Let k be an algebraically closed field of characteristic zero, and let
X/k be a projective variety. The conjectures of
Demailly--Green--Griffiths--Lang posit that every integral subvariety of X is
of general type if and only if X is algebraically hyperbolic i.e., for any
ample line bundle L on X there is a real number
Ξ±(X,L), depending only on X and L, such that for
every smooth projective curve C/k of genus g(C) and every k-morphism
f:CβX, degCβfβLβ€Ξ±(X,L)β g(C) holds.
In this work, we prove that if X/k is a projective variety such that every
integral subvariety is of general type, then for every ample line bundle
L on X and every integer gβ₯0, there is an integer
Ξ±(X,L,g), depending only on X,L, and g, such
that for every smooth projective curve C/k of genus g and every
k-morphism f:CβX, the inequality degCβfβLβ€Ξ±(X,L,g) holds, or equivalently, the Hom-scheme
Homβkβ(C,X) is projective.Comment: v2: 41 pages. Significant updates throughout to address several
mistakes in previous version. Results remain unchanged. Comments are welcome