Diophantine Approximation

and Nevanlinna Theory Lecture II

Paul Vojta University of California, Berkeley

Conjectures

Conjecture. Let k be a number field, let S ⊇ S_{∞} be a finite set of places of k, let X be a smooth projective variety over k, let D be a normal crossings divisor on X, let A be an ample divisor on X, let 𝒦 denote the canonical line sheaf on X, and let ϵ > 0 . Then there is a proper Zariski-closed subset Z of X, depending only on the above data, such that

m_{S}(D, x) + h_{𝒦 ,k} (x) ≤ ϵ h_{A,k}(x) + O(1)

for all x ∈ (X \ Z)(k) .

Conjecture. Let X be a smooth projective complex variety, let D be a normal crossings divisor on X, let A be an ample divisor on X, and let 𝒦 denote the canonical line sheaf on X. Then there is a proper Zariski-closed subset Z of X, depending only on the above data, such that for all holomorphic maps f: ℂ → X with f(ℂ) ⊈ Z,

m_{f}(D, r) + T_{𝒦 ,f} (r) ≤_{exc} O(log^{+} T_{A,f}(r)) + o(log r) .

These are proved if dim X = 1 , or if X = ℙ^{n} and D is a sum of hyperplanes. Limited results have also been shown if X is a certain type of compactification of a semiabelian variety, or if X is a completion of a Shimura variety.

In other words, we know very little if dim X > 1 .

Embeddings

For example, consider ℙ^{2} , and let D be as above. If deg D ≥ 4 then ℙ^{2} \ D should not have a dense set of integral points (or admit a Zariski-dense holomorphic curve to it).

If D is smooth of degree ≥ 4 , then there’s no hope (so far). For four lines, however, we do have results (Schmidt and Cartan).

For three lines and a conic, there are some results; e.g., if L_{1}, L_{2}, L_{3} are linear forms defining the lines and Q is a quadratic polynomial defining the conic, then all L_{i}^{2}/Q must be units (or nearly so) at integral points, so we can apply the unit lemma (or Borel’s lemma) (M. Green and V.)

Besides that, how do we get any results for a conic?

Under the 2-uple embedding ℙ^{2} ↪ ℙ^{5} , the image of a conic is contained in a hyperplane. Under the 3-uple embedding ℙ^{2} ↪ ℙ^{9} , the image of a conic spans a linear subspace of codimension 3 , so we can get 3 linearly independent hyperplanes containing it.

More generally, say you have a divisor D of degree d in ℙ^{n}, and look at its image under the r-uple embedding

ℙ^{n} ↪ ℙ^{\binom{r+n}{n}−1} .

The image of the divisor spans a linear subspace of codimension \binom{r+n−d}{n}, because there are that many monomials of degree n − d (which then get multiplied by the form defining D to get homogeneous polynomials of degree n in the original variables, hence hyperplanes in the image space).

So the inequality we’d get out of applying Schmidt’s Subspace Theorem to ℙ^{\binom{r+n}{n}−1} would look like

\binom{r + n − d}{n}m(D, x) + ⋯ ≤ (\binom{r + n}{n} · r + ϵ) h_{k}(x) + O(1)

for x ∈ ℙ^{n}(k) outside of a finite union of proper subvarieties of degree ≤ r.

So, as r → ∞, we’re interested in the ratio

\frac{\binom{r+n−d}{n}}{\binom{r+n}{n} · r} = \frac{(r − d + n) ⋯ (r − d + 1)}{(r + n) ⋯ (r + 1)r} = \frac{r^{n} + O(r^{n−1})}{r(r^{n} + O(r^{n−1}))} → \frac{1}{r} → 0 .

This is not very promising, but we can try harder. We can also consider that some hyperplanes can be made to contain D twice, or three times, etc. This gives something like

\begin{matrix}(\binom{r + n − d}{n} + \binom{r + n − 2d}{n} + …) m(D, x) + … \\ ≤ (\binom{r + n}{n} · r + ϵ) h_{k}(x) + O(1) \end{matrix}

To estimate the factor in front of m(D, x) :

\binom{r − kd + n}{n} = \frac{(r − kd + n) ⋯ (r − kd + 1)}{n!} = \frac{(r − kd)^{n} + O_{n}((r − kd)^{n−1})}{n!}

and therefore

∑_{k=1}^{[r/d]} \binom{r − kd + n}{n} = \frac{(r − d)^{n} + ⋯ + (r − [r/d]d)^{n} + O_{n,d}(r^{n})}{n!} .

As r → ∞, the ratio of interest now converges to

\frac{∑{[r/d] \atop k=1} \binom{r−kd+n}{n}}{r\binom{r+n}{n}} ≈ \frac{\frac{r^{n+1}}{(n+1)d·n!}}{\frac{r^{n+1}}{n!}} → \frac{1}{(n + 1)d} .

This would give us something like

d m_{S}(D, x) + ⋯ ≤ (n + 1 + ϵ)h_{k}(x) + O(1) .

This is best possible if d = 1 , but is still new and noteworthy if d > 1 .

But there is something wrong with this.

Suppose we have two divisors, and their images under the r-uple embedding span linear subspaces L_{1} and L_{2} of codimensions ρ_{1} and ρ_{2} , respectively. We have

codim(L_{1} ∩ L_{2}) ≤ ρ_{1} + ρ_{2}

(assuming L_{1} ∩ L_{2} ≠ ∅). If this inequality is strict then we have problems to the extent that m(L_{1}, y) and m(L_{2}, y) come from m(L_{1} ∩ L_{2}, y) (where y is the image of x under the r-uple embedding). Say you have ρ_{1} generic hyperplanes containing L_{1} and ρ_{2} generic hyperplanes containing L_{2} . If these ρ_{1} + ρ_{2} hyperplanes are in general position then this implies

codim(L_{1} ∩ L_{2}) ≥ ρ_{1} + ρ_{2} .

So we need equality. However, the usual computation of Hilbert function using short exact sequences gives (for sufficiently large n) that the codimension of the linear span of D_{1} ∩ D_{2} is

\binom{n − d_{1} + r}{n} + \binom{n − d_{2} + r}{n} − \binom{n − d_{1} − d_{2} + r}{n} ,

which is too small by \binom{n−d_{1}−d_{2}+r}{n}.