McQuillan Theory F.A.Q.

1.

What is the status of the Green-Griffiths conjecture ?

It is a theorem that for algebraic surfaces $S$ of general type with enough two jets, e.g. $13 \mathrm{c}_1^2(S)>9\mathrm{c}_2(S)$. More precisely, on such surfaces: the rational and elliptic curves form a closed proper sub-variety $P$ of $S$ and any entire map $f:\mathbb{C}\rightarrow S$ factors through $P$.

2.

Are there more quantitative theorems for such surfaces ?

Indeed there are. Notations as above, with $H$ ample on $S$ then there is a constant $\kappa > 0$ such that for $f:Y\rightarrow S$ any map from a parabolic (in the sense of Ahlfors) Riemann surface not factoring through $P$,

$${\int\!\!\!\!\!\!\nabla} _{Y(r)} f^* \mathrm{c}_1 (H) \leq_{\mathrm{exc}} -\kappa\,\chi_Y(r)$$

where the left hand side is the Ahlfors' counting function computed using the Evans' potential, and the Euler characteristic on the right should also be understood in this sense, while $r$ is excluded from a set of finite Lebesgue measure. In the related case that $f$ is a map from a compact surface of genus $g$ which does not factor through $P$,

$$\int_Y f^* \mathrm{c}_1 (H) \leq \kappa (2g-2)$$

for the same $\kappa$.

3.

What about the behaviour of discs and the Kobayashi metric ?

Again, same class of surfaces, same notation, a sequence of discs $f_n:\Delta\rightarrow S$ which is not contained in arbitrarily small neighbourhoods of $P$ has a convergent subsequence which converges to a disc with bubbles. The behaviour around $P$ may also be quantified optimally in terms of the degeneracy of the Kobayashi metric- basically exactly the reciprocal of the distance to $P$. This is a wholly general corollary for surfaces which settles a question I posed at the Peking I.C.M., [M3] 3.1. It follows from a theorem of Duval, [D], and [M4], but, since the former post dates the latter, the proof cannot be found in the downloads section. For the moment, therefore, exercise for the reader, though probably worth a stand alone version.

4.

In the particular case that $P$ is empty ?

Better again. It's immediate whether by the above theorem of Duval, [D], or another of Kleiner, [K], that Gromov's isoperimetric inequality holds.

5.

Is this just for $S$ compact ?

Not at all, there are log, stack, and log-stack versions of all of the above. The only thing that is required is the 2-jet condition. Most of the changes in the statements are the obvious, while 4 holds in the log case with respect to a complete distance, which is about the only point that requires extra work.

6.

What are the ingredients in the proof ?

Quite a few,

(a)

Resolution of foliation singularities for 3-folds, [MP1].

(b)

Minimal models for foliations by curves, [M5], i.e. the canonical $K_{\mathcal{F}}$ along the foliation $X\rightarrow [X/\mathcal{F}]$ with canonical (foliation) singularities is nef. if the foliation isn't in conics.

(c)

What's termed a refined tautological inequality, basically: $K_{\mathcal{F}}$ measures curvature iff the singularities are canonical, [M5] V.5, or [M4] §II.

(d)

The residue lemma, [M8]. This forces the intersection of $K_{[X/\mathcal{F}]}$ with an invariant measure $d\mu$ to be zero, when, functorially understood, the measure has zero winding number/Segre class $\mathrm{s}_{Z,d\mu}$ around the foliation singularities.

7.

What's the logic of how they fit together ?

It goes as follows: standard results about existence of jets on surfaces tell us that if anything in nos. 1 or 2 is false, we can find a foliation by curves $X\rightarrow [X/\mathcal{F}]$ on a variety of dimension $n$ (bounded at worst in terms of the chern numbers) leaving the curves in question invariant. Sub-sequencing appropriately, and taking the dimension of $X$ to be minimal, this gives an invariant measure $d\mu$ intersecting every effective divisor non-negatively. By (c), $K_{\mathcal{F}}.d\mu\leq 0$, so by (b) $K_{\mathcal{F}}.d\mu=0$, and, we have $0\neq K_{\mathcal{F}}^{n-1}$ parallel to $d\mu$ in homology. The residue lemma implies, however, that if $Z$ is the singular locus of the foliation then $\mathrm{s}_{Z,d\mu}\neq 0$, which for a closed positive current, $T$, let alone an invariant measure is not easy, i.e. $L.T=0$, $L$ nef of numerical Kodaira dimension $d$, $\mathrm{s}_{Y,T}\neq 0$ for $Y$ of dimension less than $d$ implies $T$ is supported in a countable union of proper algebraic sub-varieties. Given the minimal model theorem (c) this is as near to a reduction of dimension as makes no difference. For a more detailed explanation go to the overview of the proof.

