The period-index conjecture is false

Alexander Perry

REVIEW 2 major objections 4 minor 19 references

The period-index conjecture is false

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper constructs a smooth projective threefold with a Brauer class of period 2 and index 8, disproving the period-index conjecture.

desk verdict Explicit counterexamples to the period-index conjecture constructed for all d≥3; the proof is transparent, with the main caveat being the unverified cited Hodge criterion from dJP22. read the letter →

arxiv 2608.03684 v1 pith:UPASF6W2 submitted 2026-08-04 math.AG

Alexander Perry This is my paper · ORCID

classification math.AG MSC 14F2214J2814C30

keywords Brauergroupperiod-indexconjectureHodgetheoryK3surfacequotientthreefold2-torsionclassintegralclassesfunctionfields

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

The period-index conjecture predicts that for a Brauer class on a variety of dimension $d$, the index divides the period raised to the power $d-1$. This paper constructs a smooth projective threefold $X$ and a Brauer class $\alpha$ of period 2 whose index is 8, so the predicted divisibility $\mathrm{ind} \mid 4$ fails. The construction is explicit: $X$ is the quotient $(Y \times E)/G$ of a Dwork quartic K3 surface $Y$ by an elliptic curve $E$ under a $(\mathbb{Z}/4)^2$ action. The proof shows that if $\mathrm{ind}(\alpha)$ divided 4, certain integral Hodge classes would have to satisfy a congruence, and then computes that the congruence cannot hold. The same construction also yields higher-dimensional counterexamples over uncountable algebraically closed fields.

What carries the argument

The load-bearing identity is the congruence obstruction (1.3): for a threefold and a 2-torsion Brauer class with B-field $b/2$, a Hodge-theoretic index bound would force integral Hodge classes $c \in H^{1,1}(X,\mathbb{Z})$ and $d \in H^{2,2}(X,\mathbb{Z})$ with $b^2+bc+d \equiv 0 \pmod{2}$. The paper's key mechanism is a cohomology class $\bar{u}$ on $X$ such that the integral of $b^2+bc+d$ against $\bar{u}$ is identically $1 \pmod{2}$, making the congruence impossible. This computation uses the Leray–Serre spectral sequence for the quotient $X=(Y \times E)/G$, a carefully chosen equivariant elliptic K3 surface $S$, and a $G$-equivariant deformation of $S$ to a Dwork quartic that controls the invariant Néron–Severi group.

What would settle it

Check directly for integral Hodge classes $c \in H^{1,1}(X,\mathbb{Z})$ and $d \in H^{2,2}(X,\mathbb{Z})$ satisfying $b^2+bc+d \equiv 0 \pmod{2}$; the paper's Theorem 5.2 says none exist, so exhibiting one would overturn the counterexample. Independently, compute $\mathrm{ind}(\alpha)$ by a different method (e.g., an explicit cyclic algebra or a trivializing gerbe) to see whether it is 4 instead of 8.

Watch

Extended reading notes

Core claim

The main theorem (Theorem 5.3) asserts that for the threefold $X=(Y \times E)/G$, where $Y$ is a Dwork quartic with Picard rank 19 and $G=(\mathbb{Z}/4)^2$, the 2-torsion Brauer class $\alpha$ with B-field $b/2$ has $\mathrm{per}(\alpha)=2$ and $\mathrm{ind}(\alpha)=8$. Thus the unramified period-index conjecture, which would require $\mathrm{ind} \mid \mathrm{per}^{\dim-1}=4$, fails. The proof establishes that no integral Hodge classes $c,d$ satisfy $b^2+bc+d \equiv 0 \pmod{2}$, the condition that a Hodge-theoretic bound would impose if the index divided 4; combined with Matzri's bound $\mathrm{ind} \mid 8$, the index must be exactly 8.

