  The purpose of this paper is to prove a Fourier restriction estimate for
certain 2-dimensional surfaces in $\bbR^{2d}$, $d\ge 3$. These surfaces are
defined by a complex curve $\gamma(z)$ of simple type, which is given by a
mapping of the form % \[ z\mapsto \gamma (z) = \big(z, \, z^2,..., \, z^{d-1},
\, \phi(z) \big) \] % where $\phi(z)$ is an analytic function on a domain
$\Omega \subset \bbC$. This is regarded as a real mapping $z=(x,y) \mapsto
\gamma(x,y)$ from $\Omega \subset \bbR^2$ to $\bbR^{2d}$.
  Our results cover the case $\phi(z) = z^N$ for any nonnegative integer $N$,
in all dimensions $d\ge 3$. Furthermore, when $d=3$, we have a uniform
estimate, where $\phi(z)$ may be taken to be an arbitrary polynomial of degree
at most $N$. These results are analogues of the uniform restricted strong type
estimate in \cite{BOS3}, valid for polynomial curves of simple type and some
other classes of curves in $\bbR^d$, $d\ge 3$.
