If a murder were committed in Antarctica and it was unclear who did it, what country/agency would investigate? And, if the guilty party were identified, who (if anyone) would have the authority to arrest them?
Various countries have jurisdiction in Antarctica, see https://en.m.wikipedia.org/wiki/Antarctic_Treaty_System#Legal_system.
The geographic limits of each country's jurisdiction are well defined. The USA has jurisdiction if no one else does.
In the most cases, the country of the citizenship of the victim would do the investigation. If the crime was committed on an Antarctic base running within a specific jurisdiction, such jurisdiction would be also given and investigation would be probably done by that country as well.