Categorical dimension of birational automorphisms and filtrations of Cremona groups

Abstract

Using a filtration on the Grothendieck ring of triangulated categories, we define the categorical dimension of a birational map between smooth projective varieties. We show that birational automorphisms of bounded categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. In the case of threefolds, we can moreover recover the genus of a birational automorphism, and the filtration defined by Frumkin.

V4: final version, to appear on the Journal of the Mathematical Society of Japan

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• Rationality of inner twisted flags of type $A_n$