arXiv subjects
Publications and source records attributed to Pieter Belmans.
Huybrechts conjectured that for every Brauer class $α$ on a hyperkähler variety $X$ we have that $\mathop{\rm ind}(α)\mid\mathop{\rm per}(α)^{\dim(X)/2}$, strengthening the usual period-index conjecture. We show that this fails on certain hyperkähler fourfolds, both in type $\mathrm{K}3^{[2]}$ and $\mathrm{Kum}^2$.
We classify the moduli spaces of representations of subspace quivers which are Fano fourfolds, under a natural assumption on the dimension vector. These moduli spaces can also be described as GIT quotients of products of Grassmannians by the diagonal action of a projective linear group, and there are exactly four of them. They are rational, of pure Hodge-Tate type, infinitesimally rigid, and have finite automorphism groups, with Picard ranks 5, 6, 6 and 7, making them interesting examples in the classification of Fano fourfolds of large Picard rank, as they are not toric or products. Two are known varieties: Manivel's Segre cousin of the Segre cubic 3-fold, and the Fano model of the blowup of $\mathbb{P}^4$ in six points. The other two appear to be new: one is an involution surface bundle over $\mathbb{P}^2$, and the other is a "Segre cousin once-removed", whose geometry closely parallels that of the Segre cousin. Using techniques from quiver moduli, which we survey, we describe the geometry of all four fourfolds in detail.
As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.
We show that the Brauer group of any resolution of singularities of the moduli space of semistable quiver representations is trivial. We do this by extending the quiver-curve dictionary, translating a proof of the analogous result by Biswas-Hogadi-Holla for moduli of vector bundles on a curve to the setting of moduli of quiver representations, giving an algebro-geometric proof. This gives a new proof of this triviality, first proved by Le Bruyn-Schofield, building on algebraic (resp. cohomological) vanishing results due to Saltman (resp. Colliot-Thélène-Sansuc). Reversing the logic, our approach gives a new algebro-geometric proof of these vanishing results.
Bott vanishing is a strong vanishing result for the cohomology of exterior powers of the cotangent bundle twisted by ample line bundles. Buch-Thomsen-Lauritzen-Mehta conjectured that partial flag varieties (which are not products of projective spaces) do not satisfy Bott vanishing, despite all their other nice properties. The cominuscule case is an easy application of the Borel-Weil-Bott theorem, following results of Snow. We show that the (co)adjoint partial flag varieties of all classical and exceptional Dynkin types also do not satisfy Bott vanishing, thus confirming the conjecture for this class of varieties.
It is known that the semisimplicity of quantum cohomology implies the vanishing of off-diagonal Hodge numbers (Hodge--Tateness). We investigate which hyperplane sections of homogeneous varieties possess either of the two properties. We provide a new efficient criterion for non-semisimplicity of the small quantum cohomology ring of Fano manifolds that depends only on the Fano index and Betti numbers. We construct a bijection between Dynkin diagrams of types A, D or E, and complex Grassmannians with Hodge-Tate smooth hyperplane sections. By applying our criteria and using monodromy action, we completely characterize the semisimplicity of the small quantum cohomology of smooth hyperplane sections in the case of complex Grassmannians, and verify a conjecture of Benedetti and Perrin in the case of (co)adjoint Grassmannians.
To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the étale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed length) of the category $\operatorname{Perf}\mathcal{X}_V$. We use Artin's criterion to prove that, when $U$ is excellent, this is in fact an algebraic space which is moreover étale (though in general non-quasicompact and non-separated) over $U$. We moreover generalise the construction of the sheaf to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions, and conjecture it is an open and closed subspace. Along the way, we prove that for a smooth and proper family of schemes, a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an étale neighbourhood of the point.
We develop the deformation-obstruction calculus for morphisms of complexes with a fixed lift of the codomain, to derived categories of flat nilpotent deformations of abelian categories. As an application, we give an alternative proof that semiorthogonal decompositions deform uniquely in smooth proper families of schemes.
Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a natural conjecture on the structure of the moduli space, we give both general results, and discuss interesting explicit examples of the behaviour of indecomposability in families, by relating it to the behaviour of the canonical base locus in families. These examples are symmetric powers of curves, certain regular surfaces of general type with large canonical base locus, and Hilbert schemes of points on surfaces. Indecomposability for symmetric powers of curves has been settled via other means, the other cases remain open and we expect that our analysis of the base locus will prove instrumental in finding unconditional proofs.
We introduce QuiverTools, a new software package, available in both a SageMath and Julia version, to study quivers and their moduli spaces of representations. Its key features are the computation of general subdimension vectors, leading to canonical decompositions, and checking the existence of (semi)stable representations, as well as the enumeration of Harder-Narasimhan types and related calculations for Teleman quantization. Computations related to intersection theory on quiver moduli are also implemented.
We explicitly describe the Albanese morphism of a hyperelliptic variety, i.e., the quotient $X$ of an abelian variety $A$ by a finite group $G$ acting freely and not only by translations, by giving a description of the Albanese variety and the Albanese fibers in terms of $A$ and $G$. In particular, the fibers are themselves abelian or hyperelliptic varieties, and we investigate which can occur in explicit examples. As an application we show that the derived category of $X$ is indecomposable in certain cases.
There exists a dictionary between hereditary orders and smooth stacky curves, resp. tame orders of global dimension 2 and Azumaya algebras on smooth stacky surfaces. We extend this dictionary by explaining how the restriction of a tame order to a curve on the underlying surface corresponds to the fiber product of the curve with the stacky surface. By considering "bad" intersections we can start extending the dictionary in the 1-dimensional case to include non-hereditary orders and singular stacky curves. Two applications of these results are a novel description and classification of noncommutative conics in graded Clifford algebras, giving a geometric proof of results of Hu-Matsuno-Mori, and a complete understanding and classification of skew cubics, generalizing the work of Kanazawa for Fermat skew cubics.
We propose conjectural semiorthogonal decompositions for Fano schemes of linear subspaces on intersections of two quadrics, in terms of symmetric powers of the associated hyperelliptic (resp. stacky) curve. When the intersection is odd-dimensional, we moreover conjecture an identity in the Grothendieck ring of varieties and other motivic contexts. The evidence for these conjectures is given by upgrading recent results of Chen-Vilonen-Xue, to obtain formulae for the Hodge numbers of these Fano schemes. This allows us to numerically verify the conjecture in the hyperelliptic case, and establish a combinatorial identity as evidence for the stacky case.
We explain how Teleman quantization can be applied to moduli spaces of quiver representations to compute the higher cohomology of the endomorphism bundle of the universal bundle. We use this to prove Schofield's partial tilting conjecture, and to show that moduli spaces of quiver representations are (infinitesimally) rigid as varieties.
We introduce a double framing construction for moduli spaces of quiver representations. It allows us to reduce certain sheaf cohomology computations involving the universal representation, to computations involving line bundles, making them amenable to methods from geometric invariant theory. We will use this to show that in many good situations the vector fields on the moduli space are isomorphic as a vector space to the first Hochschild cohomology of the path algebra. We also show that considering the universal representation as a Fourier-Mukai kernel in the appropriate sense gives an admissible embedding of derived categories.
We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and use this to define a topological quantum field theory. A similar construction was recently introduced independently by Kontsevich--Odesskii under the name of multiplicative kernels. We end our paper by giving an efficient computational method to compute its partition function. This is the first paper in a series, and we give a survey of the applications of graph potentials in the other parts.
We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded vector space obtained by taking Hochschild homologies with coefficients in powers of the Serre functor. As applications, we obtain various consequences on the deformation theory of the Hilbert schemes; in particular, we recover and extend results of Fantechi, Boissière, and Hitchin. Our method is to compute more generally for any smooth proper algebraic variety $X$ the Hochschild-Serre cohomology of the symmetric quotient stack $[X^n/\mathfrak{S}_n]$, in terms of the Hochschild-Serre cohomology of $X$.
We describe the point class and Todd class in the Chow ring of a quiver moduli space, building on a result of Ellingsrud-Strømme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira-Spencer morphism. We illustrate the results by computing some invariants of some "small" Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker quiver moduli space is isomorphic to the zero locus of a general section of $\mathcal{Q}^\vee(1)$ on $\operatorname{Gr}(2,8)$.