Load-bearing premise

The counterexample relies on the Hodge-theoretic criterion of de Jong–Perry that, were the index to divide 4, integral Hodge classes $c$ and $d$ satisfying $b^2+bc+d \equiv 0 \pmod{2}$ would have to exist; if that criterion is wrong, the index-8 conclusion does not follow.

Editorial extensions

If this is right

The unramified period-index conjecture is false for smooth projective varieties of dimension at least 3.
The function-field period-index conjecture fails in every dimension d≥4 over uncountable algebraically closed fields of characteristic 0, with a class of period 2 and index 2^d.
The obstruction is Hodge-theoretic: it is detected by integral Hodge classes, not by ramification or by a concrete division algebra computation.
Over \overline{Q}, the three-dimensional counterexample already exists, so the failure is not a phenomenon of uncountable fields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Hodge-theoretic obstruction may yield counterexamples for other primes p≤dim−1; the paper constructs the p=2 case but suggests the pattern is general.
The conjecture might survive only for classes whose period is coprime to (dim−1)!; the paper notes no obstruction is known there.
The testing-class technique (\bar{u}) provides a concrete computational route to search for further counterexamples on other quotient varieties.
A positive-characteristic analogue may be within reach using the de Jong–Perry Hodge-theoretic extension, if the deformation arguments carry over.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper claims to settle the period-index conjecture by constructing explicit counterexamples for all dimensions ≥3, and for threefolds over \bar{Q}. I think the claim is very likely right. The construction is concrete: X = (Y×E)/G, where Y is a Dwork quartic K3 of Picard rank 19, E is an elliptic curve, and G=(Z/4)^2. The Brauer class has period 2 and index 8, against the conjectural bound 4.

What's actually new: Hotchkiss and de Jong-Perry had shown that if a threefold 2-torsion class satisfied the period-index bound, then certain integral Hodge classes c,d would have to exist with b^2+bc+d≡0 mod 2 for the B-field b. This paper is the first to find an explicit X and b where that congruence is impossible. The proof is a direct cohomology calculation: they identify H^2(X,Z) inside H^2(Y×E,Z)^G using the Leray-Serre spectral sequence, and compute the relevant intersection numbers with a cleverly chosen auxiliary class u. The lattice work in §2 is careful and includes a correction of a typo in the last statement of GS09. The paper is well-written and the computations are transparent; I spot-checked several steps and found no internal error.

The soft spots are two. First, the argument depends on the exact statement of [dJP22, Example 5.16], which is not proved or reproduced in this paper. The paper quotes it precisely in the introduction, but a referee will need to check that the hypotheses are satisfied for the b constructed here. If that example has an extra hypothesis (say b algebraic, or the classes rational rather than integral), the contradiction would not go through. This is the only load-bearing external input, and it is not independently verified in the paper. Second, the higher-dimensional bootstrap only works over uncountable fields, so the result over \bar{Q} is only in dimension 3. That is stated clearly.

If the dJP22 criterion checks out, this is a major result and a clean resolution of a long-open conjecture. It redirects the period-index problem to a modified conjecture for primes p≤d−1. I would send it to a serious referee; the referee's main job is to verify that one cited theorem and re-check the mod-2 computation. I would not desk reject it.

Referee Report

2 major / 4 minor

Summary. The paper claims to disprove the period-index conjecture by constructing, for any algebraically closed field k of characteristic 0, a smooth projective threefold X over k and a Brauer class α ∈ Br(X) with per(α) = 2 and ind(α) = 8. The construction is X = (Y × E)/G, where Y is a Dwork quartic K3 of Picard rank 19, E is an elliptic curve, and G = (Z/4)^2 acts diagonally. The proof computes the Néron–Severi and Hodge-theoretic constraints on X, proves a non-existence of integral Hodge classes satisfying a mod-2 congruence, and then invokes a Hodge-theoretic criterion of de Jong–Perry to conclude ind(α) ∤ 4; Matzri's bound then gives ind(α) = 8. The author also bootstraps the threefold counterexample to higher dimensions and to arbitrary algebraically closed fields of characteristic 0.

