Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of M are hyperplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all Hi. 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 hyperkähler manifolds with Picard group generated by a negative class z, we prove that ±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 hyperkähler manifolds.
Recall that for a variety X defined over a field k, rational points are said to be potentially dense in X if for some finite extension k ′ of k, X(k ′ ) is Zariski dense in X. For example, this is the case if X is a rational or a unirational variety.If k is a number field, a conjecture of Lang and Vojta, proved in dimension 1 by Faltings, but still open even in dimension two, predicts that varieties of general type never satisfy potential density over k. On the contrary, it is generally expected that when the canonical bundle K X is negative (i.e. X is a Fano variety) or trivial, rational points are always potentially dense in X (see for example [9] for a recent and precise conjecture).In the Fano case, there are many examples confirming this. First of all, many Fano varieties are unirational (in dimension ≤ 2, all of them are even rational over a finite extension of the definition field). Furthermore, Fano threefolds such as a three-dimensional quartic or a "double Veronese cone" (a hypersurface of degree 6 in P(1, 1, 1, 2, 3)) are birational to elliptic fibrations over P 2 , which makes it possible to prove potential density (see [13]).If the canonical bundle is trivial, the known examples are much less convincing. It is well-known that rational points on abelian varieties are potentially dense. But the simply-connected case remains largely unsolved.A number of results concerning potential density of rational points on K3 surfaces defined either over a number field or a function field appeared in the last years. Bogomolov and Tschinkel [6] proved potential density of rational points for K3 surfaces admitting an elliptic pencil, or an infinite automorphisms group. Remark that these K3 surfaces are rather special; in particular, their geometric Picard number is never equal to 1, whereas it is equal to 1 for a general projective K3.The case of K3-surfaces defined over a (complex) function field has been solved for certain types of families by Hassett and Tschinkel [14]. They prove in particular the existence of K3-surfaces defined over C(t), whose geometric Picard group is equal to Z, and which satisfy potential density. Their method can likely be adapted to produce examples over Q(t).However, no example of K3 surface defined over a number field, and with geometric Picard group equal to Z is known to satisfy potential density.In this paper, we consider the case of higher dimensional varieties which are as close as possible to K3 surfaces, namely Fano varieties of lines F of cubic 4-folds X. These are 4-dimensional varieties with trivial canonical bundles, and which even possess a non degenerate holomorphic 2-form which will play an important role in our study. Their similarity with K3 surfaces is shown by the work of Beauville and 1
Let X be an algebraic variety and let f : X X be a rational self-map with a fixed point q, where everything is defined over a number field K. 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 proved by Claire Voisin and the first author in 2007.
We prove that the characteristic foliation F on a non-singular divisor D in an irreducible projective hyperkähler manifold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and F are as above, then F can be algebraic with nonrational leaves only when, up to a finiteétale cover, X is the product of a symplectic projective manifold Y with a symplectic surface and D is the pullback of a curve on this surface. When D is of general type, the fact that F cannot be algebraic unless X is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical class of the (orbifold) base of the family of leaves is zero. This implies, in particular, the isotriviality of the family of leaves of F . We show this, more generally, for regular algebraic foliations by curves defined by the vanishing of a holomorphic (d − 1)-form on a complex projective manifold of dimension d.
Let M be a simple holomorphically symplectic manifold, that is, a simply connected compact holomorphically symplectic manifold of Kähler type with h 2,0 = 1. Assuming b2(M ) = 5, we prove that the group of holomorphic automorphisms of M acts on the set of faces of its Kähler cone with finitely many orbits. This statement is known as Morrison-Kawamata cone conjecture for hyperkähler manifolds. As an implication, we show that any hyperkähler manifold has only finitely many non-equivalent birational models. 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 {Si} an infinite set of complete, locally geodesic hypersurfaces. Then the union of Si is dense in M .
Let M be an irreducible holomorphic symplectic (hyperkähler) manifold. If b 2 (M ) 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 ) 14, similarly, we construct a deformation which admits a parabolic automorphism.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.