^[n]-type manifolds under a transparent lattice condition; expected but useful. read the letter →

arxiv 2607.09622 v1 pith:H6E7IIS3 submitted 2026-07-10 math.AG

Ljudmila Kamenova , Abhinav Kumar This is my paper · ORCID
classification math.AG MSC 14J4214C3014F05
keywords hyperkählermanifoldsK3^[n]typetranscendentallatticePicardnumberalgebraiccorrespondencesMorrison'stheoremOdaconjectureHodgeisometry
topics The Hodge Conjecture
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

Morrison proved that every algebraic K3 surface whose transcendental lattice embeds rationally into three hyperbolic planes is related by an algebraic correspondence to an abelian surface. This note extends that statement to projective hyperkähler manifolds of K3^[n] type. Under the same rational embedding condition on the transcendental lattice, such a manifold is shown to be birational to a moduli space of sheaves on a K3 surface that itself satisfies Morrison's hypothesis, and therefore inherits a correspondence with an abelian variety that identifies the rational transcendental Hodge structures. The argument assembles known results on moduli of sheaves, Hilbert schemes, and the algebraicity of rational Hodge isometries for this deformation type, and records the arithmetic conditions on the Picard rank that make the embedding automatic.

What carries the argument

The rational embedding T(X)⊗Q o U^{3}⊗Q, which forces the existence of a K3 surface S with T(S) Hodge-isometric to T(X); X is then birational to a moduli space of sheaves on S, and the known correspondence from S to an abelian surface composes with the algebraicity of rational Hodge isometries of K3^[n] type.

What would settle it

Exhibit a projective K3^[n]-type manifold whose transcendental lattice embeds rationally into U^{3} yet admits no algebraic correspondence inducing a rational Hodge isometry with the transcendental lattice of any abelian variety.

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Extended reading notes

Core claim

If X is a projective hyperkähler manifold of K3^[n] type and its transcendental lattice T(X) embeds rationally into U^{3}, then there exists an abelian variety A together with an algebraic correspondence that induces a Hodge isometry T(X)⊗Q ≅ T(A)⊗Q.

Load-bearing premise

The proof depends on the theorem that every rational Hodge isometry between projective hyperkähler varieties of K3^[n] type is algebraic; if that fails for the lattices that arise, the correspondence chain breaks.

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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

0 major / 5 minor

Summary. The paper generalizes Morrison’s solution of the modified Oda conjecture from K3 surfaces to projective hyperkähler manifolds of K3^[n] type. The main result (Theorem 1.3 / 3.1) states that if T(X) ⊗ Q embeds into U^{3} ⊗ Q, then there exists an abelian variety A and an algebraic correspondence inducing a Hodge isometry T(X) ⊗ Q ≅ T(A) ⊗ Q. The proof proceeds by Nikulin’s embedding theorem to produce a K3 surface S with T(S) Hodge-isometric to T(X), Markman/Piroddi–Ortiz realization of X as birational to a moduli space of sheaves on S (hence to S^[n]), the elementary Hilbert–Chow correspondence, Morrison’s correspondence from S to an abelian surface, and Markman’s theorem that rational Hodge isometries of projective K3^[n]-type varieties are algebraic. A short lattice-theoretic lemma and corollary give explicit Picard-number criteria under which the embedding hypothesis holds automatically.

Significance. The result cleanly extends a classical theorem of Morrison–Mukai to the most studied higher-dimensional hyperkähler deformation type, and it does so by a transparent composition of already-published tools (Nikulin, Markman, Piroddi–Ortiz, Morrison). The explicit numerical criteria in Corollary 3.4 make the statement immediately usable for concrete examples of high Picard rank. While the argument is short and largely synthetic, the uniform treatment of the correspondence chain and the lattice criteria constitute a useful contribution to the recent literature on hyperkähler manifolds with large Picard number.

