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Yohsuke Matsuzawa

Publications and source records attributed to Yohsuke Matsuzawa.

At least 19 recordsLinked to original sources
arXiv2026

On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

We give a counterexample to the following conjecture: the set of isolated periodic points of an automorphism of degree at least two on an affine space is a set of bounded height. As a positive result, we prove that any cohomologically hyperbolic dominant rational self-map on a projective variety admits a non-empty Zariski open subset on which the set of periodic points is height bounded. Concerning preperiodic points, we give an example suggesting that the same statement may fail.

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arXiv2026

Invariant Hilbert Schemes and Luna's Etale Slice Theorem

Luna's etale slice theorem is a useful theorem for the local study of quotients by reductive algebraic groups. In this article, we show that the slice theorem can also be used to study local structures of invariant Hilbert schemes. By using this method, we show some results on smoothness of invariant Hilbert schemes at closed orbits.

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arXiv2025

Arithmetic degree and its application to Zariski dense orbit conjecture

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical degree of $f$. As an application, we prove that Zariski dense orbit conjecture holds for a birational map defined over $\overline{\mathbb{Q}}$ whose first dynamical degree is strictly larger than its third dynamical degree. In particular, the conjecture holds for birational maps on threefolds whose first dynamical is degree greater than $1$.

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arXiv2025

Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

Consider a dominant rational self-map $f$ on a smooth projective variety $X$ defined over $\overline{\mathbb{Q}}$. We prove that \begin{align} \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x)) } = 0, \end{align} where $h_{Y}$ is a height associated with a closed subscheme $Y \subset X$ of codimension $c$, $h_{H}$ is any ample height on $X$, and $x \in X(\overline{\mathbb{Q}})$ is a point with well-defined orbit, under the following assumptions: (1) either $f$ is a morphism, or $Y$ is pure dimensional, regularly embedded in $X$, and contained in the locus over which all iterates of $f$ are finite; (2) the orbit of $x$ is generic; (3) $d_{c}(f)^{1/c} < α_{f}(x)$, where $d_{c}(f)$ is the $c$-th dynamical degree of $f$ and $ α_{f}(x)$ is the arithmetic degree of $x$.

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arXiv2025

Existence of arithmetic degrees for generic orbits and dynamical Lang-Siegel problem

We prove the existence of the arithmetic degree for dominant rational self-maps at any point whose orbit is generic. As a corollary, we prove the same existence for étale morphisms on quasi-projective varieties and any points on it. We apply the proof of this fact to dynamical Lang-Siegel problem. Namely, we prove that local height function associated with zero-dimensional subscheme grows slowly along orbits of a rational map under reasonable assumption. Also if local height function associated with any proper closed subscheme grows fast on a subset of an orbit of a self-morphism, we prove that such subset has Banach density zero under some assumptions.

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arXiv2024

Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps

A dominant rational self-map on a projective variety is called $p$-cohomologically hyperbolic if the $p$-th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over $\overline{\mathbb{Q}}$, we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if $f$ is an $1$-cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a $\overline{\mathbb{Q}}$-point with generic $f$-orbit is equal to the first dynamical degree of $f$; and (2) there are $\overline{\mathbb{Q}}$-points with generic $f$-orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental.

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arXiv2023

On preimages question

For a surjective self-morphism on a projective variety defined over a number field, we study the preimages question, which asks if the set of rational points on the iterated preimages of an invariant closed subscheme eventually stabilize. We prove the answer is positive for self-morphisms on $\mathbb{P}^1 \times \mathbb{P}^1$. We also prove that the answer is positive even if we allow finite extension of the number field by a bounded degree if the morphism is étale or the invariant subscheme is $0$-dimensional. We prove certain uniformity of the stabilization for self-morphisms on abelian varieties. When the subscheme is not invariant, we provide counterexamples and a result in a positive direction for products of polynomial maps on $\mathbb{P}^1 \times \mathbb{P}^1$.

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arXiv2023

Distribution of rational points on random Fano hypersurfaces

We prove an asymptotic formula for the average number of rational points on Fano hypersurfaces that are contained in a small ball centered at a given adelic point. We also prove an asymptotic formula for the number of hypersurfaces admitting adelic points that are contained in a small ball.

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arXiv2023

Non-density of points of small arithmetic degrees

Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $α_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree $α_f(x)$ is strictly smaller than the first dynamical degree $δ_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms on projective varieties including surfaces, HyperKähler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fibre spaces. We show the equivalence of the sAND Conjecture and another conjecture on the periodic subvarieties of small dynamical degree; we also show the close relations between the sAND Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on endomorphisms of projective spaces and another long standing conjecture on Uniform Boundedness of torsion points in abelian varieties.

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arXiv2022

Growth of local height functions along orbits of self-morphisms on projective varieties

In this paper, we consider the limit $ \lim_{n \to \infty} \sum_{v\in S} λ_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) $ where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field, $S$ is a finite set of places, $ λ_{Y,v}$ is a local height function associated with a proper closed subscheme $Y \subset X$, and $h_{H}$ is an ample height function on $X$. We give a geometric condition which ensures that the limit is zero, unconditionally when $\dim Y=0$ and assuming Vojta's conjecture when $\dim Y\geq1$. In particular, we prove (one is unconditional, one is assuming Vojta's conjecture) Dynamical Lang-Siegel type theorems, that is, the relative sizes of coordinates of orbits on $\mathbb{P}^{N}$ are asymptotically the same with trivial exceptions. These results are higher dimensional generalization of Silverman's classical result.

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arXiv2021

On Dynamical Cancellation

Let $X$ be a projective variety and let $f$ be a dominant endomorphism of $X$, both of which are defined over a number field $K$. We consider a question of the second author, Meng, Shibata, and Zhang, which asks whether the tower of $K$-points $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$ eventually stabilizes, where $Y\subset X$ is a subvariety invariant under $f$. We show this question has an affirmative answer when the map $f$ is étale. We also look at a related problem of showing that there is some integer $s_0$, depending only on $X$ and $K$, such that whenever $x, y \in X(K)$ have the property that $f^{s}(x) = f^{s}(y)$ for some $s \geq 0$, we necessarily have $f^{s_{0}}(x) = f^{s_{0}}(y)$. We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on $\mathbb{P}^1$ where we allow for composition by multiple different maps $f_1,\dots,f_r$.

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arXiv2021

Invariant subvarieties with small dynamical degree

Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $Σ_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $Σ_{f^{\infty}}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty}}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an optimal upper bound (of cardinality) $$\sharp P_{f^n}\le d_1(f)^n(1+o(1))$$ as $n\to \infty$, where $d_1(f)$ is the first dynamic degree of $f$. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety (with $f$ stabilizing the big torus), we give an optimal upper bound $$\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))$$ as $n \to \infty$, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.

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arXiv2020

Vojta's conjecture, heights associated with subschemes, and primitive prime divisors in arithmetic dynamics

Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomorphisms on $\mathbb{P}^{2}_{\overline{\mathbb{Q}}}$. We also discuss applications of Vojta's conjecture with truncated counting function to the problem of the existence of primitive prime divisors of coordinates of orbits of $f$

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arXiv2020

The Distribution Relation and Inverse Function Theorem in Arithmetic Geometry

We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis on versions that can be applied uniformly across families of varieties and maps. In particular, we prove two explicit versions of the inverse function theorem, the first via general distribution and separation inequalities that may be of independent interest, the second via a careful implementation of classical Newton iteration.

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arXiv2020

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.

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