# The space of metrics on an algebraic variety

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The space of (Kahler) metrics on a projective algebraic variety can be given a natural infinite dimensional Riemannian structure. This leads to the notion of a geodesic in the space of metrics. I will discuss a recent result on the optimal regularity of these geodesics and how this relates to nonlinear PDEs and canonical metrics. This is joint work with J. Chu and V. Tosatti.