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Using the large data sample accumulated by the Belle experiment at KEKBasymmetry energy $e^+e^-$ Collider, we report the preliminary measurements ofthe branching fractions of the radiative decays $D^0 \rightarrow V\gamma$,where $V$ = $\phi,\bar{K^*}^0, \rho^0$. This is the first observation of thedecay $D^0 \rightarrow \rho^0\gamma$ and also present the preliminary resultsof the first measurement of $CP$ asymmetry in these decays. We present thepreliminary result of the $CP$ asymmetry of the $D^0 \rightarrow K^0_S K^0_S$decay, which is consistent with no $CP$ violation and improves the uncertaintywith respect to the previous measurement of this quantity by more than a factorof three. Finally, we present the results of the rare charm decay $D^0\rightarrow \gamma\gamma$, resulting in the most restrictive upper limit todate.

Search for $CP$ violation in $D$ decays to radiative and hadronic decays and search for rare $D$ decay at Belle