UBC Theses and Dissertations

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UBC Theses and Dissertations

Equivariant Chow groups and multiplicities Nyenhuis, Michael


We propose a definition of equivariant Chow groups for schemes with a torus action and develop the intersection theory related to it. The equivariant intersection theories that have been considered in the past have been the Chow groups and the K-theory of the quotient scheme, as well as the equivariant K-groups of the original scheme. The equivariant Chow groups are related to all of these. At first glance, we would expect a strong relationship with the equivariant K-groups. As it turns out, the equivariant Chow groups are more closely related to the Chow groups of the quotient scheme. We chose to restrict to tori since for them the equivariant cycles are of a particularly nice form. For general groups the equivariant cycles are harder to describe, and so the intersection theory is far messier, if it even exists. By restricting to tori, we are able to define an equivariant multiplicity that behaves similarly to the degree in the projective case. In particular, we are able to show that for certain schemes, the equivariant multiplicity of an equivariant cycle in the equivariant Chow group is defined and is an invariant of that cycle. While much of this work involves generalizing the work of others, in particular the work of Fulton, Rossmann and Borho, Brylinski and Macpherson, our approach is new. The equivariant Chow groups have not been considered in the past and relating the equivariant multiplicities to the equivariant Chow groups is new as well.

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