Apollonian carpets and the boundary of the Kahler cone of a hyperkahler manifold

Abstract

The ample cone of a compact Kahler $n$-manifold $M$ is the intersection of its Kahler cone and the real subspace generated by integer (1,1)-classes. Its isotropic boundary is the set of all points $\eta$ on its boundary such that $\int_M \eta^n=0$. We are interested in the relation between the shape of the isotropic boundary of the ample cone of a hyperkahler manifold and the dynamics of its holomorphic automorphism group $G$. In this case, the projectivization of the ample cone is realized as an open, locally polyhedral subset in a hyperbolic space $H$. The isotropic boundary $S$ is realized as a subset of the hyperbolic boundary (the absolute) $A$ of $H$, which is naturally identified with a Euclidean sphere. It is clear that the isotropic boundary $S$ contains the limit set of $G$ acting on its ample cone. We prove that, conversely, all irrational points on $S$ belong to the limit set. Using a result of N. Shah about limiting distributions of curves under geodesic flow on hyperbolic manifolds, we prove that every real analytic curve in $S$ is contained in a geodesic sphere in $S$,and in presence of such curves the limit set is the closure of the union of these geodesic spheres. We study the geometry of such fractal sets, called Apollonian carpets, and establish the link between the Apollonian carpet and the structure of the automorphism group.