In this talk, I will talk about the work of mine and joint work with Xi Chen and Dingxing Zhang on automorphism and cohomology of Fano varieties of lines of cubics and complete intersections. More precisely, we prove the automorphism groups act faithfully on the co- homology groups. As a result, we classify the automorphism groups of cubic fourfolds of small Hodge groups. The poof involves equivariant Kodaira Spencer map, equivariant deformations, Hyperkaehler structure on Fano variety of lines, Bloch’s semi-regularity in mixed characterisitic, vanishing cycles and lattice theory. At the end, I will talk about the work of infinitesimal Torelli theorem in positive characterisitic.

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