In this post we prove a general result that shows, in particular, that any map from a simply connected to a curve
of genus at least
is constant.
The key preliminary result
Let us begin by discussing a nice result which mirrors a classical result in topology. For completeness, let us recall this result:
Theorem 1: Let
be a locally path connected, simply connected topological space and let
be a continuous map. Then, if
is a covering map, then there exists a lift of
to
.
This result is a standard result in a first course on topology, and underlies the key principle of simply connected spaces: they are the spaces which are unobstructed to lifting to coverings.
The same is true in the étale world for essentially the same reason. In particular we have the following:
Key lemma 2: Let
be a conected, (étale) simply connected scheme and
a morphism with
connected. Suppose that
is a finite connected étale cover. Then, there is a morphism
lifting
.
Proof: Consider the pull-back
Since is simply connected we know that
is isomorphic to
(with
copies of
) as an
-scheme. In particular, we see that there is a morphism
such that
as desired.
The general result
With this
Theorem(Main) 3: Let
be a connected, (étale) simply connected scheme. Let
be a connected locally Noetherian scheme for which there are connected finite étale covers of arbitrarily high degree. Then, there is no flat proper map
.
Proof: Suppose that is a flat proper map. Then, for any finite connected étale cover
we know, by the Key Lemma, that we can factorize
as
for some
. Note that
is necessarily flat and proper. Properness follows since
is proper (using the notation in the proof of the Key Lemma) and since the canonical inclusion
is a closed embedding their composition, is also proper. To see that
is flat we note that since
,
is flat and
is an open embedding, that the cancellation lemma implies that
is also flat.
Now, since is proper and flat (and
is locally Noetherian) we know that
is a vector bundle. Similarly, we know that since
is proper and flat (and
is locally Noetherian) that
is a vector bundle. But, by construction
But, this implies that
and so, in particular, . Since we assumed we could find
of arbitrarily high degree this is a contradiction.
Applications to curves
As an application of the main theorem we may derive the following result:
Theorem 4: Let
be a proper variety over a field
such that
. Then, for any (smooth projective integral) curve
of positive genus there are no non-constant maps
.
Here ‘variety’ means a separated, finite-type, and geometrically reduced -scheme. Of course, separatedness is not used in the proof.
Proof: Suppose that is non-constant, then
is non-constant. Thus, it suffices to assume that
. But, then we see that
non-constant implies that’s surjective (by dimension considerations) and thus (since
is reduced) a flat map. Since
is proper and
separated we know that
is automatically proper (by the cancellation lemma). Thus, we may apply the Main Theorem to arrive at a contradiction after noting that
, being positive genus, has connected étale covers of arbitrarily high degree.
As a simple application of this, we see that there are no non-constant maps from a proper smooth rational (even rationally connected!) variety to a curve of positive genus
. This follows since such varieties are smooth (see the discussion here).
Two simple examples which pop-up from this are the fact that there are no non-constant maps or
for
a positive genus curve. These, of course, both follow from elementary considerations, but this gives a conceptual way for showing the result.
The complex setting
If one is willing to consider a -analogue one can prove Theorem 4 in a slightly different way—this was pointed out to my by my friend Alex Sherman. Namely, let
be a smooth projective (integral) variety which is topologically simply connected (i.e.
) and let
be a map. Then, by composing
(by choosing some base-point) we get a map
. Then, by the universal property of the Albanese variety one obtains a map
making the following diagram commute:
But, as a complex torus
and by Hodge theory we know that
But, since is simply connected we know that
and thus
. Then, since
we conclude that
and thus
. Consequently,
and so
.
So, we see that the composition
is evidently constant. But, it’s equal to the composition
and since is injective, it follows that
must have been constant as well.
This approach should be adaptable to work over characteristic fields besides
, namely. I haven’t checked the details but it seems like the following should work. The map
is still injective. Moreover, if
is connected (and sufficiently nice) we know that
exists and has dimension
. So, we want to show that if
then
. Using standard techniques one reduces this to the case of
. Now, we claim that
implies that
is trivial. Indeed, if not then
and thus
would have a finite quotient. This then should imply that
which is a contradiction. Then, we have that
so
. One then deduces that
by Hodge theory. One could also prove this last part by something like, say,
-adic Hodge theory. The rest of the argument then goes the same.
This seems doomed to work in general though. In positive characteristic it’s not true that implies that
as an Enriques surface shows.
I don’t know much about Albanese in characteristic p, is it true that pi_1=0 would imply vanishing of Albanese (because Albanese map induces isomorphism between l-adic H^1_ét, where l is prime to p?)