Arithmetic Groups in Algebraic Geometry

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The goal of the project is to study the geometry and topology of certain special moduli spaces arising in complex algebraic geometry. Here, "special" means that they may be described as quotients of Hermitian symmetric spaces by discrete groups. (There are surprisingly many examples of these spaces.) The first specific goal is to understand the topology of a large family of these spaces that includes the moduli spaces of polarized smooth K3 surfaces and several other examples arising in singularity theory. By "understand the topology of" we mean that we beleive these spaces have contractible universal covers, and we want to prove this. The idea is to use tools from the theory of negatively-curved metric spaces that generalize the idea of a negatively-curved Riemannian manifold. The second specific goal is to complete a joint project with J. Carlson and D. Toledo, to prove that the moduli space of cubic hypersurfaces in 4-dimensional projective space is isomorphic to a quotient of the complex 10-ball by a certain discrete group. The third specific goal is to close the last open aspect of Hilbert's 14th problem, by showing that there is a representation of the 2-dimensional additive group for which the ring of invariants is not finitely generated. The project addresses concrete problems in algebraic geometry concerning the classification of various objects like curves and surfaces. Algebraic geometry as a subject deals with curves and surfaces that can be defined by means of equations, and also with higher-dimensional versions of these curves and surfaces. An example giving the flavor of the classification problem is that a circle is essentially the same as an ellipse, because one of them can be got from the other by stretching the plane in one direction. An ellipse is also essentially the same as a parabola, because you can imagine keeping one end of the ellipse fixed and pushing the other off to infinity. You can go even further and push the end off past infinity, so that it reappears on the other side of the plane, and you see a hyperbola. An algebraic geometer expresses these ideas by saying that any two conics are "projectively equivalent". What these shapes all have in common is that they are defined by simple equations--two variables, and only terms of degree two or less. A major part of modern algebraic geometry is studying similar questions but with the number of variables increased, or the complexity of the equations increased to allow higher-degree polynomials. For example, it turns out that not all curves in the plane that are defined by degree three equations are projectively equivalent. The ways in which these curves can differ from each other is very important for many fields of mathematics and even physics. This example was understood in the 19th century, but similar problems remain open. One purpose of the current proposal is to gain an understanding of how the projective equivalence classes of the degree-three "surfaces" in four-dimensional space can vary. We believe that this may be described in a beautiful and surprising way using the unit sphere in 10-dimensional complex space. This result would be similar to a classical result for degree three curves in the plane. We want to prove that our intuition is right. Another problem to address is that of finishing a part of Hilbert's 14th problem, one of the celebrated problems posed by David Hilbert at the beginning of the 20th century.