TY - JOUR
T1 - Higher arithmetic degrees of dominant rational self-maps
AU - Dang, Nguyen Bac
AU - Ghioca, Dragos
AU - Hu, Fei
AU - Lesieutre, John
AU - Satriano, Matthew
N1 - Publisher Copyright:
© 2022 Scuola Normale Superiore. All rights reserved.
PY - 2022
Y1 - 2022
N2 - Suppose that f W X 99K X is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if P 2 X.Q/ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree !1.f / of f if the orbit of P is Zariski dense in X. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of X. We begin by defining a set of arithmetic degrees of f , independent of the choice of cycles, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
AB - Suppose that f W X 99K X is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if P 2 X.Q/ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree !1.f / of f if the orbit of P is Zariski dense in X. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of X. We begin by defining a set of arithmetic degrees of f , independent of the choice of cycles, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
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U2 - 10.2422/2036-2145.201908_014
DO - 10.2422/2036-2145.201908_014
M3 - Article
AN - SCOPUS:85125162122
SN - 0391-173X
VL - 23
SP - 463
EP - 481
JO - Annali della Scuola normale superiore di Pisa - Classe di scienze
JF - Annali della Scuola normale superiore di Pisa - Classe di scienze
IS - 1
ER -