The Eisenstein-Picard modular group PU(2, 1;Z[ω]) is defined to be the subgroup of PU(2, 1) whose entries lie in the ring Z[ω], where ω is a cube root of unity. This group acts isometrically and properly discontinuously on H C , that is, on the unit ball in C2 with the Bergman metric. We construct a fundamental domain for the action of PU(2, 1;Z[ω]) on H2 C , which is a 4-simplex with one ideal...

One main problem in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the Picard group. The goal of this paper is to formulate explicit Hodge-theoretic criteria for the ampleness of line bundles on certain irreducible holomorphic symplectic manifolds. It is well known that for K3 surfaces the ample cone is governed by (−2)-curves. More generally,...

and let f = π1 · p = π2 · p2 : X → P . Schoen [S] has shown that if the surfaces are sufficiently general, then X is a smooth Calabi-Yau manifold. Let K(X) be the closure of the Kähler cone of X . By construction K(X) has infinitely many edges. Here we show that there is a fundamental domain which is a (finite) rational polyhedral cone, for the induced action of Aut(X) on K(X). Our work was ins...

T . This can now be used to find all patterns in LI(P) with special symmetries (relative to the origin) which is a standard procedure in crystallography. For repetitive, but non-crystallographic P , however, the LI-class has a much richer structure and contains uncountably many (20) translation classes. It is thus much more difficult to classify the complete symmetry structure of such classes, ...

Let {xn} be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {xn} which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {xn} can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is ...

Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean al...

Arithmetic groups are groups of matrices with integral coefficients. They first appeared in the work of Gauss, Minkowski and others on the arithmetic theory of quadratic forms. Their reduction theory consists in showing that, after a linear change of variables with integral coefficients, any quadratic form can be forced to satisfy an appropriate set of inequalities. Around 1940, Siegel develope...

The modular invariant coefficient of the D2kR4 term in the effective action of type IIB superstring theory is expected to satisfy Poisson equation on the fundamental domain of SL(2, Z). We obtain the equation satisfied by D10R4 using the tree level and one loop results for four graviton scattering in type II string theory. We find that the perturbative contributions to D10R4 vanish above three ...

We present a new method for computing the global one-dimensional unstable manifold of a hyperbolic xed point of a map. The key idea is to `grow' the manifold, where the speed of growth is determined only by the curvature of the manifold, and not by the dynamics. In other words, we do not use the standard approach of iterating a fundamental domain. Furthermore, we present two new families of pla...

Abstract. We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function j. Our algorithm makes use of the natural connection between the arithmetic-geometric mean (AGM) of complex numbers and modular functions. Through a detailed complexity analysis, we prove that for a given τ , evaluating N significative bits of j(τ) can be...