Annales scientifiques de l'Ecole normale supérieure

The geometric dynamical Northcott and Bogomolov Properties

Thomas Gauthier, Gabriel Vigny — Mon, 18 Nov 2024 11:12:49 +0100

We establish the Geometric Dynamical Northcott Property for polarized endomorphisms of a projective normal variety over a function field K of characteristic zero. This extends previous results of Benedetto, Baker and DeMarco in dimension 1, and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamics arguments and does not rely on the previous ones. We first show that, when K is the field of rational functions of a smooth complex projective variety, the canonical height of a subvariety is the mass of the appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument.

Moving from points to subvarieties, we propose, for polarized endomorphisms, a dynamical version of the Geometric Bogomolov Conjecture recently proved by Cantat, Gao, Habegger and Xie. We establish several cases of this conjecture notably non-isotrivial polynomial skew-product with an isotrivial first coordinate.

The Massey vanishing conjecture for fourfold Massey products modulo 2

Federico Scavia, Alexander Merkurjev — Mon, 18 Nov 2024 11:13:29 +0100

We prove the Massey Vanishing Conjecture for n = 4 and p = 2. That is, we show that for all fields F, if a fourfold Massey product modulo 2 is defined over F, then it vanishes over F.

Quasi-F-splittings in birational geometry

Wed, 22 Jan 2025 11:16:28 +0100

We develop the theory of quasi-$F$-splittings in the context of birational geometry. Amongst other things, we obtain results on liftability of sections and establish a criterion for whether a scheme is quasi-$F$-split employing the higher Cartier operator. As one of the applications of our theory, we prove that three-dimensional klt singularities in large characteristic are quasi-$F$-split, and so, in particular, they lift modulo $p^2$.

On the long-time behavior of scale-invariant solutions to the 2D Euler equation and applications

Ayman Rimah Said, Tarek M. Elgindi, Ryan Murray — Wed, 22 Jan 2025 11:20:20 +0100

We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on $\mathbb{S}^1$ relax to states that are piece-wise constant with \emph{finitely} many jumps.
All continuous scale-invariant solutions become singular and homogenize in infinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-time spiral and cusp formation. The main tool in our analysis is the discovery of a monotone quantity that measures the number of particles that are moving away from the origin.
This monotonicity also applies locally to solutions of the 2d Euler equation that are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point of symmetry.

Our results are also applicable to the Euler equation on a large class of surfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysis then gives \emph{generic} spiraling of trajectories and infinite-time loss of regularity for globally smooth solutions on any such smooth surface, under a discrete symmetry.

Construction of higher-dimensional ALF Calabi-Yau metrics

Daheng Min — Wed, 22 Jan 2025 11:22:53 +0100

Roughly speaking, an ALF metric of real dimension 4n should be a metric such that its asymptotic cone is 4n − 1 dimensional, the volume growth of this metric is of order 4n − 1 and its sectional curvature tends to 0 at infinity. In this paper, I will first show that the Taub-NUT deformation of a hyperkähler cone with respect to a locally free S¹−symmetry is ALF hyperkähler. Modelled on this metric at infinity, I will show the existence of ALF Calabi-Yau metric on certain crepant resolutions. In particular, I will show that there exist ALF Calabi-Yau metrics on canonical bundles of classical homogeneous Fano contact manifolds.

On affine Lusztig varieties

Xuhua He — Wed, 22 Jan 2025 11:24:08 +0100

Affine Lusztig varieties encode the orbital integrals of Iwahori--Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. In this paper, we establish a close relationship between affine Lusztig varieties and affine Deligne--Lusztig varieties. Consequently, we give an explicit nonemptiness pattern and dimension formula for affine Lusztig varieties in most cases.

Quantitative conditions for right-handedness

Anna Florio, Umberto Leone Hryniewicz — Wed, 02 Apr 2025 15:31:51 +0200

We give a numerical condition for right-handedness of a dynamically convex Reeb flow on S3. Our condition is stated in terms of an asymptotic ratio between the amount of rotation of the linearised flow and the linking number of trajectories with a periodic orbit that spans a disk-like global surface of section. As an application, we find an explicit constant δ∗ < 0.7225 such that if a Riemannian metric on the 2-sphere is δ-pinched with δ > δ∗, then its geodesic flow lifts to a right-handed flow on S3. In particular, all finite collections of periodic orbits of such a geodesic flow bind open books whose pages are global surfaces of section.

Growth of Sobolev norms in quasi-integrable quantum systems

Dario Bambusi, Beatrice Langella — Wed, 02 Apr 2025 15:32:43 +0200

We prove an abstract result giving a $ \langle t\rangle^{\epsilon}$ upper bound on the growth of the Sobolev norms of a time dependent Schr\"odinger equation of the form $\im \dot \psi = H_0 \psi + V(t) \psi $. $H_0$ is assumed to be the Hamiltonian of a { globally integrable steep quantum} system and to be a {pseudodifferential} operator of order $\td>1$; $V(t)$ is a time dependent family of pseudodifferential operators, unbounded, but of order $\tb<\td$. The abstract theorem is then applied to {perturbations of the quantum anharmonic oscillators in dimension 2 and to perturbations of the Laplacian on {some} manifold with integrable geodesic flow, {namely} Zoll manifolds, rotation invariant surfaces and Lie groups.} The proof is based a on quantum version of the proof of the classical Nekhoroshev theorem.

Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms

Lev Buhovsky, Maksim Stokić — Wed, 02 Apr 2025 15:33:34 +0200

On a closed and connected symplectic manifold, the group of Hamiltonian diffeomorphisms has the structure of an infinite dimensional Fréchet Lie group, where the Lie algebra is naturally identified with the space of smooth and zero-mean normalized functions, and the adjoint action is given by pullbacks. We show that this action is flexible: for a non-zero smooth and zero-mean normalized function u, any other smooth and zero-mean normalized function f can be written as a finite sum of elements in the orbit of u under the adjoint action. Additionally, the number of elements in this sum is dominated from above by the uniform norm of f. This result can be interpreted as a (bounded) infinitesimal version of Banyaga's theorem on simplicity of the group of Hamiltonian diffeomorphisms. Moreover, it allows to remove the C^\infty-continuity assumption from the Buhovsky-Ostrover theorem on the uniqueness of Hofer's metric.

Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

Wed, 02 Apr 2025 15:34:58 +0200

In previous work, we defined "link spectral invariants" for any compact surface and used these to study the algebraic structure of the group of area-preserving homeomorphisms; in particular, we showed that 
the kernel of Fathi's mass-flow homomorphism is never simple. A key idea for this was a kind of Weyl law, showing that asymptotically the link spectral invariants recover the classical Calabi invariant.

In the present work, we use the subleading asymptotics in this Weyl law to learn more about the algebraic structure of these homeomorphism groups in the genus zero case. In particular, when the surface has 
boundary, we show that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple; this contrasts the smooth case, where the kernel of Calabi is simple. We similarly show that the 
group of hameomorphisms of the two-sphere is not simple. Related considerations allow us to extend the Calabi homomorphism to the full group of compactly supported area-preserving homeomorphisms, answering 
an old question of Fathi.

Central to the applications is that the subleading asymptotics for smooth Hamiltonians are always O(1), and explicit computation shows that for certain autonomous maps they recover the Ruelle invariant. 
The construction of a hameomorphism with ``infinite Ruelle invariant" shows that a normal subgroup with O(1) subleading asymptotics is proper.