How should a polygynous husband resolve quarrels between his wives?

The husband must be fair, calm and impartial, and deal with the root of the conflict — usually a sense of unfairness in the rotation, maintenance or attention.
The Messenger of Allah ﷺ himself dealt gently with jealousy among his wives; when one wife broke a dish of food sent by her co-wife, he gathered up the food, said "your mother is jealous", and replaced the dish (narrated by al-Bukhari). Practical steps: keep the rotation precisely, do not speak ill of one wife to another, do not compare them, and listen to each wife separately. Separate homes reduce friction. If disputes continue, seek counselling. See Relations between co-wives and Counselling.