| Résume | Abstract: When a reductive group acts on a projective variety, a choice of (linearised) ample line bundle gives a choice of quotient. Wall-crossing (or VGIT) explains how the quotient space changes with the choice of line bundle: the quotients vary birationally, through flips, and only finitely finite birational models can occur.
Each quotient admits a natural choice of Kähler metric, through a symplectic quotient construction, and so one can ask for metrised analogues of these results. I will describe work in progress, which proves Gromov-Hausdorff convergence, and the existence of metric flips, when one suitably varies the choice of line bundle determining the quotient. I will use these results to motivate analogous conjectures governing the metric geometry of moduli spaces in wall-crossing problems.
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