8.

Seems valid in any dimension, why the emphasis on 3 folds ?

As is explained in the overview of the proof, the chain of reasoning is indeed valid, but critically 6 (a) & (d) have, so far, only been proved in dimension 3. Otherwise (b) holds in all dimensions starting from a model with canonical singularities, and (c) holds unconditionally.

9.

Fair enough, but why not cases of the Green-Griffiths conjecture for 3-folds too ?

Evidently we'd need restrictions on the chern numbers to reduce to foliations by curves, e.g. $\mathrm{s}_2$ semi-positive, $\mathrm{s}_3 > (\mathrm{s}_1)^3/3$ might work. However, even then the resulting canonical along the foliation $K_{\mathcal{F}}$ can have numerical Kodaira dimension 1, e.g. a tri-disc quotient, so one needs a better residue lemma. This is on the to do list, since it's the principle missing fact to get a classification theorem for foliations by curves on 3-folds, i.e. currently there isn't enough machinery to cover (foliated) numerical Kodaira dimension 1 in dimension 3. Nevertheless, I wholly expect not only the Green-Griffiths conjecture whenever the chern numbers of the 3-fold permit a reduction to foliation by curves, but a full classification theorem à la [M2] of foliations by curves on projective 3-folds. If, however, one only had a reduction to foliations by surfaces, or similar, then looking for a counter example may easily be the better strategy. After all, cf. no. 11, the ``Green-Griffiths conjecture'' is already false for schemes of dimension 3 in mixed characteristic.

10.

Isn't there supposed to be some easier strategy where you just compute sections on jet bundles ?

This is a common misunderstanding resulting from a lack of knowledge as to the logical limitations of what is actually possible. The above assertion is true, and a theorem of Steven Lu, [L], for surfaces of general type whenever $\mathrm{c}_1^2> 2\mathrm{c}_2$. Otherwise, by way of (an introductory) illustration consider what happens on the line $\mathrm{c}_1^2= 2\mathrm{c}_2$, where we have the following statements:

(A) Let $S$ be a surface of general type with $\mathrm{c}_1^2= 2\mathrm{c}_2$, then there are constants $a,b$ depending only on $\mathrm{c}_1^2$ and $\mathrm{c}_2$ such that for any curve $C$ on $S$,

$$K_S.C\leq a g(C) + b$$

where $g$ is the genus of the normalisation of $S$.

(B) There are infinitely many surfaces $S$ on the line $\mathrm{c}_1^2= 2\mathrm{c}_2$ such that for $p\gg 0$ with probability $1/2$, $S$ modulo $p$ has a rational curve of degree $\gg p$.

Now (B) is true, so it follows that while true in characteristic zero, (A) is false in mixed characteristic, even for the weaker statement that $a,b$ are simply bounded uniformly in $p$. Whence,

(C) Fact There is no theorem which is valid for $p\gg 0$, e.g. an $\mathrm{ACF}_0$ theorem, which together with the inequality,

$$L.C\leq 0$$

for $C$ a rational curve on $S$ and $L$ the image of $\Omega^1_S$ in differentials on its normalisation, implies even this weaker form of theorem (A) in characteristic 0.

This particular example can be understood in terms of smooth foliations on surfaces so all the steps (a)-(f) in the main strategy in characteristic zero are pretty trivial, and one knows that the surfaces giving rise to (B) are (non-integrable) bi-disc quotients. In mixed characteristic (b) fails, but not in an important way. Interestingly, an appropriate version of (c) holds, [M9], so for $H$ ample one can use duality to get on a bi-disc quotient $S$, a class $\nu$ in $\mathrm{NS}_1(S)$, in characteristic zero, such that,

(i)

$0\neq \nu$ is nef., i.e. intersects every Cartier divisor non-negatively.

