Lattices and automorphisms of compact complex manifolds

Abstract

This work makes use of well-known integral lattices to construct complex algebraic varieties reflecting properties of the lattices. In particular the automorphism groups of the lattices are closely related to the symmetries of varieties. The constructions are to two types: generalised Kummer manifolds and toric varieties. In both cases the examples are of the most interest. A generalised Kummer manifold is the resolution of the quotient of a complex torus by some finite group G. A description of the construction for certain cyclic groups G by given in terms of holomorphic surgery of disc bundles. The action of the automorphism groups is given explicitly. The most important example is a compact complex 12-dimensinoal manifold associated to the Leech lattice admitting an action of the finite simple Suzuki group. All these generalised Kummer manifolds are shown to be simply connected. Toric varieties are associated to certain decompositions of Rn into convex cones. The automorphism groups of those associated to Weyl group decompositions of Rn are calculated. These are used to construct 24-dimensional singular varieties from some Neimeier lattices. Their symmetries are extensions of Mathieu groups and their singularities closely related to the Golay codes