  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.
