Equivalences via twisted hyperholomorphic sheaves from transverse Lagrangian fibrations

Abstract: Following ideas of Kapustka-Kapustka, we use Lagrangian fibrations to construct twisted hyperholomorphic sheaves on products of hyperkähler manifolds of K3$<sup>{[n]}$- and OG10-type. As applications, we prove the Lefschetz standard conjecture and the D-equivalence conjecture for hyperkähler manifolds of OG10-type.

• The paper constructs twisted hyperholomorphic vector bundles of rank n!k^n by convolving twisted Poincaré sheaves associated with transverse Lagrangian fibrations on K3^[n]- and OG10-type hyperkähler manifolds.
• The paper uses Hodge theory, LLV-equivariant deformations, stability, and Tate–Shafarevich twists to extend Fourier–Mukai equivalences across moduli spaces and establish universal twisted derived equivalences for these deformation types.
• The paper proves the Lefschetz standard conjecture and D-equivalence conjecture for projective OG10-type hyperkähler varieties, while also producing twisted equivalences between Fano varieties of lines on related cubic fourfolds.

The paper "Equivalences via twisted hyperholomorphic sheaves from transverse Lagrangian fibrations" (2608.13403) by Moritz Hartlieb and Saket Shah develops a new construction of twisted hyperholomorphic vector bundles on products of hyperkähler manifolds of $K$-type ($\mathrm{K3}^{[n]}$) and of OG-type (OG10). The construction follows a strategy proposed by Kapustka–Kapustka and yields two principal applications: proofs of the Lefschetz standard conjecture and the D-equivalence conjecture for projective hyperkähler varieties of OG10-type.

Overview of the construction

The starting point is a projective hyperkähler manifold $Y$ of $K$- or OG-type admitting two transverse Lagrangian fibrations $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$, meaning each fiber of $\pi_1$ meets every fiber of $\pi_2$ in finitely many points. Via Hodge-theoretic input, each fibration realizes $Y$ as a Tate–Shafarevich twist of a compactified abelian scheme: in the $K$ case this uses Markman's work on Lagrangian fibrations, while in the OG case it relies on Dutta–Mattei–Shinder's characterization of Tate–Shafarevich twists of Laza–Saccà–Voisin (LSV) fibrations associated to cubic fourfolds. This produces a diagram

$$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$

where $\mathrm{K3}^{[n]}$0 realizes $\mathrm{K3}^{[n]}$1 as a Tate–Shafarevich twist of $\mathrm{K3}^{[n]}$2 and $\mathrm{K3}^{[n]}$3 as a twist of $\mathrm{K3}^{[n]}$4. Work of Arinkin on compactified Jacobians, its twisted generalization by Bottini, and Yu's analogue for LSV fibrations supply Brauer classes $\mathrm{K3}^{[n]}$5, $\mathrm{K3}^{[n]}$6 together with twisted Poincaré sheaves inducing derived equivalences $\mathrm{K3}^{[n]}$7. The Fourier–Mukai kernel $\mathrm{K3}^{[n]}$8 of the composition is the central object: the paper's main structural result states that $\mathrm{K3}^{[n]}$9 is a twisted hyperholomorphic vector bundle of rank $Y$0 on $Y$1.

Hodge-theoretic framework

The deformation argument rests on Markman's theory of hyperholomorphic sheaves, adapted to the twisted setting. The authors formulate an assumption requiring that the Brauer classes be topologically trivial with B-field lifts such that the induced Hodge isometry $Y$2 between rational twisted Hodge structures is LLV-equivariant. Using Taelman's results identifying LLV-equivariant isometries with isometries of rational extended Mukai lattices $Y$3, they show that under this assumption the untwisted arguments of Markman carry over verbatim: a slope-stable kernel deforms over generic twistor paths in the moduli space $Y$4 of pairs, and any two points of a connected component can be joined by generic twistor paths. A technical point handled explicitly is that the Kapustka–Kapustka argument that a hyperholomorphic kernel induces a derived equivalence extends to twisted kernels; the only nontrivial modification concerns polystability of Hom bundles between fibers, resolved via polystability of endomorphism bundles and an indecomposability argument. The Brauer classes of the deformed kernels are controlled by a twisted variant of Čaldăraru's theorem, proved here by comparing topological twisting classes at the level of smooth transition functions.

