Surjective endomorphisms of projective surfaces: the existence of infinitely many dense orbits

Jia Jia1, Junyi Xie2, De‐Qi Zhang1
1National University of Singapore, Singapore, Republic of Singapore
2BICMR, Peking University, Haidian District, China

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Tài liệu tham khảo

Amerik, E.: Existence of non-preperiodic algebraic points for a rational self-map of infinite order. Math. Res. Lett. 18(02), 251–256 (2011)

Amerik, E., Campana, F.: Fibrations méromorphes sur certaines variétés à fibré canonique trivial. Pure Appl. Math. Q. 4(2), 509–546 (2008)

Bell, J. P., Ghioca, D., Tucker, T. J.: The dynamical Mordell–Lang conjecture, mathematical surveys and monographs. Am. Math. Soc. 210, 978-1-4704-2408-4 (2016)

Bell, J.P., Ghioca, D., Tucker, T.J.: Applications of $$p$$-adic analysis for bounding periods of subvarieties under étale maps, 2014, 1073–7928. Int. Math. Res. Not. IMRN 11, 3576–3597 (2015)

Boucksom, S., de Fernex, T., Favre, C.: The volume of an isolated singularity. Duke Math. J. 161(8), 1455–1520 (2012)

Broustet, A., Gongyo, Y.: Remarks on log Calabi-Yau structure of varieties admitting polarized endomorphisms. Taiwanese J. Math. 21(3), 569–582 (2017)

Broustet, A., Höring, A.: Singularities of varieties admitting an endomorphism. Math. Ann. 360(1–2), 439–456 (2014)

Cascini, P., Meng, S., Zhang, D.-Q.: Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. Math. Ann. 378, 637–665 (2020)

Debarre, O.: Higher-Dimensional Algebraic Geometry, Universitext, Springer, 0-387-95227-6 (2001)

Dinh, T.-C., Nguyên, V.-A., Truong, T.T.: Equidistribution for meromorphic maps with dominant topological degree. Indiana Univ. Math. J. 64(6), 1805–1828 (2015)

Fujino, O.: Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math. Sci. 47(3), 727–789 (2011)

Fujino, O.: Minimal model theory for log surfaces in Fujiki’s class. Nagoya Math. J. 244, 256–282 (2021)

Ghioca, D., Satriano, M.: Density of orbits of dominant regular self-maps of semiabelian varieties. Trans. Am. Math. Soc. 371(9), 6341–6358 (2019)

Ghioca, D., Scanlon, T.: Density of orbits of endomorphisms of abelian varieties, 1088–6850. Trans. Am. Math. Soc. 369(1), 447–466 (2017)

Guedj, V.: Ergodic properties of rational mappings with large topological degree, 0003486X. Ann. Math. 161(3), 1589–1607 (2005)

Hartshorne, R.: Algebraic geometry, Grad. Texts in Math., Springer-Verlag, 52, 0-387-90244-9, (1977)

Iitaka, S.: Algebraic Geometry—An Introduction to Birational Geometry of Algebraic Varieties, Grad. Texts in Math., Springer-Verlag, 76, 0-387-90546-4, (1982)

Kollár, J.: Rational Curves on Algebraic Varieties, Springer-Verlag, 32, (1996)

Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics. Cambridge University Press (1998)

Matsuzawa, Y., Yoshikawa, S.: Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism. Math. Ann. 382, 1681–1704 (2022)

Medvedev, A., Scanlon, T.: Invariant varieties for polynomial dynamical systems. Ann. Math. 179, 81–177 (2014)

Meng, S., Zhang, D.-Q.: Kawaguchi-Silverman conjecture for surjective endomorphisms, Documenta Mathematica (to appear), arXiv:1908.01605

Meng, S.: Building blocks of amplified endomorphisms of normal projective varieties. Math. Z. 294(3), 1727–1747 (2020)

Meng, S., Zhang, D.-Q.: Building blocks of polarized endomorphisms of normal projective varieties, 0001–8708. Adv. Math. 325, 243–273 (2018)

Milnor, J.: Dynamics in one complex variable. (AM-160): Third Edition, Princeton University Press, 9780691124889, (2006)

Nakayama, N.: A variant of Shokurov’s criterion of toric surface. Algebraic varieties and automorphism groups. Advanced Studies in Pure Mathematics, 75, Mathematical Society of Japan (2017)

Nakayama, N.: On normal Moishezon surfaces admitting non-isomorphic surjective endomorphisms, 1923, RIMS Preprints (2020)

Nakayama, N.: On the structure of normal projective surfaces admitting non-isomorphic surjective endomorphisms, 1934, RIMS Preprints (2020)

Nakayama, N.: Ruled surfaces with non-trivial surjective endomorphisms. Kyushu J. Math. 56(2), 433–446 (2002)

Sakai, F.: Weil divisors on normal surfaces. Duke Math. J. 51, 877–887 (1984)

Wahl, J.: A characteristic number for links of surface singularities. J. Am. Math. Soc. 3(3), 625–637 (1990)

Xie, J.: The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker). J. Am. Math. Soc. (published online), arXiv:1905.07021

Xie, J.: The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane. Compos. Math. 153(8), 1658–1672 (2017)

Zhang, D.-Q.: Polarized endomorphisms of uniruled varieties with an appendix by Y. Fujimoto and N. Nakayama. Compos. Math. 146(1), 145–168 (2010)

Zhang, D.-Q.: $$n$$-dimensional projective varieties with the action of an abelian group of rank $$n-1$$. Trans. Am. Math. Soc. 368(12), 8849–8872 (2016)