Talk page
Title:
Higher index Fano varieties in positive characteristic with Bir(X) = 1
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Abstract:
For a smooth complex hypersurface X of index 1 in projective space, the Noether-Fano method was used by many authors (Fano-Segre-Iskovskikh-Manin-Pukhlikov-Cort-Cheltsov-De Fernex-Ein-Mustaţă-Zhuang) to prove birational super-rigidity; in particular, the group Bir(X) of birational automorphisms is finite. Pukhlikov proved a similar result for most index 2 hypersurfaces. To our knowledge, there are no known results in index 3 and higher. In this talk, we will revisit a construction of Kollár which produces forms on certain p-cyclic covers in characteristic p, and use it to give examples of Fano varieties of arbitrarily large index with no nontrivial birational automorphisms. The tradeoff is that we must pass to positive characteristic and allow mild singularities. Our main observation is that there is a natural Bir(X)-equivariant ample line bundle on these varieties.
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