Topology of Singular Algebraic Varieties

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AbstractAward: DMS-0705050Principal Investigator: Anatoly S. LibgoberThe project deals with the study of topological invariants forsingular algebraic varieties with many ideas motivated byphysics. One of its goals is to develop new invariants forsingular spaces extending the theory of elliptic genus ofvarieties with mild singularities and orbifolds which waspresented in joint works with Lev Borisov earlier. This willyields an extension of the previous contributions by many authorson vertex operator algebras associated to manifolds or orbifoldsand application of these invariants to geometric problems. PIplans to develop further generalizations of elliptic genusdetecting the torsion in the cobordisms of special unitarycomplex manifolds as well as use this invariant in classificationand surgery problems of singular spaces. The principalinvestigator will investigate the possibility that the ellipticgenus plays a role in singularity theory beyond classicalapplications in Landau-Ginzburg models associated with weightedhomogeneous polynomials. In particular it is planned to considerthe extensions of elliptic genus of Landau-Ginzburg models inconnection with the Arnold-Steenbrink spectrum of isolatedsingularities. In another direction, PI plans to study thehomotopy groups of non-simply connected quasiprojectivevarieties. In particular the study of characteristic varietiesassociated with homotopy groups controlling among other thingsthe cohomology of local systems will be continued. A.Libgoberwill focus on properties of the Hodge decomposition of the latterand will study the fundamental problem of realization of theseinvariants, i.e. the existence of quasiprojectve varieties withgiven characteristic varieties. As part of this work, PI willinvestigate the restrictions on the dimensions of characteristicvarieties in the case of plane reducible curves generalizingresults in the case of arrangements of lines in complexprojective plane.In general terms the proposal aims to develop new approaches tothe study of geometry of spaces appearing in a variety ofproblems in mathematics and such aspects of theoretical physicsas string theory. A new type of data, called elliptic genus,reflecting geometry and topology of these spaces which appearednaturally in physics will be investigated from a mathematicalperspective and applied to a wide range of problems in geometry,topology and singularity theory. It is planned to study furthergeneralizations of elliptic genus which should have applicationsto fundamental problems of physics. The realization of presenceof infinite dimensional aspects in geometric issues, on which ourwork depends in an essential way, is a novel feature whichemerged in the last 20 years in mathematics and physics. It isplanned to widely disseminate such new viewpoints gainingstrength in contemporary mathematics and physics.