(ii)

There is a sequence (Zariski dense in the ambient 3 -dimensional scheme) of rational curves $C_p$ (actually components of the locus of super singular abelian surfaces in terms of the modular description of $S$) for $p$ running through about half the primes, such that for any divisor $D$, in characteristic zero,

$$D.\nu = \lim_{p} \frac{i_p^* D. C_p}{H.C_p}$$

and the specialisation $i_p^* D$ is understood only to have sense for $p$ sufficiently large.

(iii)

For $\mathcal{F}$ whichever of the foliations on the bi-disc quotient one pleases, we may suppose that the $C_p$ are $\mathcal{F}$ invariant, and,

$$K_{\mathcal{F}}.\nu\leq 0$$

At this point the reasoning (d) is wholly valid, and we conclude that $\nu$ is parallel to $K_{\mathcal{F}}$ in Néron-Severi in characteristic zero. The residue lemma, however, fails, albeit not by that much since it does have variants in characteristic $p$, which since these foliations are smooth amounts to,

$$K_{[S/\mathcal{F}]}.C_p \,=\, 0\,\,\, \mathrm{mod}\,\, p$$

or better,

$$C_p\in\mathrm{Im}\bigl(\mathrm{H}^1(S_p, K_{[S/\mathcal{F}]}) \rightarrow \mathrm{H}^1(S_p, \Omega^1_{S_p})\bigr)$$

However unlike $d\mu$ of (d), $\nu$ does not belong to this image in characteristic zero, nor, unless there is a contradiction in mathematics as a whole, could it.

11.

That certainly looks like a counter-example to the ``Green-Griffiths conjecture'' in mixed characteristic, but, I'm confused. You've explained why your proof doesn't work in mixed characteristic, but the proof of (A) as stated in no. 10 by Miyaoka, [M], or the weaker statement by Bogomolov, [B], are wholly algebraic, so they should work for $p\gg 0$. What's going on ? Perhaps mathematics is inconsistent ?

ZFC or similar could be inconsistent, but an inconsistency at this level is enormously improbable. In addition the residue theorem for smooth foliations is trivial and purely algebraic as far as algebraic curves are concerned. As to Miyaoka's proof it needs closedness of differential forms. The forms in question are different for every curve of which one wants to bound the degree. Similarly, in the context of (A), Bogomolov's proof needs the Frobenius theorem, which in the example in question is true at exactly the primes where one does not find the $C_p$ appearing in no. 10, while in a more general situation it uses Jordan decomposition for log-canonical singularities, which although valid in characteristic $p$ is much less useful. Thus, Bogomolov's proof is a variant on the residue lemma, actually a less algebraic one in the context of algebraic curves, while Miyaoka's uses properties of differentiation in characteristic zero that could be considered even less algebraic again, so, in all cases, there's no inconsistency, and there's nothing to see here, move along.

12.

Ok, granted there is a logical issue, but your example comes from a smooth foliation, which, as you say, is pretty trivial, and this can be detected as an obstruction to the $K_S$ stability of $\Omega^1_S$, and lack of stability as opposed to semi-stability is all that's stopping Lu's proof from working here.

Quite true, and Lu did exactly this, but it only works because one is on the line $\mathrm{c}_1^2= 2\mathrm{c}_2$. However, under the line, one can make much more difficult variants on the above by taking bi-disc quotients with cusps. So in the region $\mathrm{c}_1^2 > (2-\varepsilon) \mathrm{c}_2$ for any $\varepsilon>0$, as many examples as one pleases, with the structure of the rational curves in mixed characteristic exactly as in no. 10. As such we can radically extend the surfaces to which Fact (C) of no. 10 applies, and all of (i)-(iii) hold too, but there is no hope of detecting from stability considerations that one is looking at such an example unless one can find the cusps first. Such examples also have serious practical consequences, e.g. in the jet bundle of order $n$ the natural foliations on the bi-disc define surfaces $S_n$- actually the blow up $n$-times in the singularities of the cusps- and the restriction of the $n$th tautological bundle $L_n$ has the form $K_{\mathcal{F}}(-E_n)$ where $E_n$ is the exceptional divisor arising from the $n$th blow up. Now $K_{\mathcal{F}}$ is nef. of numerical Kodaira dimension 1, but Kodaira dimension $-\infty$, thus, no matter what jet space one looks in $L_n$ isn't even pseudo effective. As such even formulating an Ahlfors-Schwarz lemma here, as opposed to its functorial cousin, the tautological inequality, (c), is very demanding. Of course, one could attempt a highly degenerate pseudo-metric formulation of Ahlfors-Schwarz, but the technical problems are legion, and very similar to those encountered in the classification theorem as described in no. 13 below. Finally, all of this is under the best possible hypothesis that one is looking at a model with canonical singularities, since otherwise were there to be a model of such an example such that $\omega_{S/[S/\mathcal{F}]}$ had sections, then it would be decidedly a case of barking up the wrong tree.

