We introduce the notion of maximal geometric quotient, namely a geometric quotient of a polarized pair under a torus action which has a birational contraction to any other GIT quotient. For the action of the maximal torus on a complete flag variety, we show that a maximal geometric quotient exists and it is a Mori dream space, and its Cox ring is isomorphic to the Cox ring of the flag variety. We discuss birational properties of the maximal geometric quotients, their relations with the Chow quotients and explain how to construct them for maximal torus actions on flag varieties.
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Barban et al. (2026) studied this question.
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