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…)