On the Belyi height

Belyi height of a complex curve is defined as the smallest possible degree of a Belyi function on it. For a fixed genus it is considered as a function on the moduli space; according to Belyi theorem, the Belyi height of a curve is finite if and only if the curve is defined over the field of algebraic numbers.
Belyi height will be compared with the other heights and with the Kolmogorov complexity. Some examples due to the speaker and to Leonardo Zapponi will be presented. The recent result by Ariyan Javanpeykar and John Voight on the algorithmic computability of the Belyi height will be formulated and the algorithmic aspects of the passport realizability discussed.