13.

Fair enough, looks like all of (a)-(c) are necessary for the Green-Griffiths conjecture even on surfaces close to the line $\mathrm{c}_1^2= 2\mathrm{c}_2$, but maybe these bi-disc quotients are the only foliations by curves on surfaces of general type such that the foliation itself does not have general type. If this were true, it ought to follow from fairly routine algebra, and everything else would be pretty trivial.

It is true that bi-disc quotients, both integrable and non-integrable are the only such examples. Both have numerical Kodaira dimension 1, which is also the Kodaira dimension of the former, but, as we've said, the latter is $-\infty$. As to the rest of the question this was the original motivation for the classification theorem, [M2], since it was clear even then that the residue lemma in arbitrary dimension would not be easy. The subject is dealt with at length in the introduction to op. cit., but briefly: one gets very close to proving the classification theorem without recourse to [M1], and what is required is a sufficiently regular psh. metricisation of $K_{\mathcal{F}}$ in the nef. Kodaira dimension $-\infty$ case. However I couldn't find such a thing, and Marco Brunella changed the logic. More precisely, one knows from [M1] that in this case the Poincaré metric along the leaves vanishes only on an algebraic set (so, post factum the cusps in no. 12). A priori it may not be psh., but Brunella proved that it was (a result he subsequently generalised to all dimensions), and I kicked the ball over the line by showing that it was also continuous. This gave a metric of the desired regularity, and solved the outstanding issue in the classification theorem, but it completely inverted the intended logic by deducing the classification theorem from [M1] rather than re-proving it as was intended. I couldn't say with logical certainty that this is the only way to proceed- there is some discussion of a characteristic $p$ alternative in [M2]- but I'd be very surprised if either Brunella's inversion of the logic isn't dictated by the nature of the problem or there were a substantially different strategy in characteristic 0 which escapes the logical loop that it creates.

14.

The evidence does look compelling that the strategy outlined in the overview of the proof is the way to go, but in [M6] everything is claimed modulo resolution of singularities. However, at least in dimension 3, this has been proved in [MP1] so why the new residue lemma, [M8] ? What's going on ?

Mihai Paun pointed out around may 2005 that IV.7.3 in [M6] is wrong. This amounted exactly to a mistake in the proof of the residue lemma- basically the plan was to do some harmonic theory along the foliation, and practically everything in [M6] permits blowing up ad nauseum, but not when doing the said harmonic theory. This was corrected in [M6.bis] a few months later. To achieve the correction, however, it was necessary to suppose more than just resolution of singularities, denoted LCR, i.e. a stronger variant, $\mathrm{LCR}_+$, in which the formal centre manifold converged in dimension 3 in much the same way that it does for 2 dimensional saddles. This was the motivation for [MP2], which proved something very close to $\mathrm{LCR}_+$, but also discovered that the precise form of $\mathrm{LCR}_+$ hypothesised [M6.bis] does not hold.

15.

So to prove the residue lemma 6.(d) one has to do some further work in [M8] to deduce it from the weaker version of $\mathrm{LCR}_+$ in [MP2] ?

No. The method of [M8] is completely new and has nothing to do with previous efforts to prove the residue lemma in [M6] and [M6.bis]. As is explained in the introduction to [M8], not withstanding the need for measure regularity in the residue lemma, it nevertheless achieves the calculation as a pure consequence of the formal structure around the singularities guaranteed by the resolution theorem [MP1]. As such [MP1], a.k.a. 6(a), a.k.a. LCR, is what is not only relevant, but a sine qua non.

