Russia has long been active in the Arctic and wants to get power over this land there. But I think the decision will last for several years and there will be no war. Well, at least because it's too cold and not very comfortable to fight. In addition, attention is now exacerbated by other problems.
In the land and natural resources are interested in many countries. When the number of applicants always have disputes. The world situation is so tense that it may well start a war. As for the cold it is hard for everyone, everyone will be in a difficult position. So chances will be equal.
I think that there is a ban on extracting natural resources from the Antarctic. We should have a similar ban or moratorium in the Arctic (perhaps supported by the UN), to preserve the delicate ecosystem there.
Indeed, extracting any natural resources from the Antarctic is forbidden, but I dont really think that any of the richest countries, and from those who are military strong will care about that ban.
It will be some kind of a race, race for domination on the Earth. If any country would be able to overcome the others, and have possibility to constantly exploit the Antarctic, it would have gotten such a advantage among the rest, that It would be even harder to fight against such a monstrum.
All this is science-ficition obviously, because nothing like that is going to happen in our times, I would predict that it may occur in next 50 years, when ice will molt.