arXiv subjects
Publications and source records attributed to Ekaterina Amerik.
Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_2(M)\geq 5$, we construct a deformation $M'$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_2(M) \geq 14$, similarly, we construct a deformation which admits a parabolic automorphism.
Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a hyperbolic Riemannian metric on the projectivisation of the positive cone in $H^{1,1}(M,Q)$, denoted by $H$. Torelli theorem implies that the Hodge monodromy group $Γ$ acts on $H$ with finite covolume, giving a hyperbolic orbifold $X=H/Γ$. We show that there are finitely many geodesic hypersurfaces which cut $X$ into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient $P(M')/Aut(M')$, where $P(M')$ is the projectivization of the ample cone of a birational model $M'$ of $M$, and $Aut(M')$ the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkahler birational model of a simple hyperkahler manifold of Picard number at least 5, and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism, originally deduced by Markman and Yoshioka from the Morrison-Kawamata cone conjecture.
Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmuller space of hyperkahler structures on a hyperkahler manifold, identifying any of its connected components with an open subset of the Grassmannian $SO(b_2-3,3)/SO(3)\times SO(b_2-3)$ consisting of all Beauville-Bogomolov positive 3-planes in $H^2(M, R)$ which are not orthogonal to any of the MBM classes. This is used to determine the Teichmuller space of symplectic structures of Kahler type on a hyperkahler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes $v\in H^2(M,\R)$ with $q(v,v)>0$, where $q$ is the Bogomolov-Beauville-Fujiki form on $H^2(M,\R)$.
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 $\{S_i\}$ an infinite set of locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.
This is not a research paper, but a survey submitted to a proceedings volume.
Let $M$ be an irreducible holomorphically symplectic manifold. We show that all faces of the Kahler cone of $M$ are hyperplanes $H_i$ orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all $H_i$. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkahler manifolds with Picard group generated by a negative class $z$, we prove that $\pm z$ is Q-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkahler manifolds.
Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.
This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some observations suggesting a different approach in the algebraic case.
This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.
Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this would follow immediately from the theory of canonical heights, but it does not work very well for rational self-maps. We provide an elementary proof following an argument by Bell, Ghioca and Tucker (arxiv:0808.3266).
Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).
An example of potential density of rational points on the second punctual Hilbert scheme of certain K3 surfaces is treated in detail. This is an amplification of some remarks made by O'Grady and Oguiso.
I compute the dynamical degrees in C. Voisin's example of a rational self-map of the variety of lines on a cubic fourfold.
We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where ``sufficiently general'' means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard group equal to $\mathbb{Z}$.
Motivated by an example, due to Voisin, of a smooth simply-connected projective variety with trivial canonical class and cyclic Picard group, admitting a meromorphic endomorphism of high degree, we study meromorphic fibrations on certain varieties with trivial canonical class. We show (theorem 2.1) that any nontrivial meromorphic fibration of a variety with trivial canonical class and cyclic Picard group is in varieties of general type. Therefore, Voisin's endomorphism cannot preserve a fibration, at least in the generic case. As this example is symplectic, we study meromorphic fibrations on symplectic varieties. In dimension 4 (and in all dimensions modulo minimal model program) we get some results similar to Matsushita's. The main result here is theorem 3.6. We also study the relations between meromorphic endomorphisms and meromorphic fibrations. In particular, we prove that if an endomorphism does not preserve a fibration, then its general iterated orbit is Zariski-dense. We show that some power of an endomorphism preserves the "core" (a natural fibration constructed by the second author), and ask the similar question about the Iitaka fibration.
It is proved that the degree of a morphism from a smooth projective n-fold with Picard number one to a smooth n-quadric is bounded (provided, of course, that n is at least three). Actually it has been proved some years ago, but I have never written down the proof, until J.-M. Hwang suggested, recently, that I do.
This is not a research article. This is a partial version of a mini-cours which I gave at a meeting of the ACI Jeunes Chercheurs "Dynamique des applications polynomiales" in Toulouse in november 2004. This text includes two parts of that mini-cours: the first one collects some preliminaries on Fano threefolds with cyclic Picard group, the second one discusses some methods to obtain bounds for the degree of a morphism onto such a Fano threefold (different from projective space). The mini-cours also had a third part - the boundedness theorem of Hwang and Mok for morphisms between Fano manifolds which have rational curves with trivial normal bundles; this part is not included here, as the Hwang and Mok's paper in Journal of Algebraic Geometry 12 (2003) is concise, clear and self-contained.
Let $X$, $Y$ be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let $X$ be of index one and $Y$ be of index two. It is shown that the only morphisms from $X$ to $Y$ are double coverings. In fact nearly the whole paper is the analysis of the case where $Y$ is the linear section of the Grassmannian G(1,4), since the other cases were more or less solved in another article. This remaining case is treated with the help of Debarre's connectedness theorem for inverse images of Schubert cycles.