Significance. If the main theorem is correct, this is a major negative resolution of a well-known conjecture: the unramified period-index conjecture fails in dimension 3, and the original period-index conjecture fails in all dimensions ≥ 3. The construction is explicit and the cohomological machinery — Leray–Serre spectral sequences, lattice computations, and a deformation from an elliptic K3 to a Dwork quartic — is used transparently. The paper also flags a correction to a statement in [GS09]. The computation of the spectral sequence transgressions and the lattice class u in Lemma 3.8 are detailed and appear consistent. The main risk is the exact scope and formulation of the cited external Hodge-theoretic criterion, as well as the base-change argument used to extend from C to arbitrary fields.

major comments (2)

[§5, Theorem 5.3] The step 'By Theorem 5.2 and [dJP22, Example 5.16]' is the sole bridge from the non-existence of integral Hodge classes c,d to the conclusion ind(α) ∤ 4. The manuscript does not state the precise content of [dJP22, Example 5.16] nor verify its hypotheses for the specific X and b constructed here. In particular, the paper must make explicit whether the criterion applies to a B-field b that is not a (1,1)-class (here q^*b = v − 4η_E, and v is not shown to be algebraic), whether the resulting c,d are integral rather than rational Hodge classes, and whether the mod-2 congruence is exactly b^2 + bc + d ≡ 0 (mod 2) with the present sign and normalization. Since the entire index computation depends on this external theorem, please quote the criterion and confirm that all hypotheses are satisfied.
[§5, Proof of Theorem 1.3] The assertion 'The pullback map Br(X_0)→Br(X) induces an isomorphism that preserves the period and index of Brauer classes' is not generally true and is not proved for this X. For instance, for X_0 = Spec(Q), the map Br(Q)→Br(C) is far from an isomorphism. Even if a class β∈Br(X_0) restricts to the topological class α on X_C, one must prove that per(β) and ind(β) over Q and ar Q equal the corresponding invariants over C; index can change under scalar extension. This step is needed to extend the counterexample from C to arbitrary algebraically closed fields of characteristic 0 and to feed Theorem 1.4. The C-case alone would already disprove Conjecture 1.1, so the theorem statements can be corrected by either restricting to C or supplying a genuine Galois-descent argument.

minor comments (4)

[Title] The title in the manuscript body reads 'THE PERIOD-INDEX CONJECTURE IS F ALSE'; the space is a typo for 'FALSE'.
[§4, Lemma 4.2] In the displayed condition (4.3), there is a spacing/comma issue: '4|n,and n≡ −x·f' should read '4|n and n≡ −x·f (mod 16)'.
[§1.5] The AI disclosure is transparent and does not affect the mathematics; no action needed beyond what the journal's policy requires.
[§3, Lemma 3.2] The same symbol Y is used for the total family and for the Dwork quartic fiber; consider using a different symbol for the total family to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; the construction is self-contained and the only same-author citation (dJP22) is an independent general theorem, not a re-import of the target result.

full rationale

The derivation chain is explicit and internal up to the final Hodge-theoretic step. The paper constructs a threefold X=(Y×E)/G, computes the cohomology of X via the Leray–Serre spectral sequence (Lemmas 2.2, 2.3, 4.2), produces the class b with q*b=v−4η_E and an auxiliary class u with the needed congruence properties (Lemmas 3.8, 5.1), and then proves in Theorem 5.2 by a direct degree-6 intersection on Y×E that for every integral Hodge classes c∈H^{1,1}(X,Z), d∈H^{2,2}(X,Z), one has ∫_X(b²+bc+d)ū ≡ 1 mod 2, hence b²+bc+d not ≡ 0 mod 2. This is not a fitted parameter or a renamed input; it is a computation from the explicit geometry. The paper then invokes [dJP22, Example 5.16], a same-author prior result, to conclude that if the Hodge index divided 4 then such c,d would exist; the contrapositive gives ind(α)∤4, and Matzri’s bound gives ind(α)=8. That cited criterion is a general theorem for all threefolds and 2-torsion B-fields, with stated assumptions that do not include the present conclusion, so it is independent support rather than a circular re-importation of the target theorem. No parameter is fitted to the target quantity and no prediction reduces by construction to an input. The AI disclosure is transparent and does not affect the mathematical derivation chain. Score 1 reflects only the presence of a load-bearing same-author citation; it does not indicate circularity.

