Cory doesn't really have any good nicknames.
Famous people named Cory:
Actor/singer Cory Monteith; actors Cory Hardrict and Cory Smith; rapper Cory McKay (a.k.a. Cormega); novelist Cory Doctorow; U.S. Senator Cory Booker; baseball player Cory Wade; YouTuber Cory Cotton.
If you've seen the television series "Boy Meets World" or its spinoff, "Girl Meets World," you probably associate this name with one of their main characters, Cory Matthews.