The period-index problem for hyperkähler varieties: Lower and upper bounds

It is expected that a stronger form of the period-index conjecture holds for hyperkähler varieties. Following ideas of Hotchkiss, we provide further evidence for this expectation by proving a version in which the index is replaced by the Hodge-theoretic index. We also show that the hyperkähler period-index conjecture is optimal. As an application, we prove that Mumford-Tate general hyperkähler varieties cannot be covered by families of elliptic curves passing through a fixed point. By extending work of Hotchkiss, Maulik, Shen, Yin, and Zhang, we prove the hyperkähler period-index conjecture for non-special coprime Brauer class on hyperkähler varieties of K3^n-type without any restriction on the Picard number.

💡 Research Summary

The paper investigates the period‑index problem for hyperkähler varieties, proposing a stronger version of the classical period‑index conjecture and establishing both lower and upper bounds that are shown to be optimal. The authors build on Hotchkiss’s notion of a Hodge‑theoretic index, extending it to all hyperkähler manifolds and to Brauer classes that are coprime to a universal integer depending only on the variety.

The first main result (Theorem A) proves that for any coprime Brauer class α on a hyperkähler variety X, the Hodge‑theoretic index ind _H(α) divides the product of the period per α and half the complex dimension of X, i.e.
 ind _H(α) | per α·dim X⁄2.
The Hodge‑theoretic index is defined as the least common multiple of four indices attached to natural Hodge structures on X: the Mukai lattice r H(X), its symmetric powers Sⁿ r H(X), the Verbitsky component S H(X) (the sub‑algebra generated by H²), and the full even cohomology H²·(X). Each of these carries a rank function, and after twisting by the Brauer class (via a B‑field lift) one obtains an integral Hodge structure whose rank image defines the corresponding index.

Theorem B shows that the bound in Theorem A cannot be improved: for a Mumford–Tate general hyperkähler variety with Picard number ρ > b₂ − 6, any non‑special coprime class α satisfies
 per α·dim X⁄2 | ind α.
Thus the conjectured inequality ind α | per α·dim X⁄2 is optimal, and no stronger uniform bound can hold even for coprime classes. The “non‑special” condition refers to the square of the B‑field representing α avoiding certain algebraic classes; it is automatically satisfied for a very general point in the moduli space.

The paper then focuses on hyperkähler varieties of K3ⁿ‑type. By combining Markman’s monodromy results with O’Grady’s analysis of Brauer–Severi varieties, the authors remove the Picard‑rank hypothesis present in earlier work. Theorem D proves that for any non‑special coprime Brauer class α on a K3ⁿ‑type hyperkähler manifold (including the case of Picard rank 1), one has
 ind α | per α·dim X⁄2.
If α is “special” but its period has a reduced prime factorization with distinct primes, the same divisibility also holds. Consequently, Corollary 1.4 states that for a Mumford–Tate general K3ⁿ‑type hyperkähler variety of Picard number one, the period and index are equal up to the factor dim X⁄2.

An unexpected application concerns covering families of curves. Using the optimality result, the authors prove (Theorem E) that a Mumford–Tate general hyperkähler variety X cannot be covered by a dominating family of curves all passing through a fixed point unless the geometric genus g of the curves satisfies g ≥ dim X⁄2. In particular, for dimensions ≥ 4, no covering family of elliptic curves through a single point exists. This answers a question of Voisin about the existence of elliptic curve coverings for very general hyperkähler manifolds.

Methodologically, the paper introduces a systematic framework for twisting integral Hodge structures by Brauer classes, defines rank functions on the Mukai lattice and its derived structures, and exploits the rich symmetry of hyperkähler Hodge theory (Beauville–Bogomolov–Fujiki form, Verbitsky component, monodromy invariance). The authors also make essential use of the integral Hodge conjecture for Brauer–Severi varieties: when it holds, the Hodge‑theoretic index coincides with the classical index, thereby yielding the full period‑index conjecture for the classes considered.

Overall, the work provides decisive evidence that the period‑index conjecture for hyperkähler varieties is both true and sharp, introduces the Hodge‑theoretic index as a powerful new invariant, and connects the period‑index problem to broader questions in birational geometry, such as covering gonality and irrationality measures. The results are likely to influence future research on Brauer groups of higher‑dimensional holomorphic symplectic manifolds and on the interplay between Hodge theory, derived categories, and arithmetic invariants.