arXiv subjects

Laurent Manivel

Publications and source records attributed to Laurent Manivel.

At least 19 recordsLinked to original sources

arXiv2026

An atomic criterion for irrationality without quantum computations

The birational invariants introduced by Katzarkov-Kontsevich-Pantev-Yu allows one to obtain irrationality results for varieties whose quantum cohomology is well-behaved. We observe that under certain cohomological conditions, we can deduce irrationality of a very general member from the theory of atoms without actually computing them, using only monodromy equivariance of quantum multiplication and irreducibility of the monodromy representation. Our criterion applies to the very general cubic and Gushel-Mukai fourfolds, whose irrationalities were already known, but also to the very general K{\"u}chle fourfold of type (c5), which is a Fano manifold of index one.

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arXiv2026

Quantum cohomology and birational geometry of Verra fourfolds

We compute the small quantum cohomology ring of a Verra fourfold. Using the theory of atoms recently developped by Katzarkov--Kontsevich--Pantev--Yu, and building on recent papers of the authors, we deduce that a Verra fourfold is never birational to a very general cubic fourfold, nor to a very general Gushel--Mukai fourfold, whereas it was previously known that a general Verra fourfold is birational to a general nodal Gushel--Mukai fourfold. More precisely, we show that for every smooth cubic fourfold or smooth Gushel--Mukai fourfold that is birational to some Verra fourfold, the primitive cohomology is isomorphic, as a rational Hodge structure, to the middle cohomology of some projective K3 surface.

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arXiv2026

Spinorial Fano manifolds

We construct prime Fano manifolds from spin representations of $Spin_n$ for $n\le 14$. In this range, and if $n\ne 13$, the projectivizations of these representations are prehomogeneous, and we deduce that our Fano manifolds are locally rigid and, up to a few exceptions, quasi-homogeneous under the action of their automorphism groups. For $n=13$ we obtain a non-trivial family of minimal compactifications of $SL_3\times SL_3$, modulo some finite group.

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arXiv2026

Quantum cohomology and irrationality of Gushel-Mukai fourfolds

We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring, we also deduce that a rational Gushel-Mukai fourfold has the same rational cohomology as some K3 surface.

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arXiv2025

A birational involution

Given a general K3 surface S of degree 18, lattice theoretic considerations allow to predict the existence of an anti-symplectic birational involution $ϕ$ of the Hilbert cube $S^{[3]}$. We describe this involution in terms of the Mukai model of $S$, with the help of the famous transitive action of the exceptional group $G_2(R)$ on the six-dimensional sphere. We make a connection with Homological Projective Duality by showing that the indeterminacy locus of the involution is birational to a $P^2$-bundle over the dual K3 surface of degree two. We deduce that $ϕ$ is an instance of a Mukai flop.

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arXiv2025

More birational involutions

For $S$ a very general polarized K3 surface of degree $8n-6$, we describe in geometrical terms a birational involution of the Hilbert scheme $S^{[n]}$ of $n$ points on the surface, whose existence was established from lattice theoretical considerations. In a previous work we studied this involution for $n=3$, with the help of the exceptional Lie group $G_2$, since the Mukai model of $S$ is embedded in its projectivized Lie algebra. Here we use different, more general arguments to show that some important features of the birational involution persist for $n\ge 4$. In particular, we describe the indeterminacy locus of the involution in terms of a Mori contraction, and deduce that it is birational to a $\mathbb{P}^2$-fibration over a moduli space of sheaves on $S$, that also admits a degree two nef and big line bundle and an induced birational involution.

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arXiv2025

F4 and desmic quartic surfaces

The desmic pencil of quartic surfaces is part of a beautiful, but mostly forgotten chapter of the classical theory of algebraic surfaces: it is the only non-degenerate pencil of surfaces in P3 containing at least three completely reducible members. We observe in this note that it is closely related to the Weyl group of the root system F4, and can be recovered from a series of symmetric spaces deduced from the exceptional Lie algebras. We discuss the main properties of the pencil from this Lie theoretic point of view.

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arXiv2025

BRIDGES Lectures: G2 in action, and a mathematical theory of exceptions

The BRIDGES meeting in gauge theory, extremal structures, and stability was held in June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García-Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Laurent Manivel on the geometry of G2 and the other exceptional complex Lie groups. Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information.

