Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings

De-Qi Zhang; Hsueh-Yung Lin; Keiji Oguiso

arxiv: 2104.03423 · v4 · pith:4ORQ3F3Nnew · submitted 2021-04-07 · 🧮 math.AG · math.DS· math.RA

Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings

Hsueh-Yung Lin , Keiji Oguiso , De-Qi Zhang This is my paper

classification 🧮 math.AG math.DSmath.RA

keywords boundsgkdimplovcoordinategrowthhomogeneouslog-volumepolynomial

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Let f be a zero entropy automorphism of a compact K\"ahler manifold X. We study the polynomial log-volume growth Plov(f) of f in light of the dynamical filtrations introduced in our previous work with T.-C. Dinh. We obtain new upper bounds and lower bounds of Plov(f). As a corollary, we completely determine Plov(f) when dim X = 3, extending a result of Artin--Van den Bergh for surfaces. When X is projective, Plov(f) + 1 coincides with the Gelfand--Kirillov dimensions GKdim(X,f) of the twisted homogeneous coordinate rings associated to (X,f). Reformulating these results for GKdim(X,f), we improve Keeler's bounds of GKdim(X,f) and provide effective upper bounds of GKdim(X,f) which only depend on dim X.

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An upper bound for polynomial volume growth of automorphisms of zero entropy
math.AG 2024-08 unverdicted novelty 7.0

Proves plov(f) ≤ (k/2 + 1)d with k ≤ 2(d-1) hence plov(f) ≤ d² for zero-entropy automorphisms with unipotent f* on N¹(X)_R of dim-d normal projective varieties over algebraically closed fields.