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Metric measure limits of spheres and complex projective spaces

We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf quotient of) a virtual infinite-dimensional Gaussian space, depending on the size of the spaces. These are the first discovered examples with the property that the limits are drastically different from the spaces in the sequence. For the proof, we introduce a metric on Gromov's compactification of the space of metric measure spaces.