Tate–Shafarevich twists of LSV systems

A key geometric input for the OG case is a Torelli-type theorem: if $Y$5 is a hyperkähler manifold of OG10-type with a primitive isotropic class $Y$6 such that $Y$7 is Hodge-isometric to $Y$8 for a very good cubic fourfold $Y$9, then $K$0 is birational to a Tate–Shafarevich twist of the LSV fibration $K$1; moreover, if $K$2 is nef, the birational map commutes with the Lagrangian fibrations. The proof combines the explicit period computation for $K$3, the identification of degenerate twistor deformations with Tate–Shafarevich families due to Abasheva–Rogov and Abasheva, and the birational Torelli theorem for OG10 manifolds. The nef case requires ruling out that reflections along prime exceptional divisors move the isotropic class, using a hyperbolic reflection group argument parallel to Markman's $K$4-type proof.

The authors note a genuine limitation of this characterization: when the period of $K$5 lies in the Hassett divisors $K$6 or $K$7, the cubic fourfold is not determined, and they can only conjecture the geometry — desingularized Beauville–Mukai systems $K$8 for $K$9 in the $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$0 case, and $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$1 birational to intermediate Jacobian fibrations of singular cubics in the $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$2 case.

Cohomological action of the Poincaré sheaf

On the rank-four sublattice $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$3 of the extended Mukai lattice spanned by the base class $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$4, a generalized theta divisor $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$5, and the hyperbolic plane generators, the Poincaré sheaf acts by an explicit formula involving half-integers $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$6. The crucial complementary result is that on the Lagrangian primitive cohomology $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$7, the Poincaré isometry restricts to multiplication by $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$8 integrally. The proof deforms to Noether–Lefschetz divisors where the primitive cohomology has a unique Hodge class, then propagates the sign using a lattice lemma stating that any vector in an indefinite lattice is a difference of two primitive vectors of arbitrarily large negative square. The restriction to large negative square is essential: obtaining a Poincaré sheaf requires avoiding finitely many Noether–Lefschetz divisors (in the K3 case) or the Hassett divisors $\pi_1, \pi_2 \colon Y \to \mathbb{P}^n$9 plus a codimension-two locus (in the cubic fourfold case).

Transverse fibrations from MBM classes

The examples with transverse fibrations are produced lattice-theoretically. Fixing $\pi_1$0 inside $\pi_1$1 or $\pi_1$2, surjectivity of the period map guarantees existence, and in the OG case one arranges that both quotients $\pi_1$3 correspond to very good cubic fourfolds. Since every integral algebraic class has square divisible by $\pi_1$4, choosing $\pi_1$5 larger than the uniform bound $\pi_1$6 on squares of MBM classes forces $\pi_1$7 to have no MBM classes at all; hence the Kähler cone equals the positive cone, all birational maps are biregular, and both boundary classes are nef, yielding Lagrangian fibrations whose product morphism is finite because $\pi_1$8 is ample. A Fujiki-formula computation gives the precise intersection number: transverse fibers meet in $\pi_1$9 points up to multiplicity, which also computes the rank of the convolution kernel $\pi_2$0.

