Topology of Singular Algebraic Varieties

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Abstract Award: DMS-0705050 Principal Investigator: Anatoly S. Libgober The project deals with the study of topological invariants for singular algebraic varieties with many ideas motivated by physics. One of its goals is to develop new invariants for singular spaces extending the theory of elliptic genus of varieties with mild singularities and orbifolds which was presented in joint works with Lev Borisov earlier. This will yields an extension of the previous contributions by many authors on vertex operator algebras associated to manifolds or orbifolds and application of these invariants to geometric problems. PI plans to develop further generalizations of elliptic genus detecting the torsion in the cobordisms of special unitary complex manifolds as well as use this invariant in classification and surgery problems of singular spaces. The principal investigator will investigate the possibility that the elliptic genus plays a role in singularity theory beyond classical applications in Landau-Ginzburg models associated with weighted homogeneous polynomials. In particular it is planned to consider the extensions of elliptic genus of Landau-Ginzburg models in connection with the Arnold-Steenbrink spectrum of isolated singularities. In another direction, PI plans to study the homotopy groups of non-simply connected quasiprojective varieties. In particular the study of characteristic varieties associated with homotopy groups controlling among other things the cohomology of local systems will be continued. A.Libgober will focus on properties of the Hodge decomposition of the latter and will study the fundamental problem of realization of these invariants, i.e. the existence of quasiprojectve varieties with given characteristic varieties. As part of this work, PI will investigate the restrictions on the dimensions of characteristic varieties in the case of plane reducible curves generalizing results in the case of arrangements of lines in complex projective plane. In general terms the proposal aims to develop new approaches to the study of geometry of spaces appearing in a variety of problems in mathematics and such aspects of theoretical physics as string theory. A new type of data, called elliptic genus, reflecting geometry and topology of these spaces which appeared naturally in physics will be investigated from a mathematical perspective and applied to a wide range of problems in geometry, topology and singularity theory. It is planned to study further generalizations of elliptic genus which should have applications to fundamental problems of physics. The realization of presence of infinite dimensional aspects in geometric issues, on which our work depends in an essential way, is a novel feature which emerged in the last 20 years in mathematics and physics. It is planned to widely disseminate such new viewpoints gaining strength in contemporary mathematics and physics.