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

The central claim rests on standard machinery and several cited theorems in Hodge theory and lattice theory. No free parameters or invented entities are introduced. The main new content is the explicit construction and congruence computation.

assumptions (6)

domain assumption Hodge-theoretic period-index criterion: for a smooth projective complex threefold X and a 2-torsion Brauer class α with B-field b/2, the bound ind(α)|4 implies there exist integral Hodge classes c∈H^{1,1}(X,Z) and d∈H^{2,2}(X,Z) with b^2+bc+d≡0 (mod 2).
Used in the proof of Theorem 5.3 to conclude ind(α)∤4 from the failure of the congruence. Cited from [dJP22, Example 5.16].
domain assumption Matzri's bound for 2-torsion Brauer classes on threefolds: ind(α)|8.
Used after establishing ind(α)∤4 and per(α)=2 to force the exact value ind(α)=8. Cited from [Mat16, Theorem 6.3].
domain assumption Maulik-Poonen specialization theorem: there exist λ∈A^1(Q) for the Dwork pencil with rk NS(Y)=19.
Used in Remark 3.6 and the proof of Theorem 1.3 to extend the construction from C to the algebraic closure of Q.
domain assumption Hashimoto's deformation result: there is a G-equivariant family of K3 surfaces connecting the elliptic K3 surface S to the Dwork quartic Y, with symplectic G-action.
Used in Lemma 3.2 and Remark 3.3 to transfer the cohomological and transgression data from S to Y.
domain assumption Garbagnati-Sarti lattice calculations for the elliptic K3 surface S: the invariant lattice H^2(S,Z)^G has rank 4 with the Gram matrix (2.1), and the Neron-Severi invariant lattice has rank 2, with a corrected index statement.
Used throughout Section 2 as the foundation for computing the cohomology of S and, via deformation, of Y.
standard math Leray-Serre spectral sequence for group actions and the Kunneth formula for products, used to compute the image of q^*: H^2(X,Z) -> H^2(Y×E,Z)^G.
Basis for Lemma 4.2, a central step in constructing the Brauer class.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The period-index conjecture is false." pith.science (2026). https://pith.science/paper/UPASF6W2

@misc{pith2026260803684,
  author       = {Pith},
  title        = {Pith review of: The period-index conjecture is false},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPASF6W2}},
  note         = {Machine review of arXiv:2608.03684}
}

abstract

For any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \geq 3$, we construct a variety over $k$ of dimension $d$ with a Brauer class which violates the period-index conjecture for Hodge-theoretic reasons. When $d = 3$, our construction works even without the assumption that $k$ is uncountable; in particular, the period-index conjecture fails over $\overline{\mathbf{Q}}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 14 canonical work pages

[1]

Gilberto Bini and Alice Garbagnati, Quotients of the D work pencil , J. Geom. Phys. 75 (2014), 173--198

work page 2014
[2]

Alessio Bottini and Daniel Huybrechts, The period-index problem for hyperk\" a hler varieties: L ower and upper bounds , arXiv:2512.15131 (2025)

arXiv 2025
[3]

Jean-Louis Colliot-Th\' e l\`ene, Die B rauersche G ruppe; ihre V erallgemeinerungen und A nwendungen in der arithmetischen G eometrie , arXiv:2311.02437 (2001)

work page Pith review arXiv 2001
[4]

, Exposant et indice d'alg\`ebres simples centrales non ramifi\' e es , Enseign. Math. (2) 48 (2002), no. 1-2, 127--146, with an appendix by Ofer Gabber

work page 2002
[5]

