We prove a generalization of the version, due to A. Gabrielov, of the {\L}ojasiewicz inequality for not necessarily restricted semi-Pfaffian sets. Unlike the most variants of the {\L}ojasiewicz inequality, it describes the rate of growth of a Pfaffian function on a semi-Pfaffian set $X$ not only in a neighbourhood of its zero set in the domain of $X$ but also in a neighbourhood of zeroes on the boundary of the domain. Gabrielov's version is essentially one-dimensional, we generalize it to a multidimensional case.
Let $k$ be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over $k$ and over $\mathbb{C}$ equipped with the hybrid norm. In both settings, we show an analogue of Man\'e--Sad--Sullivan's theorem about the continuity of the Julia set (in t...
We establish a Bogomolov--Gieseker-type inequality for tilt-semistable objects on smooth projective varieties of dimension $\geq 3$, in the spirit of the Bayer--Macr\`i--Toda conjecture. A key ingredient is an analysis of the action of the Frobenius morphism on tilt-semistable objects. As an application, we prove the C...
Chun-Yi Li, Zhi-Yu Liu, Alexander Perry et al.· 1 citation· ⚡1
Jurij Vol\v{c}i\v{c} conjectured that a noncommutative polynomial $g$ belongs to the two-sided ideal generated by $f_1,\ldots,f_l$ if and only if, for matrices of every size, the rank of the evaluation of $g$ is bounded by a constant times the maximum rank of the evaluations of $f_1,\ldots,f_l$. In this paper, we show...
We prove that for any $p>4$ we can find a Schatten class Hankel operator $\Ha_\phi$ defined on the Hardy space of the infinite torus $H^2(\mathbb{T}^\infty)$ that belongs to the Schatten class $S^p$ but does not admit a bounded symbol, extending the previously known range $p>(1-\log\pi /\log 4)^{-1}=5.7388...$.
We prove the uniform Mordell--Lang conjecture for semiabelian varieties, that is, for a subvariety $X$ of a semiabelian variety, its intersection with a finite-rank subgroup of rational points, can be covered by at most $c$ cosets contained in $X$, where $c$ depends only on the dimension of the semiabelian variety, the...
We prove a Fuglede-type quantitative isoperimetric inequality for the Pansu sphere, the conjectured isoperimetric set in the sub-Riemannian Heisenberg group $\mathbb{H}^1$. We show that, among suitable volume-preserving normal perturbations of the Pansu sphere that are small in a weighted $C^2$-norm, the difference bet...
Davide Massenz, R. Monti, Juli'an Pozuelo· 0 citations
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