Local supertwistors and $N=2$ conformal supergravity

$N=2$ super symmetric extension of the local twistor theory is formulated. A supertwistor superconnection determined by the superconformal structure of the base superspace is introduced on the bundle of $N=2$ local supertwistors. It is proved that the Yang–Mills equations for this superconnection coincide exactly with the Bach equations describing the dynamics of $N=2$ conformal supergravity.