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).
We prove that any compact Kähler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact Kähler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1,1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider.…
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…
We study the intersection theory of complex Lagrangian subvarieties inside holomorphic symplectic manifolds. In particular, we study their behaviour under Mukai flops and give a rigorous proof of the Plücker type formula for Legendre dual subvarieties written down by the second author before. Then we apply the formula to study projective dual varieties in projective spaces.
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.
Birational automorphism groups in families of hyper-Kähler manifolds
Universität des Saarlandes（萨尔兰州立大学）；Università degli Studi di Bologna（博洛尼亚大学）；Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学和索邦大学，法国国家科学研究中心，IMJ-PRG）；National Center for Theoretical Sciences, National Taiwan University（台湾大学理论科学中心）
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）
If $f$ is an automorphism of a compact simply connected Kähler manifold with trivial canonical bundle that fixes a Kähler class, then the order of $f$ is finite. We apply this well known result to construct compact non-Kähler manifolds. These manifolds contradict the abundance and Iitaka conjectures for complex manifolds.
In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.
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…
In this paper, we investigate automorphisms of compact Kähler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^\infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.
In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.
Let $X$ be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.
Extending well-known results on surfaces, we give bounds on the cohomological action of automorphisms of compact Kähler threefolds. More precisely, if the action is virtually unipotent we prove that the norm of $(f^n)^*$ grows at most as $cn^4$; in the general case, we give a description of the spectrum of $f^*$, and bounds on the possible conjugates over $\mathbb Q$ of the dynamical degrees $λ_1(f),λ_2(f)$. Examples on complex tori show the optimality of the results.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.
In Kähler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized Kähler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized Kähler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that…
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 $…
We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan proper…
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold $X$ whose general fiber is of dimension strictly between $0$ and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold $X$ admitting a Lagrangian fibration a Lagrangian constant cycle subvariety $Σ_H$ in $X$ which depends on a divisor class $H$ whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on $Σ_H$ in $\mathrm{CH}_0(X)$ depend neither on $H$ nor…
For the Calabi-Yau threefolds $X$ constructed by C. Schoen as fiber products of generic rational elliptic surfaces, we show that the action of the automorphism group of $X$ on the Kähler cone of $X$ has a rationally polyhedral fundamental domain. The second author has conjectured that this statement will hold in general, the example presented here being the first non-trivial case in which the statement has been checked. The conjecture was motivated by the desire to use a construction of E. Looijenga to compactify certain moduli spaces which arise in the study of conformal field theory and ``mi…
For any Kahler surface which admits no nonzero holomorphic vectorfields, we consider the group of holomorphic automorphisms which induce identity on the second rational cohomology. Assuming the canonical linear system is without base points and fixed components, C.A.M. Peters showed that this group is trivial except when the Kahler surface is of general type and either $c_1^2=2c_2$ or $c_1^2=3c_2$ holds. Moreover, this group is a 2-group in the former case, and is a 3-group in the latter. The purpose of this note is to give further information about this group. In particular, we show that $c_1…
The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of w is Zariski dense.
For any Lagrangean Kähler submanifold $M \subset T^*{\Bbb C}^n$, there exists a canonical hyper Kähler metric on $T^*M$. A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, a class of pseudo hyper Kähler manifolds of complex signature $(2,2n)$ is constructed. For any hyper Kähler manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is speci…
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.
The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of $U$ equal to the number of real embeddings of $K$. We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all non-trivial elements in $U$ are primitive in $K$, then $X$ contains no proper c…
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,…
We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kahler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, a…
We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kahler-Ricci flow on compact Kahler manifolds always form along analytic subvarieties, thus answering a question of Feldman-Ilmanen-Knopf and Campana. We also extend the second author's results about noncollapsing degenerations of Ricci-flat Kahler metrics on Cala…
We study Nakai-Moishezon type question and Donaldson's "tamed to compatible" question for almost complex structures on rational four manifolds. By extending Taubes' subvarieties--current--form technique to $J-$nef genus $0$ classes, we give affirmative answers of these two questions for all tamed almost complex structures on $S^2$ bundles over $S^2$ as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.
Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as…
Given a Lagrangian fibration $π: X \to \mathbb{P}^n$ of a compact hyper-Kähler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, $\text{OG10}$ or $\text{OG6}$-type, we construct a natural compactification of its dual torus fibration. Specifically, this compactification is given by a quotient of $X$ by certain automorphisms acting trivially on the second cohomology and respecting the Lagrangian fibration. It is a compact hyper-Kähler orbifold with identical period mapping behavior as $X$.
We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.
On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in $C^\infty$-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.
Let $X$ be a compact complex manifold in Fujiki's class $\mathcal{C}$, i.e., admitting a big $(1,1)$-class $[α]$. Consider $\text{Aut}(X)$ the group of biholomorphic automorphisms and $\text{Aut}_{[α]}(X)$ the subgroup of automorphisms preserving the class $[α]$ via pullback. We show that $X$ admits an $\text{Aut}_{[α]}(X)$-equivariant Kähler model: there is a bimeromorphic holomorphic map $σ\colon \widetilde{X}\to X$ from a Kähler manifold $\widetilde{X}$ such that $\text{Aut}_{[α]}(X)$ lifts holomorphically via $σ$. There are several applications. We show that $\text{Aut}_{[α]}(X)$ is a Lie …
Let (X,L) be a polarized manifold. Assume that the automorphism group is finite. If the height discrepancy of (X,L) is O(d^2) then (X,L) admits a csck metric in the first chern class of L if and only if (X,L) is asymptotically stable.
Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperkähler manifold, up to finite group factor. We also discuss relevant topics, especially, new counterexamples of Kodaira's problem about algebraic approximation of a compact Kähler manifold.
Let M be a compact, holomorphically symplectic Kahler manifold, and $η$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $η$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $η$ are coisotropic. When M is generic, this is used to show that all Lelong numbers of $η$ vanish. We prove that any hyperkahler manifold with Pic(M) of rank 1 has non-trivial coisotropic subvarieties, if a generator of Pic(M) is parabolic.
Given any connected topological space $X$, assume that there exists an epimorphism $ϕ: π_1(X) \to \mathbb{Z}$. The deck transformation group $\mathbb{Z}$ acts on the associated infinite cyclic cover $X^ϕ$ of $X$, hence on the homology group $H_i(X^ϕ, \mathbb{C})$. This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring $\mathbb{C}[t,t^{-1}]$, which is a finite dimensional $\mathbb{C}$-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When $X$ is a compact Kähler manifold, we show that all the Jordan …
In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kahler metrics on the projectivization of the dual vector bundle.
In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group.
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun struct…
Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probab…
We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Aubin's conjecture on the relation between the anticanonical volume and the existence of a Kähler-Einstein metric. Finally, we consider the case that $\dim X=3$; we give a complete answer for slope (semi)stability along divisors of Fano threefolds.
In this article we show how to calculate the group of automorphisms of flat Kähler manifolds. Moreover we are interested in the problem of classification of such manifolds up to biholomorphism. We consider these problems from two points of view. The first one treats the automorphism group as a subgroup of the group of affine transformations, while in the second one we analyze it using automorphisms of complex tori. This leads us to the analogues of the Bieberbach theorems in the complex case. We end with some examples, which in particular show that in general the finiteness of the automorphism…
