Annales scientifiques de l'Ecole normale supérieure

Hausdorff dimension and exact approximation order in R^n

Nicolas de Saxcé, Prasuna Bandi — Wed, 02 Apr 2025 15:30:15 +0200

Given a non-increasing function $\psi:\mathbb{N}\to\mathbb{R}^+$ such that $s^{\frac{n+1}{n}}\psi(s)$ tends to zero as $s$ goes to infinity, we show that the set of points in $\mathbb{R}^n$ that are exactly $\psi$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\geq 2$, this answers questions of Jarnı́k and of Beresnevich, Dickinson and Velani.

Foliated Plateau problems and asymptotic counting of surface subgroups

Graham Andrew Smith, Sébastien Alvarez, Ben Lowe — Wed, 02 Apr 2025 15:38:13 +0200

In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a higher-dimensional analogue of the geodesic flow when the ambient manifold is negatively curved. In this paper, following the recent work [5] of Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups in terms of areas of $k$-surfaces. We determine a lower bound, and we prove rigidity when this bound is achieved. Our work differs from that of [5] in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5] breaks down in the present case, we require a different approach. Following ideas outlined by Labourie in [19], we prove rigidity by solving a general foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build on Labourie's theory of $k$-surface dynamics, and propose a number of new constructions, conjectures and questions.

An obstruction to the local lifting problem

Aristides Kontogeorgis, Alexios Terezakis — Fri, 13 Jun 2025 12:17:14 +0200

We are investigating the lifting problem for local actions involving semidirect products of a cyclic $p$-group with a cyclic group prime to $p$, where $p$ represents the characteristic of the special fiber. We establish a criterion based on the Harbater-Katz-Gabber compactification of local actions, enabling us to determine whether a given local action can be lifted or not. Specifically, in the case of the dihedral group, we present an example of a local dihedral action that cannot be lifted. This instance provides a more potent obstruction than the KGB obstruction.

Rigid currents on compact hyperkähler manifolds

Andrey Soldatenkov, Nessim Sibony, Misha Verbitsky — Fri, 13 Jun 2025 12:26:48 +0200

A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kähler manifold X all eigenvectors of hyperbolic automorphisms acting on H^{1,1}(X) that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kähler cone and have vanishing volume. We study parabolic (1,1)-classes on compact hyperkähler manifolds with b_2 > 6. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkähler manifold is rigid.

A motivic circle method

Margaret Bilu, Tim Browning — Fri, 13 Jun 2025 12:33:15 +0200

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.

Topology of irrationally indifferent attractors

Davoud Cheraghi — Wed, 02 Jul 2025 09:17:55 +0200

We study the post-critical set of a class of holomorphic systems with an irrationally indifferent fixed point.

We prove a trichotomy for the topology of the post-critical set based on the arithmetic of the rotation number

at the fixed point. The only options are Jordan curves, a one-sided hairy Jordan curve, and Cantor bouquet.

This explains the degeneration of the closed invariant curves inside the Siegel disks, as one varies the

rotation number.

Hyperbolic models of transitive topological Anosov flows in dimension three

Mario Shannon — Wed, 02 Jul 2025 09:18:53 +0200

We prove that every transitive topological Anosov flow on a closed $3$-manifold is orbitally equivalent to a smooth Anosov flow, preserving an ergodic smooth volume form.

Hecke actions on loops and periods of iterated Shimura integrals

Richard Hain, (Pham Huu Tiep) — Wed, 02 Jul 2025 09:23:51 +0200

In this paper we show that the action of the classical Hecke operators $T_N$ ($N>0$) act on the free abelian groups generated by the conjugacy classes of the modular group $\SL_2(\Z)$ and the conjugacy classes of its profinite completion. We show that this action induces a dual action on the ring of class functions of a certain relative unipotent completion of the modular group. This ring contains all iterated integrals of modular forms that are constant on conjugacy classes. It possesses a natural mixed Hodge structure and, after tensoring with $\Ql$, a natural action of the absolute Galois group. Each Hecke operator preserves this mixed Hodge structure and commutes with the action of the absolute Galois group. Unlike in the classical case, the algebra generated by these Hecke operators is not commutative.

In the appendix, Pham Tiep proves that for all primes $p\ge 5$, every irreducible character of $\SL_2(\Z/p^n)/(\pm \id)$ appears in its conjugation action on the complex group algebra of $\SL_2(\Z/p^n)$, a result needed in the body of the paper.

Algebraic vector bundles and p-local A1-homotopy theory

Jean Fasel, Aravind Asok, Michael Hopkins — Wed, 02 Jul 2025 09:25:39 +0200

We construct many ``low rank" algebraic vector bundles on ``simple" smooth affine varieties of high dimension. In a related direction, we study the existence of polynomial representatives of elements in the classical (unstable) homotopy groups of spheres. Using techniques of A1-homotopy theory, we are able to produce ``motivic" lifts of elements in classical homotopy groups of spheres; these lifts provide interesting polynomial maps of spheres and algebraic vector bundles.

A degenerate Arnold diffusion mechanism in the restricted 3-body problem

Jaime Paradela, Marcel Guàrdia, Tere M. Seara — Wed, 02 Jul 2025 09:27:53 +0200

A major question in dynamical systems is to understand the mechanisms driving global instability in the 3-body problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0,m_1$. We consider the Restricted Planar Elliptic 3-body problem (RPE3BP) where the primaries revolve in Keplerian ellipses. We prove that the RPE3BP exhibits topological instability: for any values of the masses $m_0,m_1$ (except $m_0=m_1$), we build orbits along which the angular momentum of the massless body experiences an arbitrarily large variation provided the eccentricity of the orbit of the primaries is positive but small enough.

In order to prove this result we show that a degenerate Arnold diffusion mechanism, which moreover involves exponentially small phenomena, takes place in the RPE3BP. Our work extends the one of Delshams, Kaloshin, de la Rosa, and Seara (2019) for the a priori unstable case $m_1/m_0\ll1$, to the case of arbitrary masses $m_0,m_1>0$, where the model displays features of the so-called a priori stable setting.

Erratum to "The Local Lifting Problem for Actions of Finite Groups on Curves"

Ted Chinburg, Robert Guralnick, David Harbater — Wed, 23 Jul 2025 14:28:39 +0200

In view of examples found by B. Weaver, we correct Theorem 1.5 of the article referenced in the title.

Non-symmetric quantum loop groups and K-theory

Michela Varagnolo, Eric Vasserot — Wed, 23 Jul 2025 14:29:43 +0200

We realize the quantum loop groups and shifted quantum loop groups of arbitrary
types, possibly non symmetric, using critical K-theory. This generalizes the Nakajima
construction of symmetric quantum loop groups via quiver varieties to non symmetric types.
We also give a new geometric construction of some simple modules of both quantum loop
groups and shifted quantum loop groups.