Antonyms for "search high heaven" could include terms such as "ignore," "overlook," or "neglect." If someone does not even attempt to search high heaven for something, they may simply be disregarding it altogether. Similarly, if they are aware of an issue or item that needs to be found but choose to ignore it, they can be said to have neglected it. Other antonyms could include "disregard," "miss," or "forget," indicating a lack of attention or effort. While searching high heaven implies a thorough, exhaustive search, these contrasting terms suggest an opposite or lack of action.