Supersingular Loci of Unitary Shimura Varieties

The supersingular locus of a unitary Shimura variety parametrizes supersingular abelian varieties with an action of a quadratic imaginary field, meeting a certain signature condition. In low-dimensional cases these supersingular loci have a particular kind of structure: for example, every irreducible component is isomorphic to a Deligne-Lusztig variety. We’ll consider these low-dimensional cases for motivation, and then see how the structure of the supersingular locus can be very different in general.

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