Discrete Groups and Algebraic Geometry

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The problems comprising the project are (1) to study moduli spaces inreal algebraic geometry as real hyperbolic orbifolds, which this ispossible, with particular emphasis on the real forms of theDeligne-Mostow quotients of the complex hyperbolic space; (2) to provethat the moduli space of K3 surfaces with a polarization of given degreehas contractible universal cover, and a similar result for thecomplement of the complex hyperplane arrangement associated to an arbitrary Coxeter group; (3) to try to construct complex hyperbolicreflection 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 inalgebraic geometry of discrete groups acting on Hermitian symmetricspaces like the ball and the type IV domains. Besides these discretegroups, there are a number of other important groups involved, like thefundamental groups of hyperplane complements, and certain quotients ofthese fundamental groups (perhaps including the monster).The study of symmetry is called group theory; a group is the collectionof 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 apoint 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 somethat are difficult of visualize but are still very important in physics and mathematics. One of these difficult-to-visualize objects is calledcomplex hyperbolic space, and the possible ways that groups can act oncomplex hyperbolic space have been studied by many mathematicians sincethe 19th century. The easiest-to-understand transformations of complexhyperbolic space are called "complex reflections", which (despite thename) are a sort of rotation around an axis. Groups that are generatedby this kind of transformation play a privileged role in the field ofalgebraic geometry, helping to explain certain patterns which appearwhen studying some important objects, including what are called "binaryquantics" and "K3 surfaces". The investigator has also noticed somemore patterns, currently unexplained, which may connect complexreflections to a famous group called the "monster" simple group. Thespecific technical problems to be addressed by the project all attemptto advance our understanding of groups generated by complex reflections,acting on complex hyperbolic space.