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Yohsuke Matsuzawa

Publications and source records attributed to Yohsuke Matsuzawa.

24 records · Page 2Linked to original sources
arXiv2020

A note on Kawaguchi-Silverman conjecture

We collect some results on endomorphisms on projective varieties related with the Kawaguchi-Silverman conjecture. We discuss certain condition on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi-Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.

math.AG↗
arXiv2019

Int-amplified endomorphisms on normal projective surfaces

We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.

math.AG↗
arXiv2018

On upper bounds of arithmetic degrees

Let $X$ be a smooth projective variety over $ \overline{\mathbb Q}$, and $f:X -rightarrow X$ be a dominant rational map. Let $δ_{f}$ be the first dynamical degree of $f$ and $h_{X}:X( \overline{\mathbb Q})\to [1,\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X (f^n(P)))_{n\geq0}$ where $P$ is a point of $X( \overline{\mathbb Q})$ whose forward orbit by $f$ is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $ \overlineα_{f}(P) \leq δ_{f}$. Furthermore, if the Picard number of $X$ is one, $f$ is algebraically stable and $δ_{f}>1$, we prove that the limit defining canonical height $\lim_{n\to \infty} h_{X} (f^{n}(P)) \big/ δ_f^n$ converges.

math.AG↗
arXiv2017

Arithmetic degrees for dynamical systems over function fields of characteristic zero

We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are interpreted into geometric ones. We give another proof of the theorem that the arithmetic degree at any point is smaller than or equal to the dynamical degree. We give a sufficient condition for an arithmetic degree to coincide with the dynamical degree, and prove that any self-map has so many points whose arithmetic degrees are equal to the dynamical degree. We study dominant rational self-maps on projective spaces in detail.

math.AG↗
arXiv2017

Arithmetic degrees and dynamical degrees of endomorphisms on surfaces

For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points having disjoint orbits if the endomorphism is an automorphism.

math.AG↗