Let be a variety over an algebraically closed field . is said to be **rational** if is birational to . In general, it is difficult to determine when a variety in higher dimensions is rational, although there are numerical invariants in dimensions one and two.

- Let be a smooth projective curve. Then is rational if and only if its genus is zero.
- Let be a smooth projective surface. Then is rational if and only if there are no global 1-forms on (i.e., ) and the second plurigenus vanishes. This is a statement about the negativity of the cotangent bundle (or, equivalently, of the positivity of the tangent bundle) which is a birational invariant and which holds for . The result is a criterion of Castelnuovo, extended by Zariski to characteristic .

In higher dimensions, it is harder to tell when a variety is rational. An easier problem is to determine when a variety is **unirational**: that is, when there is a dominant rational map

or, equivalently, when the function field has a finite extension which is purely transcendental. In dimensions one and two (and in characteristic zero), the above invariants imply that a unirational variety is rational. In higher dimensions, there are many more unirational varieties: for example, a theorem of Harris, Mazur, and Pandharipande states that a degree hypersurface in , is always unirational.

The purpose of this post is to describe a theorem of Serre that shows the difficulty of distinguishing rationality from unirationality. Let’s work over . The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial .

Theorem 1 (Serre)A unirational (smooth, projective) variety over has trivial .

The reference is Serre’s paper “On the fundamental group of a unirational variety,” in J. London Math Soc. 1959.

**1. Finiteness of **

To prove this theorem, let’s fix a unirational (smooth, projective) variety over of dimension , and a dominant rational map

There is a subvariety of codimension such that the map actually extends to an honest morphism (via the valuative criterion).

Let’s form the universal cover ; here is only a complex manifold, not necessarily an algebraic variety. The goal is to show that is actually an isomorphism. To start with, though, let’s show that it is a finite map: in other words, that is finite. To see this, observe that the map is generically finite and étale, and so we may choose an open subset such that

is a finite covering map. We now have a diagram

Now is a cover of , and since we’ve thrown away codimension one subsets of , it is a connected covering space. (Equivalently, the map is a surjection.)

The claim is that we can get a map of covering spaces as in the diagram. In fact, we will do better: we will produce a lift in the diagram

We get this map simply by noting that is simply connected, since we have thrown away a subset of codimension .

In view of this, we conclude that there is a map from the *finite* cover to the connected cover (where the latter is a -Galois cover). Since any map of connected covers is necessarily surjective, we must conclude that is finite.

**2. is trivial**

We need to go a bit further to conclude that is actually trivial, though. First, since has finite fundamental group, the universal cover acquires the structure of an algebraic variety, and we get a finite étale map

We’re going to show that in fact is unirational. In fact, consider the diagram

where is the pull-back. Now is a dominant rational map. However, since is a covering map, we conclude that is a disjoint union of copies of itself; taking any one produces the dominant rational map

We conclude that is also unirational as claimed, and from here will get a contradiction by looking at the arithmetic genus.

A basic property of a unirational variety (over ) is that certain Hodge numbers vanish:

because Hodge theory gives equalities

and a nonzero holomorphic -form on would pull back to one on (where there are none). In particular, we conclude that the holomorphic Euler characteristic is :

Now, as we’ve seen, the holomorphic Euler characteristic of a unirational variety is one. Therefore,

However, the Hirzebruch-Riemann-Roch formula allows one to compute the holomorphic Euler characteristic of a complex manifold via a polynomial in the Chern numbers and implies that it is multiplicative in finite covers. That is:

Proposition 2Let be a finite cover of compact complex manifolds of degree . Then

Putting this together, we conclude that the degree of the universal cover is equal to one, so that as desired.

March 19, 2013 at 7:45 am

This is true much more generally, cf. results of Koll\’ar (fundamental group of any smooth, projective rationally chain connected variety is finite; fundamental group of any smooth, projective, separably rationally connected variety is trivial), results of Mingmin Shen (fundamental group of any smooth, projective freely rationally connected variety is trivial) and Ekedahl (p-power part of every smooth, projective, rationally chain connected variety is trivial). There are examples due to Shioda of unirational varieties in positive characteristic (even surfaces, if memory serves) with nontrivial fundamental group (necessarily prime-to-p, thanks to Ekedahl).

By the way, I love your blog.

March 19, 2013 at 8:36 am

Thank you! I wasn’t aware of these results.