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Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

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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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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Correspondence to Fabio Perroni.

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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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Mathematics Subject Classification (2000)

  • 32H50
  • 14J50
  • 32M05
  • 37B40