Pieter Belmans

New paper: The hyperkähler period-index conjecture is false

Tue, 11 Aug 2026 00:00:00 +0000

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.

Hyperelliptic varieties: quotients of complex tori by finite groups

Tue, 21 Jul 2026 00:00:00 +0000

There is a new website, hyperelliptic.ncag.info, on the classification of hyperelliptic (or generalized hyperelliptic) varieties in complex dimensions 2, 3 and 4. For now it lives as a subdomain of ncag.info.

A hyperelliptic variety is a quotient $X = T/G$ of a complex torus $T$ by a finite group $G$ acting freely and without translations. In dimension 1 these are the elliptic curves, in dimension 2 the seven bielliptic surfaces of Bagnera and De Franchis; the word has nothing to do with hyperelliptic curves. The remarkable thing is that all the numerical invariants (the Hodge diamond, the order of the canonical bundle, the number of moduli, the irregularity, the polyvector fields, the twisted Hodge numbers, the Hochschild cohomology) depend only on the tangent representation $\rho\colon G \to \mathrm{GL}(V)$, and can be read off from its character theory. The website computes them all with OSCAR, for every group in the classifications of Uchida–Yoshihara, Lange and Catanese–Demleitner (dimension 3) and of Demleitner (the 79 groups in dimension 4).

This all started from a collaboration with Andreas Demleitner and Pedro Núñez on the Albanese morphism for these varieties. Working on that paper I learned a great many things about hyperelliptic varieties from Andreas, and this website is in a sense a place to keep all of it: the invariants we computed by hand, and many more, now generated automatically and cross-checked against the literature.

As with my other websites it is a static site built (using LLMs) with Hugo. Feature requests, corrections and contributions are very welcome, on GitHub or by email.

New paper: Fano 4-fold quiver moduli from subspace quivers

Wed, 15 Jul 2026 00:00:00 +0000

Whilst it might look like I am hyperproductive these few weeks, with a new blogpost coming up almost every day about something I have done, I am mostly just putting things online that were finished before, in preparation of the summer holidays.

This post is about Fano 4-fold quiver moduli from subspace quivers, a new paper joint with Markus Reineke. Let's discuss the origin story of this paper in a way that is not appropriate for the paper's introduction, but that is perfect for a blog post.

Origin

It all started from Laurent Manivel's A four-dimensional cousin of the Segre cubic. In this paper, he studies a Fano fourfold, defined as the zero locus of a vector bundle on the product of two Grassmannians. Looking at the properties he finds for this Fano fourfold, my mind was applying the duck test in the following form:

If it looks like a quiver moduli space, swims like a quiver moduli space, and quacks like a quiver moduli space, then it probably is a quiver moduli space.

Quiver moduli are after all very special varieties: rational, Hodge–Tate, rigid. Laurent's Segre cubic cousin had all these properties. Through its relation to the Segre cubic, I also quickly guessed the quiver and dimension vector that should do the trick: the 6-subspace quiver, and $\mathbf{d}=(1,1,1,1,1,2;3)$. However, I couldn't pin down an isomorphism.

Then in February Markus came to Utrecht as the Springer visiting chair. Markus realized there is actually a fun classification problem hiding here: which other subspace quivers give rise to Fano 4-folds? The reason to be interested in subspace quivers is that, besides their moduli spaces being rigid, their moduli spaces also have no infinitesimal automorphisms, making them extra special.

It turns out that up to natural identifications, there are precisely 4 such Fano 4-fold subspace quiver moduli: one being the (expected) Segre cousin, another being the Fano model of $\mathop{\rm Bl}_6\mathbb{P}^4$, which is also $(\mathbb{P}^1)^7//\mathrm{PGL}_2$, the moduli space of 7 points on $\mathbb{P}^1$ (let's not spell out stability).

The other two are also very interesting: one admits a natural map to $\mathbb{P}^2$, exhibiting it as something called an involution surface bundle (the first time I encountered one in the wild), and another which looks a lot like the Segre cubic cousin.

And in the end, we also managed to prove that Manivel's Segre cubic cousin was indeed the subspace quiver moduli space we expected it to be all along. All's well that ends well.

Working with quiver moduli

What I like about this paper is that it is so very explicit with quiver moduli, and applies so many of the tools that have been developed for them. I consider it a bit of an advertisement, and I hope that it gets picked up by people who, a priori, would not work with quiver moduli, but now realize how useful this perspective can be!

Related to this is also the use of QuiverTools, which makes it possible to compute many invariants of quiver moduli. For the 4 cases in this paper, the code is available as FanoFourfoldSubspaceQuiverModuli.jl. And the tools we have used in this paper were also the inspiration for some of the new features for QuiverTools, more about those later.

Note that the duck test for quiver moduli doesn't always work so well! We are also working on a paper that explains how well the duck test works for Fano 3-folds. Stay tuned for that.

Also note that there is another work-in-progress (don't expect either of them before the start of my summer break, though!) where we take the involution surface bundle Fano 4-fold subspace quiver moduli, and take its connection to the extended Dynkin quiver $\widetilde{\mathrm{D}}_4$, to find 5 infinite series of even-dimensional Fano varieties attached to extended Dynkin quivers, with lots of interesting properties. Stay tuned for that too!

