In order to hover directly above the location he took off, the pilot would have to gain speed as he ascended in the precises direction of the earth’s rotation, maintain that precise vector while hovering, then lose that speed slowly on the way down.

If a spinning earth was for some unknown reason able to drag a 10,000 pound helicopter along with it as it ascended, then this would reduce the rotation speed of the earth by a very very small amount due to the energy required to accelerate the helicopter to the velocity needed while overcoming the inertia of the aircraft and also overcoming drag.