An algebraic $C_2$-equivariant B\'{e}zout's theorem

TL;DR

This paper generalizes Be9zout's theorem to the $C_2$-equivariant setting, calculating the Euler class of line bundles over complex projective space using equivariant cohomology theories.

Contribution

It introduces an algebraic $C_2$-equivariant version of Be9zout's theorem, extending classical results to the equivariant context with explicit cohomological calculations.

Findings

Computed the $C_2$-equivariant Euler class in terms of bundle rank and degrees.

Compared results using ordinary $C_2$-cohomology and Borel cohomology.

Extended the grading to define the Euler class in the equivariant setting.

Abstract

B\'ezout's theorem, nonequivariantly, can be interpreted as a calculation of the Euler class of a sum of line bundles over complex projective space, expressing it in terms of the rank of the bundle and its degree. We give here a generalization to the $C_2$-equivariant context, using the calculation of the cohomology of a $C_2$-complex projective space from an earlier paper. We use ordinary $C_2$-cohomology with Burnside ring coefficients and an extended grading necessary to define the Euler class, which we express in terms of the equivariant rank of the bundle and the degrees of the bundle and its fixed subbundles. We do similar calculations using constant $\mathbb{Z}$ coefficients and Borel cohomology and compare the results.