Let f by an automorphism of zero entropy of a smooth projective variety X. The polynomial log-volume growth plov(f) of f is a natural analog of Gromov's log-volume growth of automorphisms (of positive entropy), formally introduced by Cantat and Paris-Romaskevich for slow dynamics in 2020. A surprising fact noticed by Lin, Oguiso, and Zhang in 2021 is that this dynamical invariant plov(f) essentially coincides with the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with (X, f), introduced by Artin, Tate, and Van den Bergh in the 1990s. It was conjectured by them that plov(f) is bounded above by d², where d = dim X.

 

We prove an upper bound for plov(f) in terms of the dimension d of X and another fundamental invariant k of (X, f) (i.e., the degree growth rate of iterates fⁿ with respect to an arbitrary ample divisor on X). As a corollary, we prove the above conjecture based on an earlier work of Dinh, Lin, Oguiso, and Zhang. This is joint work with Chen Jiang.

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