16.

So [MP2] has no relevance to the residue lemma ?

In a strict logical sense that's correct, or, better the residue lemma can be proved without any reference to [MP2]. It has, however, been used occasionally for ease of exposition and avoiding repetition of very similar calculations. It is also essential reading if one really wants to understand what one is up against, and the problems that the new method of almost holonomy in [M8] has solved.

17.

I do category theory and the appendix A.3 in [M5.bis] which isn't in [M5] looks really interesting, but are you sure that's right that everything can be done with bounded Borel functions rather than continuous ones ?

In effect, it can't, one has to work with continuous functions. The appendix was there since it was used in the proof of residue lemma conditional on $\mathrm{LCR}_+$, cf. no. 14. Since the proof of the residue lemma is now what it should be, i.e. relies only on LCR, this section no longer has any logical relevance to the proof. Nevertheless, it is pretty interesting, and even more so when one replaces Borel by continuous. There'll be a stand alone version shortly, [M10].

18.

That's two things you've flagged in nos. 14 & 17. Any other caveats before I start reading ?

Basically No. Obviously more polished versions are going to be ready sooner rather than later, but the basic rule with the current versions is just ignore everything that either [M6] or [M6.bis] says about the residue lemma, except the final statement, and defer to [M8] for the proof in dimension 3. As such, even though for example [M6.bis] IV.2.4 is wrong, and the use of Stokes' in IV.6.7 may not be justifiable without full convergence of the centre manifold relative to the measure, it's wholly irrelevant, and the whole of §IV be it of [M6] or [M6.bis] should just be ignored. There will shortly be a new version in which this section will simply be cut. Further technical precisions: Sibony pointed out that there are examples which oblige the use of ``a'', as opposed to ``the'', Segre class, while Rousseau observed that the idea behind [M6.bis] VI.1.5 is correct, but the execution is wrong, so $n$ should be replaced by $\log n$, or thereabouts. None of which has any adverse consequences in practice beyond replacing $2/3$ by $9/13$ in the 2-jet condition.

19.

I'm interested in arithmetic, and it's claimed that there's some analogy between hyperbolicity and Mordellicity. As such, any relation between the project and questions of finiteness of rational points on surfaces ?

If by arithmetic one means over $\mathbb{Z}$ then I'd say next to none. One needs to be able to differentiate, although the need for close to arbitrary dimension near the Castelnuovo line, and the need for arbitrarily many factors to get $2+\varepsilon$ in Roth's theorem are curiously reminiscent. Function field arithmetic in characteristic 0 is another matter, however, and in principle [M7] §I solves the differentiation problem, and the basic strategy should bound the degree of rational points. However, in the presence of bad reduction there's an issue that I haven't been able to get to the bottom of, so the best so far is Mordellicity for surfaces with $\mathrm{c}_1^2> 2\mathrm{c}_2$ which relies on ideas from classification, but differentiates in the classical way, [M7] §2.

20.

Reading between the lines, you seem confident about resolution of foliation singularities in arbitrary dimension, and you're mentioning new versions of what's available already. Don't you think, therefore, that we should be able to read about such up dates on facebook and twitter ?

Coincidentally there has been a significant revision of resolution of singularities in dimension 3, [MP1.bis]. As for updates in general, however, there are two problems: cookies on facebook and twitter which makes their use a non-starter, and, even were it otherwise, the rule in mathematics is whatever estimate you have for doing something, double it and move to the next unit of time, so, 1 hour becomes 2 days, 2 days 4 weeks, 4 weeks 8 months, etc.. As such, a dedicated twitter account rather than a home page may be an embrace too far of the interweb.

21.

I've no interest in mathematics, but I'm deeply impressed by these html pages. They're written to the highest standard, and are showing no errors whatsoever on my bug metre. Even an average google page shows half a dozen, and google news several hundred. How did you do it ?

Mostly with latex2html on an OpenBSD-4.6 box after I edited the make file in the ports tree to compile against my teTeX installation as opposed to the TeX Live monster, although, curiously, it's not working as well on my 4.8 box. Irrespectively even on the 4.6 box it was making syntax errors in the internal linking, so I did that bit my hand. I'm impressed too.

home