BRST and the ghost number supercurrents in the two-dimensional $N=2$ supergravity. We follow the path integral prescription for anomaly calculation \[\]. In the case of $N=1$ supergravity, the superfield path integral is known to be an appropriate tool to compute those anomalies \[,\]. For $N=2$ case, however, it is well-known that the action of the matter multiplet (A.4) cannot be written directly in terms of the real scalar superfield (A.8) \[\]. Therefore it is not clear how the superfield path integral is useful in the present context of the anomaly calculation.[^3][For the ghost and anti-ghost multiplet, the action can be written directly by the superfield as in (A.12). Thus the superfield path integral may work.]{}

Here we consider the path integral in the component fields. The classical gauge symmetries of the $N=2$ supergravity are the general coordinate, the local Lorentz, the $N=2$ supersymmetry, the Weyl, the super-Weyl, and the chiral transformations \[\]. We assume under our regularization, the Weyl, the super-Weyl and the chiral transformations are anomalous at quantum level. Using the non-anomalous transformations, we can fix the $N=2$ supergravity multiplet in the superconformal gauge as $$\eqalign{
  &e_\mu^a=e^{\sigma/2}\delta^a_\mu,
\cr
  &\chi^\pm_\mu=\gamma_\mu\phi^\pm,
\cr
  &A_\mu=\epsilon_{\mu\nu}\partial^\nu\phi,
\cr
}
\ee$$ where $\sigma$ is the Liouville 