In topology, there are many tools for distinguishing smooth manifolds. The most common are the Euler characteristic and the signature. In complex geometry, many invariants can be expressed in terms of Chern classes. This naturally leads to the following question: which invariants are applicable to singular spaces, such as algebraic varieties, i.e. loci defined by polynomial equations?
Many such varieties arise naturally as orbit closures of algebraic group actions. The best-known examples are Schubert varieties in Grassmannians. Other examples include nilpotent cones, i.e. sets of nilpotent matrices with prescribed properties, such as the set of matrices satisfying X^2 = 0.
The usual procedure for defining invariants of such objects is to find a resolution of singularities, i.e. a smooth variety equipped with a map to the singular one that is an isomorphism away from the singular locus. One may then hope that a given invariant of the resolution captures information about the underlying singular space. However, this strategy works only for a very restricted class of invariants. With suitable modifications, it is possible to define at most a four-parameter family of invariants, namely the Borisov–Libgober elliptic genus. Nevertheless, this invariant is rich enough to give rise to interesting algebraic structures when applied to Schubert varieties or nilpotent orbits.
