Geometric realizations of cyclic actions on surfaces: II

Geometric Realizations of Substitutions

Geometric Realizations of Polyhedral Complexes

We discuss two problems in combinatorial geometry. First, given a geometric polyhedral complex in R (a family of 3-polytopes attached face-to-face), can it always be realized over Q? We give a negative answer to this question, by presenting an irrational polyhedral complex with 1278 convex polyhedra. We then present a universality theorem on the space of realizations of such complexes. Second, ...

Group Actions, Cyclic Coverings and Families of K3-surfaces

In this paper we describe six pencils of K3-surfaces which have large Picardnumber (15 ≤ ρ ≤ 20) and contain precisely five singular fibers: four have A-D-E singularities and one is non-reduced. In particular we describe these surfaces as cyclic coverings of the K3-surfaces of [BS]. In many cases using this description and latticetheory we are able to compute the exact Picard-number and to desc...

Geometric Realizations of Hermitian Curvature Models

We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant ⋆-scalar curvature which satisfies the Kaehler condition at the point in question.

Combinatorial realizations of crystals via torus actions on quiver varieties

Let V (λ) be a highest-weight representation of a symmetric Kac–Moody algebra, and let B(λ) be its crystal. There is a geometric realization of B(λ) using Nakajima’s quiver varieties. In many particular cases one can also realize B(λ) by elementary combinatorial methods. Here we study a general method of extracting combinatorial realizations from the geometric picture: we use Morse theory to in...