Let f be a holomorphic automorphism of a compact Kahler manifold (X,ω) of dimension k>1. We study the convex cones of positive closed (p,p)-currents T_p, which satisfy a functional relation $f^*(T_p)=λT_p, λ>1,$ and some regularity condition (PB). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy h(f) of f is positive, then for some m, there is a positive closed (m,m)-current T_m which satisfies the relation $f^* T_m=\exp(h(f)) T_m$. Moreover,every quasi-p.s.h. function is integrable with respect to the trace measure of T_m. When the dynamical degrees of f are all distinct, we construct an invariant measure $μ$ as an intersection of closed currents. We show that this measure is mixing and gives no mass to pluripolar sets and to sets of small Hausdorff dimension.
We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.
Let $f$ be a holomorphic automorphism of a compact Kähler manifold. Assume moreover that $f$ admits a unique maximal dynamic degree $d_p$ with only one eigenvalue of maximal modulus. Let $μ$ be its equilibrium measure. In this paper, we prove that $μ$ is exponentially mixing for all d.s.h.\ test functions.
Algebraic entropy of birational automorphism groups in families of hyper-Kähler manifolds
Universität des Saarlandes（萨尔兰州立大学）；The University of Tokyo（东京大学）；National Taiwan University（台湾大学）；Università della Calabria（卡拉布里亚大学）；Université de Strasbourg（斯特拉斯堡大学）
We prove that there is a projective K3 surface admitting a (fixed point) free automorphism of positive entropy and that no smooth compact Kähler surface other than projective K3 surfaces and their blow up admits such an automorphism.
We study the automorphisms of compact Kähler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions $2$ and $3$. We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the $C^{\infty}$ context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension $2$ and prove they exist only on tori. We conjecture that this is true for any dim…
For a compact Kahler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most rank NS(X) of g-periodic prime divisors B_i (i.e., g^s(B_i) = B_i for some s > 0). When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic itself (a result in the spirit of the Dynamic Manin-Mumford conjecture).
For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact Kähler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus T and G is mostly descended from the symmetries on the torus T, provided that both X and the pair (X, G) are minimal.
We study zero entropy automorphisms of a compact Kähler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $…
Let $X$ be a compact Kähler manifold of dimension 3 and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $λ_1(f)^2>λ_2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be…
Let $X$ be a compact Kähler manifold of dimension $k\leq 4$ and $f:X\rightarrow X$ a pseudo-automorphism. If the first dynamical degree $λ_1(f)$ is a Salem number, we show that either $λ_1(f)=λ_{k-1}(f)$ or $λ_1(f)^2=λ_{k-2}(f)$. In particular, if $\mbox{dim}(X)=3$ then $λ_1(f)=λ_2(f)$. We use this to show that if $X$ is a complex 3-torus and $f$ is an automorphism of $X$ with $λ_1(f)>1$, then $f$ has a non-trivial equivariant holomorphic fibration if and only if $λ_1(f)$ is a Salem number. If $X$ is a complex 3-torus having an automorphism $f$ with $λ_1(f)=λ_2(f)>1$ but is not a Salem number,…
Let G be a group of automorphisms of a compact Kähler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N(G) is a free abelian group of rank < n-1, or G/N(G) is a free abelian group of rank n-1 and X is a complex torus, or G is a free abelian group of rank n-1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X is a projective manifold of dimension n > 2 and is no…
We give an algebro-geometric approach towards the dynamics of automorphisms/endomorphisms of projective varieties or compact Kähler manifolds, try to determine the building blocks of automorphisms /endomorphisms, and show the relation between the dynamics of automorphisms/endomorphisms and the geometry of the underlying manifolds.
We classify invariant probability measures for non-elementary groups of automorphisms, on any compact Kähler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also prove that except in certain rigid situations known as Kummer examples, there are only finitely many invariant, ergodic, probability measures with a Zariski dense support. If X is a K3 or Enriques surface, and the group does not preserve any algebraic subset, this leads to a complete description of orbit closures.
We study the dynamics of a bimeromorphic selfmap of a compact complex Kähler surface $X$. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of $\mathbb{C}^2$. This extends recent results by E. Bedford and J. Diller.
On the virtual invariants of zero entropy groups of compact Kähler manifolds
National University of Singapore（新加坡国立大学）；National Taiwan University（台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；the University of Tokyo（东京大学）
In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\geq 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\rightarrow X_0$ is a finite blowup along smooth centers, here $X_0$ is a projective manifold of interest. Here we allow $X_0$ to be either one of the following manifolds: it has Picard number 1, or a Fano manifold, or it is a projective hyper-Kähler manifold. We also allow the centers of blowups to have large dimensions relative to that of $X_0$ …
We classify compact Kähler manifolds $M$ of dimension $n\geq 3$ on which acts a lattice of an almost simple real Lie group of rank $\geq n-1$. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.
Let $X$ be a projective variety of dimension $n\ge1$ over an algebraically closed field of arbitrary characteristic. We prove a Fujiki-Lieberman type theorem on the structure of the automorphism group of $X$. Let $G$ be a group of zero entropy automorphisms of $X$ and $G_0$ the set of elements in $G$ which are isotopic to the identity. We show that after replacing $G$ by a suitable finite-index subgroup, $G/G_0$ is a unipotent group of the derived length at most $n-1$. This result was first proved by Dinh, Oguiso and Zhang for compact Kähler manifolds.
A meromorphic transform between complex manifolds is a surjective mutivalued map with an analytic graph. Let $F_n$ be a sequence of meromorphic transforms from a compact Kahler manifold X into compact Kahler manifolds X_n. We give conditions which imply that the behavior of the sequence of preimages $F_n^{-1}(x_n)$ of x_n does not depend on the generic sequence of points (x_1,x_2,....). Using this formalism, we obtain sharp results on the limit distribution of common zeros, of $l$ random holomorphic sections of high powers $L^n$ of a positive holomorphic line bundle L over a projective manifol…
Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits, whenever $b_2(M)\neq 5$. This is a version of the Morrison-Kawamata cone conjecture for hyperkahler manifolds. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and $\{S_…
