This post is about The hyperkähler period-index conjecture is false, joint with James Hotchkiss.

If I were still posting fortnightly links, Alexander Perry: The period-index conjecture is false would have prominently featured, because it disproves (as the title suggests) the period-index conjecture, which is famously a theorem due to de Jong in dimension 2, with a beautiful proof. But Alex found counterexamples in every dimension at least 3, with some assistance from an LLM.

The period-index conjecture

To understand the statement of the period-index conjecture, let us consider a field $K$ of transcendence degree $d$ over an algebraically closed field $k$. We attach two integers to a Brauer class $\alpha\in\mathop{\rm Br}(K)$:

  • the index $\mathop{\rm ind}(\alpha)$ (which measures the size of the unique division algebra in the Brauer class)
  • the period $\mathop{\rm per}(\alpha)$ (which is the order of $\alpha$ in the torsion abelian group $\mathop{\rm Br}(K)$)

The period-index conjecture, attributed to Colliot-Thélène, then predicts that \[ \mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{d-1}. \]

Using a bit of Galois cohomology, one can prove that

  • $\mathop{\rm per}(\alpha)\mid\mathop{\rm ind}(\alpha)$
  • the period and the index have the same prime factors

so certainly for every individual $\alpha$ there is some exponent for which the index divides the corresponding power of the period. The conjecture claims there is a uniform bound, which is moreover given in terms of the transcendence degree.

As a special case of this we can consider a smooth projective variety $X$ together with its function field $k(X)$, for which $\mathop{\rm Br}(X)\hookrightarrow\mathop{\rm Br}(k(X))$. Johan de Jong proved that it suffices to prove the period-index conjecture for $k(X)$ for the classes in the image of this inclusion, the unramified classes.

Alex constructed unramified classes in every dimension at least 3 which violate the period-index conjecture.

The hyperkähler period-index conjecture

Hyperkähler varieties (see also hyperkaehler.info) are a special kind of variety. They have rich geometry, strong constraints, and many tools to study them. In particular, their geometry is controlled to a large extent by $\mathrm{H}^2(X,\mathbb{Z})$, together with its Beauville–Bogomolov–Fujiki form.

Because of this, and various pieces of evidence, Huybrechts conjectured the stronger hyperkähler period-index conjecture, predicting that \[ \mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{(\dim X)/2} \] for unramified Brauer classes on smooth projective hyperkähler varieties.

Given that Alex started from an incorrect LLM-assisted counterexample, James and I independently tried constructing an LLM-assisted counterexample for the hyperkähler period-index conjecture, ending up with examples in different deformation types of hyperkähler fourfolds. Given the similarity in tools used (namely James's Hodge-theoretic index), it felt natural to write a joint paper.

So what are the counterexamples? First, observe that for the fourfolds we consider (whose integral cohomology is torsion-free) it is possible to write every Brauer class in terms of $\mathrm{H}^2(X,\mathbb{Z})$, as $\mathop{\rm Br}(X)\cong(\mathrm{H}^2(X,\mathbb{Z})/\mathop{\rm NS}(X))\otimes\mathbb{Q}/\mathbb{Z}$.

Then it becomes a matter of writing down classes in the lattice, giving us the polarization, the Brauer class, and a way to check the conditions for our two obstruction lemmas. For more details one is referred to the paper.

The results are

  • a period-2 Brauer class on certain hyperkähler fourfolds of type $\mathrm{Kum}^2$, whose index is divisible by 8
  • a period-2 Brauer class on certain hyperkähler fourfolds of type $\mathrm{K3}^{[2]}$, whose index is divisible by 8
  • a period-5 Brauer class on certain hyperkähler fourfolds of type $\mathrm{K3}^{[2]}$, whose index is divisible by 125

In each case the index is one factor too big: the conjecture predicts $\mathop{\rm ind}(\alpha)\mid 4$ resp. $\mathop{\rm ind}(\alpha)\mid 25$. The classical period-index conjecture only predicts $\mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^3$, so these are not, as far as we know, counterexamples to it.

This construction lives just outside the known cases of the hyperkähler period-index conjecture:

  • Huybrechts proved it for $X$ admitting a Lagrangian fibration, for classes whose period is coprime to an integer depending on $X$.
  • Huybrechts proved it for $X$ the Hilbert scheme of points on a K3 surface, for all classes, with a variant for generalised Kummer varieties.
  • Hotchkiss–Maulik–Shen–Yin–Zhang proved it for $X$ of $\mathrm{K3}^{[n]}$-type of Picard rank at least 2, for non-special coprime classes.
  • Bottini–Huybrechts removed the condition on the Picard rank, and also cover the special coprime classes whose period is squarefree.