Cubic 4-folds: special cubic fourfolds and their K3 surfaces

Mon, 13 Jul 2026 00:00:00 +0000

Since early 2023 there has been a website at cubics.fanography.info on the relationship between cubic fourfolds and K3 surfaces, but I never announced it I think: a first draft version was written in just 2 days and I didn’t know what to do next. I’m not sure it is in any state or form “finished”, but given that I’m on a website spree lately, I might as well get this one out too. So this is that announcement, three years late.

Hassett divisors

A smooth cubic fourfold $X\subseteq\mathbb{P}^5$ is special if it contains a surface not homologous to a complete intersection; these form the Hassett divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds.

The website is a table with one row for every discriminant $d$, collecting some things that are known about $\mathcal{C}_d$: the various notions of an associated (twisted) K3 surface, rationality, the Kodaira dimension, Fourier–Mukai partners, and which surfaces the generic member contains, with clickable references. It goes back to a chart of Nicolas Addington, whose comments shaped the website.

As with my other websites it is a static site built with Hugo. Feature requests, corrections and contributions are very welcome, on GitHub or by email.

LLMs were used to build this, which made the whole process much faster.

le superficie algebriche: a table and more surfaces

Sun, 12 Jul 2026 00:00:00 +0000

Back in 2015, Johan Commelin and I made le superficie algebriche, an interactive picture of the geography of minimal complex algebraic surfaces, plotted by their Chern numbers $\mathrm{c}_1^2$ and $\mathrm{c}_2$ (see the original post and a 2019 update). It has just had its biggest update since.

A table. Next to the plane of Chern numbers there is now a table: every class of surface in the database with its Kodaira dimension and numerical invariants, sortable by any column, searchable by name, and with descriptions you can unfold in place.

More surfaces, with descriptions and references. A literature sweep added a long list of named classes, from fake quadrics and the ball quotients on the Bogomolov–Miyaoka–Yau line (Ishida, Cartwright–Steger, Yeung’s surface of maximal canonical degree) to product-quotient and Beauville-type surfaces. Each now comes with a short description, what it is and why it is interesting, and full citations resolved to MathSciNet, arXiv or DOI, using the same bibliography machinery I built for mgnbar.

The map, filled in. The whole interior between the Noether line and the signature-zero line is now populated (following Persson), so the picture genuinely shows which Chern numbers occur and, just as clearly, the thin strip below the Bogomolov–Miyaoka–Yau line that is still an open problem.

Smaller things: the holomorphic Euler characteristic of the tangent bundle $\chi(\mathrm{T}_S)$ is shown too, and every surface now has its own URL, so you can link straight to one, like superficie.info/2/4-8/fake-quadrics. As with my other websites it is now a static site built with Hugo.

Corrections and, especially, more surfaces are very welcome, on GitHub or by email. If you know a surface that belongs on the map, it is easy to add.

LLMs were used to build this, which made the whole process much faster.

DOIs for my websites

Fri, 10 Jul 2026 00:00:00 +0000

Some quick housekeeping: several of my websites now have a Zenodo DOI, so they are easier to cite. This currently covers fanography.info, grassmannian.info, hyperkaehler.info and mgnbar.info, with more to follow. Each site's "how to cite" section now lists the DOI, and I use the version-independent one, so it always resolves to the latest release.

If you use any of these in your work, please do cite them: it helps me explain to the people who fund my work that building these things is useful.

Mgnbar.info: two new layers

Thu, 09 Jul 2026 00:00:00 +0000

Back in 2023, together with Ignacio Barros, I made Mgnbar.info, a website about the geometry of $\overline{\mathrm{M}}_{g,n}$, the moduli space of stable $n$-pointed genus $g$ curves. For a long time it showed a single invariant, the Kodaira dimension. It now has two more layers, which you can pick from the selector above the table, and the Kodaira dimension layer itself has gained some extra information.

The tautological ring. For each $(g,n)$ the table records whether the Chow ring of $\overline{\mathrm{M}}_{g,n}$, and of the open $\mathrm{M}_{g,n}$, is generated by the tautological classes, the natural $\psi$- and $\kappa$-classes and boundary strata, or whether there are genuinely more mysterious cycles.

Point counts. Whether the number of points of $\overline{\mathrm{M}}_{g,n}$ and $\mathrm{M}_{g,n}$ over a finite field $\mathbb{F}_q$ is a polynomial in $q$. The first place where this fails is $\overline{\mathrm{M}}_{1,11}$, where the culprit is the Ramanujan $\tau$-function, the coefficients of the weight-12 cusp form.

Both layers are transcribed from Hannah Larson’s recent survey arXiv:2606.29656, which collects the state of the art on precisely these questions. It is a wonderful read, and any errors in the transcription are of course mine.

More on the Kodaira dimension. Kodaira dimension $-\infty$ is only the coarsest sign that a moduli space is not of general type. The finer notions rational $\Rightarrow$ unirational $\Rightarrow$ rationally connected $\Rightarrow$ uniruled all imply it, and are now shown, as shaded blues, on the cells where the Kodaira dimension is $-\infty$. The baseline is Benzo’s 2014 survey table, extended with the more recent results of Agostini–Barros, Keneshlou–Tanturri, and (for genus 11-15) Verra and Bruno–Verra.

