Discrete Groups and Algebraic Geometry

University Of Texas At Austin, Austin TX

Investigators

Abstract

The problems comprising the project are (1) to study moduli spaces in real algebraic geometry as real hyperbolic orbifolds, which this is possible, with particular emphasis on the real forms of the Deligne-Mostow quotients of the complex hyperbolic space; (2) to prove that the moduli space of K3 surfaces with a polarization of given degree has contractible universal cover, and a similar result for the complement of the complex hyperplane arrangement associated to an arbitrary Coxeter group; (3) to try to construct complex hyperbolic reflection groups in arbitrarily large dimensions, or prove that they do not exist; and (4) to see if the monster simple group manifests itself in a certain simple way in terms of a certain quotient of complex hyperbolic 13-space. The common thread in these projects is the role in algebraic geometry of discrete groups acting on Hermitian symmetric spaces like the ball and the type IV domains. Besides these discrete groups, there are a number of other important groups involved, like the fundamental groups of hyperplane complements, and certain quotients of these fundamental groups (perhaps including the monster). The study of symmetry is called group theory; a group is the collection of all self-transformations of a picture, pattern, etc., that leave the picture, pattern, etc. alone. The focus is on the transformation, for example, the act of rotating a picture around a point in the plane. If the picture looks exactly the same before and after the rotation, then it has rotational symmetry. To emphasize this point of view we say that the group "acts" on the picture. There are lots of different sorts of objects with group actions, including some that are difficult of visualize but are still very important in physics and mathematics. One of these difficult-to-visualize objects is called complex hyperbolic space, and the possible ways that groups can act on complex hyperbolic space have been studied by many mathematicians since the 19th century. The easiest-to-understand transformations of complex hyperbolic space are called "complex reflections", which (despite the name) are a sort of rotation around an axis. Groups that are generated by this kind of transformation play a privileged role in the field of algebraic geometry, helping to explain certain patterns which appear when studying some important objects, including what are called "binary quantics" and "K3 surfaces". The investigator has also noticed some more patterns, currently unexplained, which may connect complex reflections to a famous group called the "monster" simple group. The specific technical problems to be addressed by the project all attempt to advance our understanding of groups generated by complex reflections, acting on complex hyperbolic space.