arXiv subjects
Publications and source records attributed to Simon Brandhorst.
We give a short proof that every supersingular K3 surface (except possibly in characteristic $2$ with Artin invariant $σ=10$) has an automorphism of Salem degree 22. In particular an infinite subgroup of the automorphism group does not lift to characteristic zero. The proof relies on the case $σ=1$ and the cone conjecture for K3 surfaces.
We characterize conjugacy classes of isometries of odd prime order in unimodular Z-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.
Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical Néron-Severi lattice of $Y$ induced by $f$. We apply a modification of McMullen's method to prove that the modulo-$2$ reduction $(p_f(x) \bmod 2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials $Φ_m$, where $m \leq 9$ and $m$ is odd. We study Enriques surfaces that realize modulo-$2$ reductions of $Φ_7$, $Φ_9$ and show that each of the five polynomials $(Φ_m(x) \bmod 2)$ is a factor of the modulo-$2$ reduction $(p_f(x) \bmod 2)$ for a complex Enriques surface.
We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.
In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that $\log λ_d$, where $λ_d$ is the minimal Salem number of degree $d$, is realized in characteristic $5$ if and only if $d\leq 22$ is even and $d\neq 18$. In the complex projective setting we settle the case of entropy $\log λ_{12}$ left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, i.e. preserves a chamber of the positive cone.
In this note we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes p congruent 3 mod 4 in a systematic way. Automorphisms of Salem degree 22 do not lift to any characteristic zero model.
We consider classes T of topological spaces (referred to as T-spaces) that are stable under continuous images and frequently under arbitrary products. A local T-space has for each point a neighborhood base consisting of subsets that are T-spaces in the induced topology. A general necessary and sufficient criterion for a product of topological spaces to be a local T-space in terms of conditions on the factors enables one to establish a broad variety of theorems saying that a product of spaces has a certain local property (like local compactness, local sequential compactness, local σ-compactness, local connectedness etc.) if and only if each factor has that local property, almost all have the corresponding global property, and not too many factors fail a suitable additional condition. Many of the results admit a point-free formulation; a look at sum decompositions into components of spaces with local properties yields product decompositions into indecomposable factors for certain classes of frames like completely distributive lattices or hypercontinuous frames.