Abstract
We use a concise method to construct pseudo-automorphisms \(f_n\) of the first dynamical degree \(d_1(f_n) > 1\) on the blowups of the projective \(n\)-space for all \(n \ge 2\) and more generally on the blowups of products of projective spaces. These \(f_n\), for \(n=3\) have positive entropy, and for \(n\ge 4\) seem to be the first examples of pseudo-automorphisms with \(d_1(f_n) > 1\) (and of non-product type) on rational varieties of higher dimensions.
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References
Bedford, E., Kim, K.: Periodicities in linear fractional recurrences: degree growth of birational surface maps. Michigan Math. J. 54(3), 647–670 (2006)
Bedford, E., Kim, K.: Pseudo-Automorphisms of 3-space: Periodicities and Positive Entropy in Linear Fractional Recurrences. arXiv: 1101.1614
Dinh, T.-C., Nguyen, V.-A.: Comparison of dynamical degrees for semi-conjugate meromorphic maps. Comment. Math. Helv. 86(4), 817–840 (2011)
Dinh, T.-C., Sibony, N.: Une borne supérieure pour l’entropie topologique d’une application rationnelle. Ann. Math. (2) 161(3), 1637–1644 (2005)
Dolgachev, I.V.: Reflection groups in algebraic geometry. Bull. Am. Math. Soc. (N.S.) 45(1), 1–60 (2008)
Dolgachev, I.V.: Cremona special sets of points in products of projective spaces. In: Complex and Differential Geometry, pp. 115–134. Springer Proceedings of Mathematics, vol. 8. Springer, Heidelberg (2011)
Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions, Astérisque 165 (1988)
Gromov, M.: On the entropy of holomorphic maps. Enseign. Math. (2) 49, 217–235 (2003)
Humphreys, J.E.: Reflection groups and Coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
McMullen, C.T.: Coxeter groups, Salem numbers and the Hilbert metric. Publ. Math. Inst. Hautes Études Sci. 95, 151–183 (2002)
McMullen, C.T.: Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes Études Sci. 105, 49–89 (2007)
Mukai, S.: Conterexample to Hilbert’s fourteenth problem for the 3-dimensional additive group, RIMS. Kyoto University preprint 1343 (2001)
Mukai, S.: Geometric realization of \(T\)-shaped root systems and counterexamples to Hilbert’s fourteenth problem. In: Algebraic Transformation Groups and Algebraic Varieties, pp. 123–129. Encyclopaedia of Mathematical Sciences, vol. 132. Springer, Berlin (2004)
Oguiso, K., Perroni, F.: Automorphisms of rational manifolds of positive entropy with Siegel disks. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22(4), 487–504 (2011)
Oguiso, K., Truong, T.T.: Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive, entropy, arXiv:1306.1590
Yomdin, Y.: Volume growth and entropy. Israel J. Math. 57(3), 285–300 (1987)
Acknowledgments
The present work took place when D.-Q. Zhang was visiting Bayreuth in October 2011 and in the realm of the DFG Forschergruppe 790 Classification of algebraic surfaces and compact complex manifolds and was partly supported by an ARF of NUS. We express our thanks to Professor Catanese for his interest, warm encouragement and hospitality, and Professor Dolgachev for bringing the very interesting paper [6] to our attention. The second author would like to thank Max Planck Institute for Mathematics, Bonn, for the warm hospitality, Professor T. -C. Dinh for clarifying the relation between entropy and dynamical degrees and Professor McMullen for the discussion on the Salem numbers.
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Perroni, F., Zhang, DQ. Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces. Math. Ann. 359, 189–209 (2014). https://doi.org/10.1007/s00208-013-0992-4
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DOI: https://doi.org/10.1007/s00208-013-0992-4