L2 de Rham and Dolbeault cohomology groups on singular complex analytic spaces

Georgetown University, Washington DC

Investigators

Abstract

The principal investigator will examine problems in the area of several complex variables that involve differential equations on singular spaces. The spaces of interest in this proposal are algebraic varieties (zero sets of polynomials) and complex analytic sets (zero sets of holomorphic functions). In particular, the principal investigator will address several questions concerning local and global L2-existence and regularity of solutions to the de Rham and inhomogeneous Cauchy-Riemann equations on the smooth part of compact complex spaces with arbitrary singularities. Results of this type are useful to algebraic geometers working on the interplay between L2-cohomology and intersection cohomology. Many physical phenomena are modeled by systems of differential equations and one would like to understand the space of solutions to these equations. In the early 60's Atiyah and Singer showed that the number of solutions to certain equations on some spaces can be completely determined by the shape of the space on which these equations are defined. This result led to the advancement of quantum field theory and the development of string theory in physics. Singularities are ubiquitous in nature from the atomic scale (crystals) to the cosmic scale of the universe (black holes). The proposed research hopes to contribute in the understanding of how the geometry of singular spaces influences the existence and behavior of solutions to certain natural equations on these spaces.