Let {X} be a variety over an algebraically closed field {k}. {X} is said to be rational if {X} is birational to {\mathbb{P}_k^n}. 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 {X} be a smooth projective curve. Then {X} is rational if and only if its genus is zero.
  • Let {X} be a smooth projective surface. Then {X} is rational if and only if there are no global 1-forms on {X} (i.e., {H^0(X, \Omega_{X/k}) = 0}) and the second plurigenus {H^0(X, \omega_{X/k}^{\otimes 2}) } 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 {\mathbb{P}^2_k}. The result is a criterion of Castelnuovo, extended by Zariski to characteristic p.

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

\displaystyle \mathbb{P}_k^n \dashrightarrow X;

or, equivalently, when the function field {k(X)} 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 {d} hypersurface in {\mathbb{P}^N}, {N \gg 0} 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 {\mathbb{C}}. The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial {\pi_1}.

Theorem 1 (Serre) A unirational (smooth, projective) variety over {\mathbb{C}} has trivial {\pi_1}.

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

I edited this post to fix some sign issues.  (11/29)

I now want to discuss a result of Myers, which I can summarize as follows:

If {M} is a complete Riemannian manifold with positive, bounded-below curvature, then {M} is compact.

This is a very loose summary—Myers’ theorem actually gives a lower bound for the diameter of {M}. Moreover, I haven’t explained what “bounded below curvature” actually means. To say that the sectional curvature is bounded below is sufficient, but we can do better.

I will now outline how the proof works.

The first thing to notice is that any two points {p,q \in M} can be joined by a length-minimizing geodesic {\gamma}, by the Hopf-Rinow theorems. In particular, if we can show that every sufficiently long geodesic (of length {>L}, say) doesn’t minimize length, then {M} is necessarily of diameter at most {L}.

All the same, the length function as a map {\mathrm{Curves} \rightarrow \mathbb{R}} is not so easy to work with; the energy integral is much more convenient. Moreover, we know that if {\gamma} minimizes length and is a geodesic, it also minimizes the energy integral.

If {\gamma} is a geodesic that minimizes the energy integral (among curves with fixed endpoints), then in particular we can consider a variation {\gamma_u} of {\gamma}, and consider the function {E(u):=E(\gamma_u)}; this necessarily has a minimum at {u=0}. It follows that {E''(0) \leq 0}. If we apply the second variation formula, we find something involving the curvature tensor that looks a lot like sectional curvature.

Before turning to the details, I will now define the refinement of sectional curvature we can use:

Ricci curvature

Given a Riemannian manifold {M} with curvature tensor {R} and {p \in M}, we can define a linear map {T_p(M) \rightarrow T_p(M)},

\displaystyle X \rightarrow -R(X,Y)Z  

that depends on {Y,Z \in T_p(M)}. The trace of this linear map is defined to be the Ricci tensor {\rho(Y,Z)}. This is an invariant definition, so we do not have to do any checking of transformation laws.

A convenient way to express this is the following: If {E_1, \dots, E_n} is an orthonormal basis for {T_p(M)}, then by linear algebra and skew-symmetry

\displaystyle \boxed{\rho(Y,Z) = \sum R(E_i, Y, E_i, Z) .}

The Ricci curvature has many uses. Considered as a {(1,1)} tensor (by the functorial isomorphism {\hom(V \otimes V, \mathbb{R}) \simeq \hom(V, V)} for any real vector space {V} and the isomorphism {T(M) \rightarrow T^*(M)} induced by the Riemannian metric), its trace yields the scalar curvature, which is just a real-valued function on a Riemannian manifold. It is also used in defining the Ricci flow, which led to the recent solution of the Poincaré conjecture.  I may talk about these more advanced topics (much) later if I end up learning about them–I am finding this an interesting field, and may wish to pursue geometry further.

Statement of Myers’ theorem

Theorem 1 Let {M} be a complete Riemannian manifold whose Ricci tensor {\rho} satisfies \displaystyle \rho(X,X) \geq C|X|^2   

for all {X \in T_p(M), p \in M}. Then

 \displaystyle \mathrm{diam}(M) \leq \pi \sqrt{ \frac{n-1}{C}}.  (more…)