University of Georgia
Mathematics Department Colloquium
Klaus Hulek
University of Hannover, Germany and Harvard University
September 30, 1999
Modular forms and the geometry of Siegel modular varieties
Siegel modular varieties parametrize g-dimensional abelian
varieties (possibly with some extra data such as a level structure).
These varieties arise as the quotient of the Siegel upper half space
H_g of genus g by some arithmetic group G which depends on the given
moduli problem. Modular functions give rise to sections in Q-line
bundles on the quotient A(G)=H_g/G. The spaces A(G) are
quasi-projective and can be compactified in various ways, e.g. by the
Satake compactification or by toroidal compactifications such as the
second Voronoi compactification. In order to understand how the Siegel
modular varieties A(G) (respectively, their compactifications) fit into
the classification theory of algebraic varieties one wants to answer
questions such as the following:
1. Construct projective models of Siegel modular varieties for some special
groups G.
2. Determine the Kodaira dimension of the varieties A(G).
3. Determine the nef cone of suitable compactifications of
A(G).
In this talk we want to explain how modular forms can be used to solve
(some of) these questions.