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We classify smooth projective surfaces that are quotients of abelian surfaces by finite groups., Comment: 12 pages

We study finite quotients of abelian varieties (fqav for short) i.e. quotients of abelian varieties by finite groups. We show that Q-abelian varieties (i.e. fqav's with $\mathbb Q$-linearly trivial canonical divisors) are characterized by the existence of quasi\'etale polarized (or int-amplified) endomorphisms. We show that every fqav has a finite quasi\'etale cover by the product of an abelian variety and a $\mathbb Q$-Fano fqav. Using such coverings, we give a characterization of $\mathbb Q$-Fano fqav's, and show that $\mathbb Q$-Fano fqav's and Q-abelian varieties are ``building blocks'' of general fqav's., Comment: 18 pages. Comments welcome

We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we prove the potential density for projective varieties with int-amplified endomorphisms in dimension $\leq 3$. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field., Comment: 12 pages; comments are welcome!

Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $\alpha_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $\bar{\kappa}(X)$ is non-negative (resp. the \'etale fundamental group $\pi_1^{\text{\'et}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with ''larger'' first dynamical degree (resp. every dominant endomorphism of any semiabelian variety)., Comment: 35 pages; comments are welcome!

We prove that any surjective self-morphism with $\delta_f > 1$ on a potentially dense smooth projective surface defined over a number field $K$ has densely many $L$-rational points for a finite extension $L/K$., Comment: 10 pages

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this article, we show that there are densely many $\overline{\mathbb Q}$-rational points with maximal arithmetic degree (i.e. whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the appendix., Comment: 22 pages

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 $\Sigma_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $\Sigma_{f^{\infty}}$, 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 $f$., Comment: Minor revision, 31 pages, International Mathematics Research Notices (to appear)

Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $\alpha_f(x)$ 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 $\alpha_f(x)$ is strictly smaller than the first dynamical degree $\delta_f$ 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, HyperK\"ahler 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., Comment: 45 pages, minor editing

We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell--Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety., Comment: 41 pages. The previous version has a serious mistake on the proof of the main conjecture for simple abelian varieties, but the present version gives a renewed proof that works for arbitrary abelian varieties

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., Comment: 28 pages

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., Comment: 26 pages

We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf., Comment: 18 pages