minor comments (5)
  1. The abstract and introduction speak of “known examples of hyperkähler manifolds such as pointed Hilbert schemes,” yet the body of the paper proves the statement only for K3^[n] type; the remaining types are relegated to a brief sketch in Remark 3.5. A single clarifying sentence in the abstract would avoid overstatement.
  2. In the proof of Theorem 3.1 the authors write “primitive vector v” while noting that the cited theorem does not require primitivity; a parenthetical reference to the construction of a primitive vector would remove the slight inconsistency.
  3. Lemma 3.2 is stated for quadratic spaces over Q of signature (2,k); the parenthetical remark that k=4 is impossible for signature reasons is correct but could be made more precise by recalling that the ambient space U^{3} has signature (3,3).
  4. Typographical inconsistencies appear throughout: “hyperk¨ahler” vs. “hyperkähler,” “CORREPONDENCES” in the running header, and occasional missing spaces after periods. A light copy-edit would clean these up.
  5. The appendix collects standard lattice facts; a one-sentence pointer to Nikulin’s original papers or to a modern survey would help readers who are not lattice specialists.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.1 is a linear chain of external lattice, period-map, moduli, and correspondence theorems with no self-definitional or fitted steps.

full rationale

The claimed derivation of Theorem 3.1 (and its restatement as Theorem 1.3) proceeds by: (i) Nikulin’s embedding theorem to place T(X) primitively into the K3 lattice L; (ii) surjectivity of the period map to produce an algebraic K3 surface S with a Hodge isometry T(S) ≅ T(X); (iii) Markman / Piroddi–Ortiz identification of X with a moduli space of sheaves on S, hence birational to S^[n]; (iv) the elementary Hilbert–Chow correspondence from S^[n] to S; (v) Morrison’s theorem relating S to an abelian surface A; and (vi) Markman’s algebraicity theorem for rational Hodge isometries of projective K3^[n]-type varieties. None of these steps is defined in terms of the final correspondence, none is a parameter fit renamed as a prediction, and none rests on a uniqueness theorem or ansatz from the present authors. The only self-contained lattice lemma (Lemma 3.2) is an elementary Witt-cancellation argument over Q that does not feed back into its own hypotheses. The paper therefore contains no circular reduction; the result stands or falls with the cited external theorems.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper is a pure-math note that rests entirely on standard lattice theory, the global Torelli theorem for K3 surfaces, and a short list of recent theorems on moduli of sheaves and algebraicity of Hodge isometries. No free parameters are fitted; no new geometric entities are postulated. The only non-standard inputs are the external theorems listed as domain assumptions.

assumptions (6)
  • standard math Nikulin’s embedding theorem (Thm A.8): an even lattice of signature (t+,t−) with l(A_M) ≤ rank(L)−rank(M)−2 embeds uniquely and primitively into an even unimodular lattice of larger signature.
    Used at the beginning of the proof of Theorem 3.1 to embed T(X) into the K3 lattice.
  • domain assumption Surjectivity of the period map for K3 surfaces (Morrison Cor. 1.9 / Thm 1.7).
    Produces the K3 surface S with T(S) ≅ T(X) as Hodge structures.
  • domain assumption Markman / Piroddi–Ortiz: a projective hyperkähler of K3^[n] type is induced by T(S) iff it is birational to a moduli space of sheaves on S (Thm 2.4 / [PRO25, Thm 3.7]).
    Central step that realises X as a moduli space on the auxiliary K3 surface S.
  • domain assumption Markman [Mar24, Thm 1.1]: every rational Hodge isometry between projective hyperkähler varieties of K3^[n] type is algebraic.
    Converts the rational Hodge isometry between X and S^[n] into an algebraic correspondence.
  • domain assumption Morrison’s solution of the modified Oda conjecture for K3 surfaces (Thm 1.1).
    Supplies the final correspondence from the auxiliary K3 surface S to an abelian surface A.
  • standard math Witt cancellation for quadratic spaces over Q (Lemma A.11).
    Used in the proof of the embedding criteria of Lemma 3.2.

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Pith. "Pith review of Correspondences for hyperk\"ahler varieties with large Picard numbers." pith.science (2026). https://pith.science/paper/H6E7IIS3

@misc{pith2026260709622,
  author       = {Pith},
  title        = {Pith review of: Correspondences for hyperk\"ahler varieties with large Picard numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6E7IIS3}},
  note         = {Machine review of arXiv:2607.09622}
}
read the original abstract

In this note, we explore the connection between hyperk\"ahler manifolds with large Picard numbers and abelian varieties. In particular, we are interested in Morrison's solution to the (modified) Oda's conjecture: every K3 surface whose Picard group is large enough (in a certain precise sense) must be related via an algebraic correspondence to an abelian surface. We generalize this theorem to the case of known examples of hyperk\"ahler manifolds such as pointed Hilbert schemes on K3 surfaces.

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Works this paper leans on

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