Perry: period-index is false

I was blown away last night to read that Alex Perry had disproved the period-index conjecture with the help of AI, specifically some version of GPT [17]. This problem is what my PhD thesis was about, so it is much more than a curiosity to me.

The period-index conjecture

The conjecture asserts a relationship between the order of Brauer classes and the sizes of the division algebras which represent them. The period is the order in the Brauer group; the index is the square root of the $k$-dimension of the division algebra (a twisted form of a matrix algebra). One has that $\per(\alpha)$ divides $\ind(\alpha)$ and they have the same prime divisors.

Conjecture. Let $K=\bC(X)$ be the function field of a $d$-dimensional variety over the complex numbers. If $\alpha\in\Br(K)$, then $\ind(\alpha)$ divides $\per(\alpha)^{d-1}$.

For $d=0$, $K=\bC$ so $\Br(K)=0$. For $d=1$, Tsen’s theorem implies that $\Br(K)=0$. The $d=2$ case is a famous result of Johan de Jong [8]. Work of Lieblich over $\bF_q$ [15] and AAIKL over $p$-adic fields [2] gives additional results for surfaces. But, there is no known example of a higher-dimensional ($d>3$) function field over any base field for which there is a uniform bound $N$ such that $\ind(\alpha)|\per(\alpha)^N$. Work of Krashen and Matzri [14] uses the Bloch–Kato conjectures to give bounds that depend on the prime divisors of the period.

This conjecture is folklore. It was first raised in print by Jean-Louis Colliot-Thélène in 2001 [6]. Work of de Jong and Starr [10] also reduces the general conjecture to unramified classes, i.e., classes $\alpha$ which lift to $\Br(X)\subseteq\Br(\bC(X))$ for a smooth proper model $X$.

Theorem (Perry). There is a smooth complex $3$-fold $X$ and a class $\alpha\in\Br(X)$ such that $\per(\alpha)=2$ and $\ind(\alpha)=8$.

This disproves the conjecture and it can be used to disprove it in all dimensions bigger than $3$ as well.

Surprised, but not surprised

In my PhD thesis, I introduced a more computable variant of the index, the étale index, and showed that it divides the index and is divisible by the period. I gave general upper bounds on this index and showed later with Ben Williams that it is not equal in all cases to the period; see [1], [3].

Later, Ben Williams and I introduced and studied the period and index problem for topological Brauer classes, elements in $\H^3(X,\bZ)_\tors$ of finite CW complexes. We gave general upper bounds. In particular, we showed that if $X$ has dimension $2d$ and $\per(\alpha)$ is prime to $(d-1)!$, then

\[\ind(\alpha)|\per(\alpha)^{d-1},\]

exactly the expected bound in algebraic geometry. We did this first in [4] for $d\leq 3$ and in a later paper for all $d$ [5], and we proved upper bounds in general without the coprimality condition. However, when $d=3$, we also proved that there were $6=3\cdot 2$-dimensional finite CW complexes and Brauer classes $\alpha$ with $\per(\alpha)=2$ and $\ind(\alpha)=8$ [4]. In fact, the $6$-skeleton of any CW structure on the Eilenberg–Mac Lane space $K(\bZ/2,2)$ works. Later, my PhD student Xing Gu carried this program out through dimension $8$, where the naive conjecture fails for $p=3$ as well [11], [12]. It remains an interesting problem to compute the exact topological period-index bounds in higher dimensions. This depends on getting a better understanding of the integral cohomology of spaces like $\mathrm{BPU}_n$.

The work with Williams suggested that period-index should be false in exactly the spot that Perry found it was. Indeed, when is a topological problem harder than an algebraic one when it comes to bundles?

However, Crowley–Grant later proved in [7] that one could not realize our topological obstructions in the cohomology of a smooth projective $3$-fold. Later, Hotchkiss showed (in private correspondence to me) that one could not ever get topological obstructions in any dimension.

Speaking of Hotchkiss, $\ldots$

In his thesis, Hotchkiss produces another flavor of index, which he calls the Hodge-theoretic index and uses intersection theory [13]. This perspective was further developed by de Jong and Perry [9]. As with the étale index, one has

\[\per(\alpha) | \ind_{\mathrm{Hdg}}(\alpha) | \ind(\alpha).\]

Moreover, Hotchkiss proved that, like the topological index, for a $d$-dimensional smooth proper variety $X$ and $\alpha\in\Br(X)$ one has

\[\ind_{\mathrm{Hdg}}(\alpha) | \per(\alpha)^{d-1}\]

if $\per(\alpha)$ is prime to $(d-1)!$ [13]. More specifically, Hotchkiss proved that

\[\ind_{\mathrm{Hdg}}(\alpha) | \per(\alpha)^{d-1}((d-1)!)^{d-2}.\]

When $d=3$, this collapses to the precise upper bound Ben Williams and I produced in the topological setting.

