If you hit the boss, it comes after you and the countdown begins. If you just hit the obelisk (it's possibleto snipe it with ice or aim/nox or axe), the countdown begins but boss is not coming after you.

I remember from the test server arena that you must break the obelisk before the one minute to stop the boss from enraging. Although with the current targetability issues that the obelisk has, it's much easier to just focus on taking out the boss fast.