Stability of $\pi_2$1 is established fiberwise: each restriction $\pi_2$2 is $\pi_2$3-stable with respect to a suitable polarization, by Bottini's proposition producing twisted locally free sheaves from finite covers, and stability is then extended from ample to Kähler classes via a lemma observing that stability depends only on pairings with rational Hodge classes, hence is insensitive to transcendental perturbations of the polarization. Combining with Markman's deformation criterion yields hyperholomorphicity, and consequently: for any hyperkähler manifold $\pi_2$4 of $\pi_2$5- or OG-type there exists another manifold $\pi_2$6 of the same type and a twisted vector bundle on $\pi_2$7 realizing a twisted derived equivalence. This universality statement is what powers the applications.

Applications

Lefschetz standard conjecture. For projective hyperkähler manifolds of OG10-type, the degree-reversing Hodge isometry induced by the positive-rank Fourier–Mukai kernel, combined with Markman's Lemma 1.6, proves Grothendieck's Lefschetz standard conjecture. This strictly generalizes earlier results of LSV-based and FFZ-type papers that covered lower-dimensional strata of the moduli space.

$\pi_2$8-cyclic isometries and Fano varieties. With explicit markings, the B-fields of the convolution kernel are computed exactly: $\pi_2$9 and $Y$0. The resulting isometry $Y$1 restricts on Lagrangian primitive cohomology to a $Y$2-cyclic isometry in Buskin's sense, realized as reflection across $Y$3 composed with an integral isometry. Via the Beauville–Donagi isometry $Y$4, this produces a $Y$5-cyclic Hodge isometry between the second cohomologies of the Fano varieties of lines $Y$6 and $Y$7, which by Markman's criterion yields a twisted derived equivalence $Y$8.

D-equivalence conjecture. Following the strategy of Maulik–Shen–Yin–Zhang, the paper proves that birational projective hyperkähler manifolds of OG10-type are derived equivalent. The proof reduces to adjacent chambers separated by an MBM class wall. Two lattice-theoretic lemmas replace their $Y$9-type analogues: an Eichler-criterion argument producing a divisibility-one class $K$0 with prescribed pairing against the MBM class $K$1 (using that the discriminant of $K$2 is $K$3), and a construction modifying $K$4 by a multiple of a transcendental-direction vector to obtain an isotropic divisibility-one class $K$5. Choosing the model manifold with $K$6 and transporting along an isometry sending $K$7 to $K$8, the B-field component $K$9 transports to $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$0, which represents the trivial Brauer class since $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$1 is algebraic and $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$2 integral. Chaining equivalences through the auxiliary manifold gives $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$3.

The authors remark that a fully twisted version of the MSYZ theorem should hold, contingent on whether a Brauer class admits a B-field lift isotropic for the BBF form — a question left open.

Limitations and open questions

Several caveats qualify the results. The construction requires the parameter $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$4 to exceed the deformation-type-dependent MBM bound $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$5, so the initial models are special points of the moduli space; the passage to arbitrary manifolds proceeds by twistor deformation rather than by direct construction. The identification of Tate–Shafarevich twists with LSV fibrations fails over the Hassett divisors $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$6 and $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$7, where the geometry is only conjectural. The paper also leaves open the comparison with Markman's original construction: is the twisted hyperholomorphic bundle $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$8 deformation equivalent along diagonal twistor lines to the bundle $$X \to \mathbb{P}^n \leftarrow Y \to \mathbb{P}^n \leftarrow Z,$$9? Finally, the twisted strengthening of the MSYZ theorem remains unproven, tied to the isotropic B-field lift question noted above.

Conclusion

The paper establishes that transverse Lagrangian fibrations on hyperkähler manifolds of $\mathrm{K3}^{[n]}$00- and OG-type produce twisted hyperholomorphic vector bundles on products, via convolution of twisted Poincaré sheaves on compactified abelian fibrations. The construction transfers the known $\mathrm{K3}^{[n]}$01-type technology — Lefschetz standard conjecture, cyclic Hodge isometries, and the D-equivalence conjecture — to OG10-type, resolving both conjectures in that deformation type and yielding new twisted derived equivalences between Fano varieties of lines on related cubic fourfolds.