Abstract: 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, every irreducible component of the supersingular locus is isomorphic to a Deligne-Lusztig variety, and the intersection combinatorics are governed by a Bruhat-Tits building. 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. I'll also present some topics generally relevant to the study of supersingular loci, in hopes of finding common interest with other members of the department.
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