Publicación

Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

Anderson, Dave · Gonzales, Richard · Payne, Sam · Vezzosi, Gabriele
2021 Algebra and Number Theory DOI: 10.2140/ant.2021.15.341 15 vistas

Resumen

We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X whose equivariant K-theory of vector bundles does not surject onto its ordinary K-theory, and describe the operational K-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational K-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic K-theory of relatively perfect complexes to bivariant operational K-theory.

Autores y colaboradores

Authors

Anderson, Dave
Payne, Sam
Vezzosi, Gabriele

Palabras clave

Mathematics Equivariant map Vector bundle Pure mathematics Riemann hypothesis

Proyectos

Research Projects

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Item type:Research Project,
Group embeddings and Cox rings
P_GE-COX
The purpose of this project is to investigate the cone of projective group embeddings with the same Cox ring R(X). The ring R(X) is a unique factorization domain. Q-factorial projective varieties with a finitely generated Cox ring are characterized using Mori theory. If X is a spherical variety, then R(X) is a finitely generated algebra, and it carries a natural grading by the classes of effective divisors. The Cox ring of spherical varieties has been described by Brion and Gagliardi. In the case when X is a G-embedding, we can associate a certain cone H, in the class group Cl(X), so that any element y of H defines a group embedding X_y satisfying R(X_y) = R(X). It seems that H corresponds to the movable cone of X. The purpose of this project is to calculate H explicitely, using the knowledge of NE(X), the cone of effective curves on X, and nef(X), the cone of numerically effective divisors on X, and certain monoid data. Moreover, we would like to understand, and quantify, how the cones H and H', associated to different G-embeddings, are related.
P_GE-COX
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