# Fano hypersurfaces and differential forms via positive characteristic

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Fano hypersurfaces and differential forms via positive characteristic

**Abstract: **Holomorphic forms are an important birational invariant for studying the geometry of a variety. In characteristic 0, Fano varieties do not have any holomorphic forms. Surprisingly, Koll\'{a}r showed that in positive characteristic certain (singular) Fano varieties admit many global (n-1)-forms, and he combined this with a specialization method to prove nonrationality of many complex Fano hypersurfaces. In this talk, we will revisit this construction and use it to address several related questions for Fano hypersurfaces in certain ranges: (1) how can one further measure their nonrationality, (2) what are their possible rational endomorphisms, and (3) is their birational automorphism group infinite or finite? Parts of this will be joint with David Stapleton as well as with Lena Ji-Stapleton.