Work

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

Year: 2024

Type: article

Abstract: Let f be a zero entropy automorphism of a compact Kähler manifold X . We study the polynomial log-volume growth \mathrm{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 \mathrm{Plov}(f) . As a corollary, we completely determine \mathrm{Plov}(f) when \dim X = 3 , extending a result of Artin–Van den Bergh for surfaces. When X is projective, \mathrm{Plov}(f) + 1 coincides with the Gelfand–Kirillov dimensions \mathrm{GKdim}(X,f) of the twisted homogeneous coordinate rings associated to (X,f) . Reformulating these results for \mathrm{GKdim}(X,f) , we improve Keeler’s bounds of \mathrm{GKdim}(X,f) and provide effective upper bounds of \mathrm{GKdim}(X,f) which only depend on \dim X .

Source: Journal of Noncommutative Geometry

DOI: 10.4171/jncg/595

Authors: Hsueh-Yung Lin, Keiji Oguiso, De-Qi Zhang

Institutions: National Taiwan University, National Center for Theoretical Sciences, The University of Tokyo, National University of Singapore

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science and Technology Council, Japan Society for the Promotion of Science

Awards: 代数多様体の自己写像に関する多角的研究, G80590013