Thom form in equivariant Čech–de Rham theory

In the present paper, we provide the foundation of a [Formula: see text]-equivariant Čech–de Rham theory for a compact Lie group [Formula: see text] 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 [Formula: see text]-equivariant Thom form of [Formula: see text], which deforms the classical Bochnor–Martinelli kernel. Also, we discuss a version of equivariant Riemann–Roch formula.