I am wondering why turbines are not implemented with controllable pitch blades (on the rotor system)? I assume the reason is that the coupling between the stator and rotor systems posed by typical controllable pitch mechanisms poses an unacceptable frictional loss for such a high speed/ low torque application.
There are turbos with variable geometry, in fact my car is fitted with one.
This is obvious with the actuator visible which, when the ignition is switched on, the actuator is cycled from min to max and back to min.
There are two classic issues that put the engine error light on - one is the actuator fails (usually soldered joints apparently) and two is that the vanes seize due to lots of low speed use clogging the vanes (a good regular "italian service" on the motorway is one solution.
Here is a link to some further info.
To add the info based on the comments, turbos for cars tend to be around 2" to 2.5" max and the turbine blades very thin (rotating around 180000rpm), trying to put a control system to control the pitch of those blades would be a challenge due to size and the effects of play in the mechanisms.
The images below are from the Jaguar x type manual pages 1129 and 1130, showing low and high engine speed operation and the effects of the vane position:
Here is an excellent answer to my question.