Chern numbers for singular varieties and elliptic homology

By B. Totaro. 2000

95 citations recorded by the source corpus.

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 Hohn [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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c00d53525662a60c3ce69f7a4ca933f422ee624

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Chern classes of automorphic vector bundles (2002, background): “6.5. Proof. Let t ⊂ h be the Lie algebra of a Cartan subgroup T ⊂ H and let W denote its Weyl group.”
A survey of characteristic classes of singular spaces (2005, background): “First of all ma ny authors (like [BL1, BL2, To , Wang]) use −y instead of y so that their elliptic genus is related to the χ−y-genus.”
Elliptic genera, real algebraic varieties and quasi-Jacobi forms (2009, background): “[47]) The kernel of the homomorphism Ell W ̋U ̋Q ! F taking an almost complex manifold X to its elliptic genusEll.”
Characteristic classes of mixed Hodge modules (2009, background): “[51] B. Totaro, Chern numbers for singular varieties and elliptic homology, Ann. of Math.”
Small resolutions of minuscule Schubert varieties (2006, methodology): “re not as well ﬁtted as IH-small resolutions for the computation of Kazhdan-Lusztig polynomials. We also describe all relative canonical models of X(w). Furthermore, the following result of B. Totaro [To] using a key result of J. Wisniewski [Wi] tells us to look for IH-small resolutions in the class”
HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES (2005, methodology): “It was already used by Totaro [77] in his study of the relation between Chern numbers for singular complex varieties and elliptic homology.”
Hodge genera of algebraic varieties I (2006, background): “Then, by the stability by intermediate direct images, the complex ICV̄ (L) is a 6Finding numerical invariants of complex varieties, more precisely Chern numbers that are invariant under small resolutions, was Totaro’s guiding principle in his paper [28].”
McKay correspondence for elliptic genera (2002, background): “Alternatively, the elliptic genus can be written as Ell(X; z, τ) = ∫ X ch(ELLz,τ )td(X)(3) where ELLz,τ := y− dimX 2 ⊗n≥1 ( Λ−yqn−1 T ∗X ⊗Λ−y−1 qnTX ⊗SqnT ∗X ⊗SqnTX ) .”