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  • Free rational points on smooth hypersurfaces
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-12-07
    Tim D. Browning; Will Sawin

    Motivated by a recent question of Peyre, we apply the Hardy–Littlewood circle method to count “sufficiently free” rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rationals.

    更新日期:2020-12-08
  • Essential dimension of representations of algebras
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-12-07
    Federico Scavia

    Let $k$ be a field, $A$ be a finitely generated associative $k$-algebra and Rep$_A[n]$ be the functor Fields$_k\to$ Sets, which sends a field $K$ containing $k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n) := \mathrm {ed}_k$ (Rep$_A[n])$, as $n\to\infty$

    更新日期:2020-12-08
  • The Euler characteristic of Out($F_n$)
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-12-07
    Michael Borinsky; Karen Vogtmann

    We prove that the rational Euler characteristic of Out($F_n$) is always negative and its asymptotic growth rate is $\Gamma (n-\frac{3}{2})/\sqrt{2\pi}\log^2n$. This settles a 1987 conjecture of J. Smillie and the second author. We establish connections with the Lambert $W$-function and the zeta function.

    更新日期:2020-12-08
  • On Borel Anosov representations in even dimensions
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-12-07
    Konstantinos Tsouvalas

    We prove that a word hyperbolic group which admits a $P_{2q+1}$-Anosov representation into $\mathsf{PGL}(4q+2, \mathbb{R})$ contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative answer to Sambarino's question for Borel Anosov representations into $\mathsf{SL}(4q+2,\mathbb{R})$.

    更新日期:2020-12-08
  • On the decomposability of mod 2 cohomological invariants of Weyl groups
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-12-07
    Christian Hirsch

    We compute the invariants of Weyl groups in mod 2 Milnor $K$-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the coefficient module. All basis elements are induced either by Stiefel–Whitney classes or specific invariants in the Witt ring. The proof is based on Serre’s splitting

    更新日期:2020-12-08
  • Involutions and Chern numbers of varieties
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-12-07
    Olivier Haution

    Consider an involution of a smooth projective variety over a field of characteristic not two. We look at the relations between the variety and the fixed locus of the involution from the point of view of cobordism.We show in particular that the fixed locus has dimension larger than its codimension when certain Chern numbers of the variety are not divisible by two, or four. Some of those results, but

    更新日期:2020-12-08
  • Algebraic flows on commutative complex Lie groups
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Tien-Cuong Dinh; Duc-Viet Vu

    We recover results by Ullmo–Yafaev and Peterzil–Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax–Lindemann–Weierstrass theorem for commutative complex

    更新日期:2020-09-16
  • The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Daniel Cristofaro-Gardiner; Marco Mazzucchelli

    A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has

    更新日期:2020-09-16
  • The Liouville property and random walks on topological groups
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Friedrich Martin Schneider; Andreas Thom

    We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich–Vershik and Rosenblatt, confirms a

    更新日期:2020-09-16
  • Hyperbolic surfaces with sublinearly many systoles that fill
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Maxime Fortier Bourque

    For any $\varepsilon > 0$, we construct a closed hyperbolic surface of genus $g=g(\varepsilon)$ with a set of at most $\varepsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most $\varepsilon g$ for the systole function, disproving the lower bound of $2g-1$ posited by Schmutz Schaller.

    更新日期:2020-09-16
  • Squeezing Lagrangian tori in dimension 4
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Richard Hind; Emmanuel Opshtein

    Let $\mathbb C^2$ be the standard symplectic vector space and $L(a,b) \subset \mathbb C^2$ be the product Lagrangian torus, that is, a product of two circles of areas $a$ and $b$ in $\mathbb C$. We give a complete answer to the question of finding the minimal ball into which these Lagrangians may be squeezed by a Hamiltonian flow. The result is that there is full rigidity when $a \leq b \leq 2a$, which

    更新日期:2020-09-16
  • Rigidity of center Lyapunov exponents and $su$-integrability
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Shaobo Gan; Yi Shi

    Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. This implies every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov

    更新日期:2020-09-16
  • Fractal geometry of the complement of Lagrange spectrum in Markov spectrum
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-09-15
    Carlos Matheus; Carlos Gustavo Moreira

    The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ remained

    更新日期:2020-09-16
  • Simple groups of birational transformations in dimension two
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-06-16
    Christian Urech

    We classify simple groups that act by birational transformations on compact complex Kähler surfaces. Moreover, we show that every finitely generated simple group that acts nontrivially by birational transformations on a projective surface over an arbitrary field is finite.

    更新日期:2020-07-20
  • Poisson brackets of partitions of unity on surfaces
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-06-16
    Lev Buhovsky; Alexander Logunov; Shira Tanny

    Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant of the cover measures how much the functions from such a partition of unity can become close to being Poisson commuting. We introduce a new approach to this invariant, which enables us to prove the lower bound conjectured

    更新日期:2020-07-20
  • Singular genuine rigidity
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-06-16
    Luis A. Florit; Felippe Guimarães

    We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain newglobal rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer–Gromoll theorems by showing that any compact $n$-dimensional submanifold of $\mathbb R^{n+p}$ is singularly genuinely rigid in $\mathbb R^{n+q}$, for any $q

    更新日期:2020-07-20
  • Rational equivalence and Lagrangian tori on K3 surfaces
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-06-16
    Nick Sheridan; Ivan Smith

    Fix a symplectic K3 surface $X$ homologically mirror to an algebraic K3 surface $Y$ by an equivalence taking a graded Lagrangian torus $L \subset X$ to the skyscraper sheaf of a point $y \in Y$. We show there are Lagrangian tori with vanishing Maslov class in $X$ whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about

    更新日期:2020-07-20
  • Algebraic varieties are homeomorphic to varieties defined over number fields
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-06-16
    Adam Parusiński; Guillaume Rond

    We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by carefully choosing a small deformation of the coefficients of the original equations. This deformation preserves all polynomial relations over $\mathbb Q$ satisfied by these coefficients

    更新日期:2020-07-20
  • Connected components of strata of Abelian differentials over Teichmüller space
    Comment. Math. Helv. (IF 0.854) Pub Date : 2020-06-16
    Aaron Calderon

    This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups

    更新日期:2020-07-20
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