# Thom form in equivariant Cech-de Rham theory

## TL;DR

This paper develops an elementary $G$-equivariant Cech-de Rham theory using the Cartan model, providing explicit formulas for Thom forms and discussing an equivariant Riemann-Roch formula, advancing the understanding of equivariant differential geometry.

## Contribution

It introduces a new elementary approach to $G$-equivariant Cech-de Rham theory and derives explicit formulas for Thom forms without relying on the Mathai-Quillen framework.

## Key findings

- Explicit formula for $U(l)$-equivariant Thom form of $C^l$
- Elementary construction of $G$-equivariant Cech-de Rham theory
- Discussion of an equivariant Riemann-Roch formula

## Abstract

In the present paper, we provide the foundation of a $G$-equivariant Cech-de Rham theory for a compact Lie group $G$ by using the Cartan model of equivariant differential forms. Our approach is quite elementary without referring to the Mathai-Quillen framework. In particular, by a direct computation, we give an explicit formula of the $U(l)$-equivariant Thom form of C^l, which deforms the classical Bochnor-Martinelli kernel. Also we discuss a version of equivariant Riemann-Roch formula.