Abstract
We study zero entropy automorphisms of a compact Kähler manifold X. Our goal is to bring to light some new structures of the action on the cohomology of X, in terms of the so-called dynamical filtrations on \(H^{1,1}(X,{{\mathbb {R}}})\). Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations \((g^m)^* \circlearrowleft H^2(X,{{\mathbb {C}}})\) where g is a zero entropy automorphism, in terms of \(\dim X\) only. We also give an upper bound for the (essential) derived length \(\ell _{\mathrm{ess}}(G, X)\) for every zero entropy subgroup G, again in terms of the dimension of X only. We propose a conjectural upper bound for the essential nilpotency class \(c_\mathrm{ess}(G,X)\) of a zero entropy subgroup G. Finally, we construct examples showing that our upper bound of the polynomial growth (as well as the conjectural upper bound of \(c_\mathrm{ess}(G,X)\)) are optimal.
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Notes
This is a generalization of the classical primitive subspace \(\mathrm{Ker}(c^{n-1} \mathbin {\smile }\bullet ) \subset H^{1,1}(X,{{\mathbb {R}}})\) for some Kähler class c.
\(\alpha \) can be regarded as a square root of a class which is bounded below by \(-M_j\beta \) in some sense.
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Acknowledgements
The authors are supported by NUS and MOE grants R-146-000-248-114 and MOE-T2EP20120-0010; Taiwan Ministry of Education Yushan Young Scholar Fellowship (NTU-110VV006), and Taiwan Ministry of Science and Technology (110-2628-M-002-006-); a JSPS Grant-in-Aid (A) 20H00111 and an NCTS Scholar Program; and an ARF of NUS, respectively. We would like to thank Fei Hu and Jun-Muk Hwang for helpful discussions; Serge Cantat for various comments and for bringing the references [CX18, ET79] to our attention; the referees for very detailed and constructive suggestions; and KIAS, National Center for Theoretical Sciences (NCTS) in Taipei, NUS and the University of Tokyo for the hospitality and support during the preparation of this paper.
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Dinh, TC., Lin, HY., Oguiso, K. et al. Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations. Geom. Funct. Anal. 32, 568–594 (2022). https://doi.org/10.1007/s00039-022-00599-3
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DOI: https://doi.org/10.1007/s00039-022-00599-3
Keywords and phrases
- Automorphisms of compact Kähler manifolds
- Zero entropy automorphisms
- Dynamical filtrations
- Polynomial growth
- Derived length
- Nilpotency class