Rational circle-equivariant elliptic cohomology of CP(V)

TL;DR

This paper computes the rational circle-equivariant elliptic cohomology of complex projective spaces associated with T-representations, reducing complex calculations to known sphere cohomology results through a splitting property.

Contribution

It proves that T-equivariant elliptic cohomology theories are 1xT-split, enabling simplified computations of elliptic cohomology for CP(V) from sphere cases.

Findings

Established 1xT-splitting of elliptic cohomology theories

Reduced computation of EC_T(CP(V)) to T^2-elliptic cohomology of spheres

Provided explicit methods for calculating T-equivariant elliptic cohomology

Abstract

We compute rational $T$-equivariant elliptic cohomology of CP(V), where $T$ is the circle group, and CP(V) is the $T$-space of complex lines for a finite dimensional complex $T$-representation V. Starting from an elliptic curve C over the complex numbers and a coordinate data around the identity, we achieve this computation by proving that the $T$-equivariant elliptic cohomology theory $EC_T$ built in [Gre05], and the $T^2$-equivariant elliptic cohomology theory $EC_{T^2}$ built in [Bar22] are 1x$T$-split. This result allows us to reduce the computation of $EC_T(CP(V))$ to the computation of $T^2$-elliptic cohomology of spheres of complex representations, already performed in [Bar22].