Proof.
Let 
and let 
be the induced map. The map 
is
dominant and has irreducible general fiber. We claim that Z is a variety of
general type. If either
of 
is generically finite (and therefore birational) then this is clear.
By Viehweg's additivity theorem ([V1], Satz III), it
suffices to show that the generic fiber of 
is of general type. This
follows since the fibers of 
sweep 
. (Specifically, let d be the
dimension of the generic fiber of 
. Choose a
general codimension-d plane section 
, then 
is generically finite and dominant, therefore 
is of general type,
therefore the generic fiber of 
is of general type.)
Proof. Given an extension 
let 
be a
dominant rational map. By the lemma above with 
there
exists a variety Z and a dominant rational map with irreducible general fiber
dominating both 
and 
. By maximality 
and since the general fibers of 
are irreducible, 
is
birational. The map 
gives the required
dominant rational map.
Proof of the theorem.
Using Stein factorization we may restrict attention
to maps with irreducible general fibers. As above, let 
,
and let 
be an extension such that 
.
We need to show that 
can be descended to k.
First, we may assume that
K is finitely generated over k, since both Y and 
require only
finitely many coefficients in their defining equations.
Next, we descend 
to an algebraic extension of k. Choose a model B for K, and a model
for Y. We have a dominant rational map 
over
B. There exists a point 
with 
finite, such
that 
is a variety of general type of dimension l and such that the
rational map 
exists. The lemma above shows that 
is
birational to Y. Alternatively, this step follows
since by theorems of Maehara (see
[Mor]) and Kobayashi - Ochiai (see [MD-LM]) the set of rational
maps to varieties of general type 
is discrete, therefore
each 
is birationally equivalent to a map defined
over a finite extension of k.
We may therefore replace K by an algebraic Galois extension of k, which we
still call K. Let 
. For any 
we have a rational map
. Applying lemma
1.3 we obtain a birational map
There are open sets 
over which these
maps are regular isomorphisms, giving rise to descent data for 
to
k.
Is there a way to describe the fibers of the Lang map 
? A first
approximation is provided by the following:
Proof. Let 
be the generic point and let 
be the lang map of the generic fiber. Let 
be a model of
. By definition, the generic fiber 
of M is of
general type, therefore by Viehweg's additivity theorem M is of general type,
and by definition M is birational to 
.
We will see that the answer is yes, if one assumes the following inspiring conjecture of higher dimensional classification theory:
This conjecture allows us to ``construct'' the Lang map ``from above'':
The proof is obvious.
