# Algebra Seminar

Title: Studying Fano hypersurface with holomorphic forms

Abstract: In characteristic 0 Fano varieties never admit holomorphic forms. In characteristic p, Kollár showed that Fano varieties that are p-cyclic covers can admit differential forms. This is a powerful tool for studying these varieties. By specializing to characteristic p one can use the positivity of these forms to show that the birational geometry complex Fano hypersurfaces can have quite different behavior from the birational geometry of projective space. For example, Kollár used these forms to prove that there are Fano hypersurfaces which are not rational (or even ruled), and Totaro showed that these Fano hypersurfaces are not even stably rational. In this talk we give new applications of these degeneration techniques. We explain that Fano hypersurfaces can have arbitrarily large degrees of irrationality and we show that the degrees of rational endomorphisms of Fano hypersurfaces must satisfy certain congruence constraints modulo p. This is joint work with Nathan Chen.