Chern Numbers for Singular Varieties and Elliptic Homology

Abstract

A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so.Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology [11]. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: "complex bordism modulo flops equals elliptic homology."This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn [13] were also an important influence. Thanks to Dave Bayer, Mike Stillman, John Stembridge, Sheldon Katz, and Stein Stromme for their computer algebra programs Macaulay, SF, and Schubert, which helped in guessing the right answer. Contents 1. Statements 2. Weakly complex manifolds 3. The complex elliptic genus 4. Complex bordism modulo flops 5. Complex bordism modulo flops, continued 6. SU-bordism modulo flops 7. Saito's homology classes χ n−k i 8. The Chern numbers c k 1 χ n−k i 9. The twisted χ y genus

Higher Todd Classes and Holomorphic Group Actions

“…Presumably there is some simultaneous generalization that applies to higher elliptic genera. Some useful references that discuss elliptic genera in the context of birational geometry are [47], [9], [10], [11], [13], [45]. (2) Are there examples that show that our vanishing theorem for higher Todd genera are false for finite π, if one does not rationalize?…”

McKay correspondence for elliptic genera

“…Elliptic genus appears to be a key tool for a solution to this problem. In [30] it was shown that the Chern numbers invariant in small resolutions are determined by the elliptic genus of such a resolution. In [7] the elliptic genus was defined for singular varieties with Q-Gorenstein, Kawamata-logterminal singularities and its behavior in resolutions of singularities was studied.…”

“…Since the elliptic genus depends only on the Chern numbers, it is a cobordism invariant. Totaro [30] found a characterization of the elliptic genus (2) of SU-manifolds from the point of view of cobordisms as the universal genus invariant under classical flops.…”

Functional Equations and the Generalised Elliptic Genus

“…The proof in [31], that there is no other genus preserved by flops (or equivalently, satisfying φ(E) = 0 for any fibre bundle E CP (3)…”