Along the way each layer got its own URL, and clicking a cell now updates the address bar too, so you can link straight to a single entry like mgnbar.info/tautological/#9,4. There is a small about page explaining how to cite the website, and, as with my other websites, Mgnbar.info is now a static site built with Hugo. I’m a big fan of Hugo nowadays, can you tell?

Suggestions and corrections are very welcome, on GitHub or by email.

LLMs were used to build this, which made the whole process much faster.

New paper: Brauer groups of resolved quiver moduli via gerbes

Wed, 08 Jul 2026 00:00:00 +0000

I'm happy to announce a new paper, Brauer groups of resolved quiver moduli via gerbes, joint with Gianni Petrella and Sebastian Torres.

In this paper we give a new and very different proof of an old result due to Le Bruyn and Schofield (two mathematicians I admire greatly!), by translating a proof (due to Biswas–Hogadi–Holla) in the context of moduli of vector bundles on curves to that of quiver moduli. It is no secret that the quiver-curve dictionary is a favourite topic of mine, and this paper can be seen as a contribution to this dictionary.

The result in question says that the Brauer group of a resolution of any moduli space of quiver representations vanishes. If you believe that all quiver moduli (and moduli of vector bundles) are rational, then certainly this Brauer group has to vanish. Along the way of (re)proving this vanishing, we explain some interesting properties of quiver moduli, highlighting parallels (and differences) between the curve and quiver case.

This project was the result of a working group back in Luxembourg, which was a lot of fun!

Congratulations Dr. Gianni Petrella!

Fri, 03 Jul 2026 00:00:00 +0000

I want to congratulate Gianni Petrella, for his PhD defense today at the University of Luxembourg, and the wonderful thesis Windows, walls and invariants of quiver moduli that he has written. The thesis will be available online at some point, but you can also check out the papers it builds on:

Rigidity and Schofield's partial tilting conjecture for quiver moduli, a joint paper with Ana-Maria Brecan, Hans Franzen, Markus Reineke, and myself, published in the Journal de l'École Polytechnique
Partial semiorthogonal decompositions for quiver moduli, his solo paper, published in the Journal of Symbolic Computation
Finding the walls for quiver moduli, a joint paper with Hans Franzen and Rachel Webb, published in the Journal of Pure and Applied Algebra
The QuiverTools package for SageMath and Julia, a joint paper with Hans Franzen and myself, published in the Journal of Software for Algebra and Geometry
Brauer groups of resolved quiver moduli via gerbes, a joint paper with Sebastian Torres and myself, which will appear on the arXiv very soon

Gianni was my PhD student (although FNR regulations prevented an official recognition of this fact) and I have greatly enjoyed our mathematical conversations. I look forward to our future ones, as he will take up a postdoc at the University of Amsterdam, working with Raf Bocklandt. Congratulations!

Fanography is now a static website (and other improvements)

Thu, 02 Jul 2026 00:00:00 +0000

Almost eight years ago I launched Fanography. It has just received its biggest technical overhaul since then: fanography.info is now a static website, generated with Hugo and served from GitHub Pages.

From Flask to Hugo

Before, Fanography was a Flask application: a little Python server that assembled each page on the fly from the underlying classification data. That worked well, but it meant there was always a server that needed to keep running, be kept up to date, and be paid for in one way or another. So I rebuilt it from scratch with Hugo:

The old Flask application has been retired. As with moving my own website to Hugo, the migration should be invisible to visitors—please let me know if something broke!

New information on Fanography

Whilst I was busy with Fanography, I decided to add a bit of data that I've wanted to be there for a long time.

The Mori–Mukai determinant: Each entry page now displays the Mori–Mukai determinant $d(X)$, and it is included for all 105 families. I wrote about this invariant, and how I computed it for every family, in a separate blogpost.
Holomorphic Poisson structures: There is a new card recording the holomorphic Poisson structures on $X$, that is, the global sections $\pi$ of $\wedge^2\mathrm{T}_X$ for which the Schouten–Nijenhuis bracket with itself vanishes, i.e., $[\pi,\pi]_{\mathrm{NS}}=0$. For Fano 3-folds of Picard rank 1 these were classified by Loray–Pereira–Touzet, and the card lists the irreducible components of $\mathbb{P}\mathrm{H}^0(X,\wedge^2\mathrm{T}_X)$ together with their dimensions. For higher Picard rank the classification is, as far as I know, still open. This sounds like a fun challenge!

Other improvements

A small quality-of-life improvement: you can now use the left and right arrow keys to move between consecutive families, which makes browsing the classification in order much more pleasant.

A handful of smaller additions came along for the ride:

several rank-1 Fano 3-folds are now referred to by name;
del Pezzo surface pages show the number of exceptional lines, and their polyvector parallelogram.

Maybe more updates to come soon!

Oh, and hyperkaehler.info is now also a static website! But that one didn't get any improvements, so no separate blogpost.