The counterexample

The Hodge-theoretic index is a much finer invariant than the topological index, but it remains computable in practice from the cohomology ring. Perry gives an example of a $3$-fold where $\per(\alpha)=2$ and $\ind_{\mathrm{Hdg}}(\alpha)=8$. Of course, this implies that $\ind(\alpha)\geq 8$ as well; Perry notes that in fact $\ind(\alpha)=8$ (see more on that in the next section).

The example is a quotient $(Y\times E)/(\bZ/4)^2$, where $Y$ is a Dwork quartic K3 surface of Picard rank $19$ and $E$ is an elliptic curve. The group $G=(\bZ/4)^2$ acts diagonally: by translation through $4$-torsion on $E$, and on $Y$ through a $G$-equivariant deformation of an elliptic K3 surface on which $G$ acts by translation through $4$-torsion sections [17].

The future

At least for $3$-folds and period $2$, the answer is now complete: Matzri’s general bound gives $\ind(\alpha)\mid 8,$ and Perry’s example shows that this is sharp [16].

The next obvious step is to construct a $4$-fold with a $3$-torsion class $\alpha$ where $\ind(\alpha)\geq 3^4$. Then, a $(p+1)$-fold with a $p$-torsion Brauer class with $\ind(\alpha)\geq p^{p+1}$. After that, one should prove some positive results.

References

[1] Benjamin Antieau, Cohomological obstruction theory for Brauer classes and the period-index problem, J. K-Theory 8 (2011), no. 3, 419–435. arXiv:0909.2352.

[2] Benjamin Antieau, Asher Auel, Colin Ingalls, Daniel Krashen, and Max Lieblich, Period-index bounds for arithmetic threefolds, Invent. Math. 216 (2019), no. 2, 301–335. arXiv:1704.05489.

[3] Benjamin Antieau and Ben Williams, Serre-Godeaux varieties and the étale index, J. K-Theory 11 (2013), no. 2, 283–295. arXiv:1205.1279.

[4] Benjamin Antieau and Ben Williams, The topological period-index problem over 6-complexes, J. Topol. 7 (2014), 617–640. arXiv:1208.4430.

[5] Benjamin Antieau and Ben Williams, The topological period-index conjecture, Math. Res. Lett. 28 (2021), no. 5, 1307–1317. arXiv:2003.10539.

[6] Jean-Louis Colliot-Thélène, Die Brauersche Gruppe; ihre Verallgemeinerungen und Anwendungen in der arithmetischen Geometrie (2001), arXiv:2311.02437.

[7] Diarmuid Crowley and Mark Grant, The topological period-index conjecture for $\mathrm{spin}^c$ $6$-manifolds, Ann. K-Theory 5 (2020), 605–620. arXiv:1802.01296.

[8] Aise Johan de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), no. 1, 71–94.

[9] Aise Johan de Jong and Alexander Perry, The period-index problem and Hodge theory, arXiv:2212.12971.

[10] Aise Johan de Jong and Jason Starr, Almost proper GIT-stacks and discriminant avoidance, Doc. Math. 15 (2010), 957–972.

[11] Xing Gu, The topological period-index problem over 8-complexes, I, J. Topol. 12 (2019), no. 4, 1368–1395. arXiv:1709.00787.

[12] Xing Gu, The topological period-index problem over 8-complexes, II, Proc. Amer. Math. Soc. 148 (2020), 4531–4545. arXiv:1803.05100.

[13] James Hotchkiss, Hodge theory of twisted derived categories and the period-index problem, arXiv:2212.10638.

[14] Daniel Krashen and Eliyahu Matzri, Diophantine and cohomological dimensions, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2779–2788. arXiv:1305.5295.

[15] Max Lieblich, Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), no. 1, 1–31.

[16] Eliyahu Matzri, Symbol length in the Brauer group of a field, Trans. Amer. Math. Soc. 368 (2016), no. 1, 413–427. doi:10.1090/tran/6326.

[17] Alexander Perry, The period-index conjecture is false, arXiv:2608.03684.