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arXiv2025

Göpel Varieties

We show that the Coble hypersurfaces, uniquely characterized by the remarkable property that their singular loci are an abelian surface and a Kummer threefold, respectively, belong to a family of hypersurfaces exhibiting similar behavior, but defined in various types of homogeneous spaces. With the help of Jordan-Vinberg theory, we show how these hypersurfaces can be parametrized by G{ö}pel type varieties inside projectivized representations of complex reflection groups.

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arXiv2025

Fano fourfolds of K3 type

We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of flag manifolds. We study the geometry of these Fano fourfolds in some detail, and we find the origin of their K3 structure by relating most of them either to cubic fourfolds, Gushel-Mukai fourfolds, or actual K3 surfaces. Their main invariants and some information on their rationality and on possible semiorthogonal decompositions for their derived categories are provided.

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arXiv2025

On linear sections of the spinor tenfold II

Following previous work by A. Kuznetsov, we study the Fano manifolds obtained as linear sections of the spinor tenfold in $\mathbb{P}^{15}$. Up to codimension three there are finitely many such sections, up to projective equivalence. In codimension four there are three moduli, and this family is particularly interesting because of its relationship with Kummer surfaces on the one hand, and a grading of the exceptional Lie algebra $\mathfrak{e}_8$ on the other hand. We show how the two approaches are intertwined, and we prove that codimension four sections of the spinor tenfolds and Kummer surfaces have the very same GIT moduli space. The Lie theoretic viewpoint provides a wealth of additional information. In particular we locate and study the unique section admitting an action of $SL_2\times SL_2$; similarly to the Mukai-Umemura variety in the family of prime Fano threefold of genus 12, it is a compactification of a quotient by a finite group.

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arXiv2024

Even nodal surfaces of K3 type

We study Fano fourfolds of K3 type with a conic bundle structure. We construct direct geometrical links between these fourfolds and hyperKähler varieties. As a result we describe families of nodal surfaces that can be seen as generalisations of Kummer quartic surfaces. Each of these families actually arises through two families of Fano fourfolds, whose conic bundle structures are related by hyperbolic reduction.

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arXiv2023

Hecke cycles on moduli of vector bundles and orbital degeneracy loci

Given a smooth genus two curve $C$, the moduli space SU$_C(3)$ of rank three semi-stable vector bundles on $C$ with trivial determinant is a double cover in $\mathbb{P}^8$ branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of $C$. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of $GL_9$, initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}. Starting from a general trivector $v$ in $\wedge^3\mathbb{C}^9$, we construct a Fano manifold $D_{Z_{10}}(v)$ in $G(3,9)$ as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SU$_C(3)$. We deduce that $D_{Z_{10}}(v)$ is isomorphic to the odd moduli space SU$_C(3, \mathcal{O}_C(c))$ of rank three stable vector bundles on $C$ with fixed effective determinant of degree one. We deduce that the intersection of $D_{Z_{10}}(v)$ with a general translate of $G(3,7)$ in $G(3,9)$ is a K3 surface of genus $19$.

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arXiv2023

Matrices of linear forms of constant rank from vector bundles on projective spaces

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Dr{é}zet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Dr{é}zet bundles related either to plane quartics or to instanton bundles on the three-dimensional projective space.

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arXiv2022

A four-dimensional cousin of the Segre cubic

This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona-Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group $S_5$. From this construction we are able to describe the Chow ring explicitly.

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arXiv2022

Four-Dimensional Lie Algebras Revisited

The projective variety of Lie algebra structures on a 4-dimensional vector space has four irreducible components of dimension 11. We compute their prime ideals in the polynomial ring in 24 variables. By listing their degrees and Hilbert polynomials, we correct an earlier publication and we answer a 1987 question by Kirillov and Neretin.

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arXiv2022

Discriminants of Theta-Representations

Tevelev has given a remarkable explicit formula for the discriminant of a complex simple Lie algebra, which can be defined as the equation of the dual hypersurface of the minimal nilpotent orbit, or of the so-called adjoint variety. In this paper we extend this formula to the setting of graded Lie algebras, and express the equation of the corresponding dual hypersurfaces in terms of the reflections in the little Weyl groups, the associated complex reflection groups. This explains for example why the codegree of the Grassmannian $G(4, 8)$ is equal to the number of roots of $\mathfrak{e}_7$ .

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arXiv2022

On the automorphisms of Mukai varieties

Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for 7 $\le$ g $\le$ 10. In the Appendix by Y. Prokhorov, the latter statement is extended to g = 12.

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