This is not specific to 12.1, I've been seeing recently since about 12.03+.
I've had the cursor go missing after various actions. eg: A rename refactoring; Pasting text using the multi-clipboard.
Is anybody else seeing this behavior. Its getting on my nerves as it is constantly forcing me to reach for the mouse. Which BTW requires that I click outside of the IDEA editor (say another app or even just the tool panel) and then click back.