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| Chern–Schwartz–MacPherson classes | |
|---|---|
| Name | Chern–Schwartz–MacPherson classes |
| Field | Algebraic geometry |
| Introduced | 1950s–1970s |
| Discoverer | Marie-Hélène Schwartz; Robert MacPherson; Mark Goresky |
| Related | Chern class, Todd class, Segre class, Hirzebruch–Riemann–Roch theorem |
Chern–Schwartz–MacPherson classes are characteristic classes assigning homology classes to singular algebraic varieties and complex analytic spaces, extending the notion of total Chern class from smooth manifolds to singular settings. They provide a functorial theory compatible with pushforwards and Euler characteristic, connecting work of Marie-Hélène Schwartz, Robert MacPherson, and later contributors like Luc Illusie and M.-H. Schwartz. These classes link intersection-theoretic invariants used in the Hirzebruch–Riemann–Roch theorem and enumerative geometry to singularity theory studied by researchers at institutions such as IHÉS and CNRS.
The classes were motivated by attempts to generalize the classical Chern class from smooth projective varieties studied by mathematicians at École Normale Supérieure and Université Paris-Sud to singular spaces encountered in the work of Jean-Pierre Serre, Alexander Grothendieck, and practitioners of scheme theory. Parallel advances in topology by René Thom and John Milnor and in stratified Morse theory by Mark Goresky and Robert MacPherson influenced their formulation. The resulting invariant reconciles perspectives from algebraic geometry, differential topology, and singularity theory, and has been applied in contexts including the Grothendieck–Riemann–Roch theorem and calculations related to the Todd class.
One approach constructs these classes via a natural transformation from the functor of constructible functions on a variety to homology theories, following the program initiated by Robert MacPherson and formalized in the language of sheaf theory and derived category methods linked to work by Alexander Grothendieck. For complex algebraic varieties, the transformation sends the indicator function of a subvariety to a homology class whose degree recovers the topological Euler characteristic studied by Henri Poincaré and Élie Cartan. An alternative analytic construction uses local Euler obstructions introduced by MacPherson and further developed by Marie-Hélène Schwartz and Tadao Oshima, relating to characteristic cycles in the cotangent bundle as in studies by Masaki Kashiwara and Lê Dũng Tráng.
Chern–Schwartz–MacPherson classes satisfy functoriality under proper morphisms, echoing the pushforward properties examined by Grothendieck in his formulation of K-theory and the Grothendieck group. They are uniquely characterized by their normalization on smooth, compact, complex manifolds to the Poincaré dual of the total Chern class and by additivity on decompositions into constructible subsets, a property reminiscent of additivity in Euler characteristic theory employed by Poincaré and Lefschetz. Compatibility with stratified maps studied by Goresky and MacPherson ensures behavior under resolutions of singularities such as those constructed by Heisuke Hironaka.
These classes relate to the Todd class appearing in the Hirzebruch–Riemann–Roch theorem and to Fulton’s Chern class and Segre classes studied by William Fulton. Connections to the Milnor number from singularity theory and to L-classes investigated by Hirzebruch reflect interactions with signature theorems of Atiyah–Singer type, with further ties to the theory of perverse sheaves advanced by Joseph Bernstein and Marie-France Vignéras. Comparisons with Stiefel–Whitney class and Pontryagin class in differential topology show analogies and contrasts when extending characteristic classes across categories developed by Serre and Grothendieck.
Explicit computations exist for Schubert varieties within flag manifolds central to work of Hermann Schubert and modern Schubert calculus by William Fulton and Andrei Zelevinsky, using techniques from equivariant cohomology studied by Michael Atiyah and Raoul Bott. For hypersurfaces, formulas express the classes in terms of the Jacobian ideal and Chern classes of ambient smooth varieties as in treatments by Aluffi and Fulton–MacPherson. Calculations for toric varieties connect to combinatorial data investigated by David Cox, John Little, and Henry Schenck, while examples involving isolated singularities invoke invariants developed by John Milnor and Vladimir Arnold.
Applications span enumerative geometry problems historically pursued at Institut des Hautes Études Scientifiques and computational projects at Mathematical Sciences Research Institute; they include counting singular points in families of hypersurfaces, refining characteristic numbers in moduli problems studied by Deligne and Mumford, and informing string-theoretic compactifications considered by researchers at CERN and Caltech. The classes provide tools for formulating and proving generalizations of Riemann–Roch type theorems and for comparing invariants obtained from mixed Hodge theory developed by Pierre Deligne and Wilfried Schmid.
The concept evolved from independent contributions: Marie-Hélène Schwartz developed local Euler obstruction techniques in the 1950s, while Robert MacPherson provided a global functorial construction in the 1970s, influenced by collaborations with Mark Goresky on intersection homology. Subsequent refinements and algebraic formulations were advanced by William Fulton, Paolo Aluffi, Luc Illusie, and others working within frameworks established by Alexander Grothendieck and Jean-Pierre Serre. Ongoing work by researchers at institutions like IHÉS, CNRS, Princeton University, and University of California, Berkeley continues to expand computational methods and applications in modern algebraic geometry and topology.