Authors:	Javanpeykar, Ariyan Kamenova, Ljudmila
Title:	Demailly’s notion of algebraic hyperbolicity : geometricity, boundedness, moduli of maps
Online publication date:	17-May-2021
Year of first publication:	2020
Language:	english
Abstract:	Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties.
DDC:	510 Mathematik 510 Mathematics
Institution:	Johannes Gutenberg-Universität Mainz
Department:	FB 08 Physik, Mathematik u. Informatik
Place:	Mainz
ROR:	https://ror.org/023b0x485
DOI:	http://doi.org/10.25358/openscience-5862
Version:	Published version
Publication type:	Zeitschriftenaufsatz
License:	CC BY
Information on rights of use:	https://creativecommons.org/licenses/by/4.0/
Journal:	Mathematische Zeitschrift 296
Pages or article number:	1645 1672
Publisher:	Springer
Publisher place:	Berlin u.a.
Issue date:	2020
ISSN:	1432-1823
Publisher URL:	https://doi.org/10.1007/s00209-020-02489-6
Publisher DOI:	10.1007/s00209-020-02489-6
Appears in collections:	JGU-Publikationen