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
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.