, Alg\`ebres simples centrales sur les corps de fonctions de deux variables (d'apr\`es A . J . de J ong) , no. 307, 2006, S\' e minaire Bourbaki. Vol. 2004/2005, pp. Exp. No. 949, ix, 379--413

work page 2006
[6]

3, 369--374

Alexandru Dimca, Monodromy and B etti numbers of weighted complete intersections , Topology 24 (1985), no. 3, 369--374

work page 1985
[7]

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

work page 2004
[8]

Aise Johan de Jong and Alexander Perry, The period-index problem and H odge theory , arXiv:2212.12971 (2022)

work page Pith review arXiv 2022

Show all 19 references

[9]

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

2010
[10]

Algebra 37 (2009), no

Alice Garbagnati and Alessandra Sarti, Elliptic fibrations and symplectic automorphisms on K3 surfaces , Comm. Algebra 37 (2009), no. 10, 3601--3631

2009
[11]

Kenji Hashimoto, Finite symplectic actions on the K3 lattice , Nagoya Math. J. 206 (2012), 99--153

2012
[12]

James Hotchkiss, Davesh Maulik, Junliang Shen, Qizheng Yin, and Ruxuan Zhang, The period-index problem for hyper- K \" a hler varieties via hyperholomorphic bundles , arXiv:2502.09774 (2025)

2025 arXiv
[13]

James Hotchkiss, Hodge theory of twisted derived categories and the period-index problem, arXiv:2212.10638 (2022)

2022 arXiv
[14]

James Hotchkiss and Alexander Perry, The period-index conjecture for abelian threefolds and D onaldson-- T homas theory , arXiv:2405.03315 (2024)

2024 arXiv
[15]

Daniel Huybrechts, The period-index problem for hyperk\" a hler manifolds , arXiv:2411.17604 (2025)

2025 arXiv
[16]

Ruoxi Li, Minimal trivializing isogenies of G_m -gerbes over abelian varieties and period-index problem , arXiv:2605.30530 (2026)

2026 arXiv
[17]

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

2008
[18]

Eliyahu Matzri, Symbol length in the B rauer group of a field , Trans. Amer. Math. Soc. 368 (2016), no. 1, 413--427

2016
[19]

Davesh Maulik and Bjorn Poonen, N\'eron- S everi groups under specialization , Duke Math. J. 161 (2012), no. 11, 2167--2206

2012

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Gilberto Bini and Alice Garbagnati, Quotients of the D work pencil , J. Geom. Phys. 75 (2014), 173--198

work page 2014

[2] [2]

Alessio Bottini and Daniel Huybrechts, The period-index problem for hyperk\" a hler varieties: L ower and upper bounds , arXiv:2512.15131 (2025)

arXiv 2025

[3] [3]

Jean-Louis Colliot-Th\' e l\`ene, Die B rauersche G ruppe; ihre V erallgemeinerungen und A nwendungen in der arithmetischen G eometrie , arXiv:2311.02437 (2001)

work page Pith review arXiv 2001

[4] [4]

, Exposant et indice d'alg\`ebres simples centrales non ramifi\' e es , Enseign. Math. (2) 48 (2002), no. 1-2, 127--146, with an appendix by Ofer Gabber

work page 2002

[5] [5]

, Alg\`ebres simples centrales sur les corps de fonctions de deux variables (d'apr\`es A . J . de J ong) , no. 307, 2006, S\' e minaire Bourbaki. Vol. 2004/2005, pp. Exp. No. 949, ix, 379--413

work page 2006

[6] [6]

3, 369--374

Alexandru Dimca, Monodromy and B etti numbers of weighted complete intersections , Topology 24 (1985), no. 3, 369--374

work page 1985

[7] [7]

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

work page 2004

[8] [8]

Aise Johan de Jong and Alexander Perry, The period-index problem and H odge theory , arXiv:2212.12971 (2022)

work page Pith review arXiv 2022